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OR:

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, Lagrangian mechanics is a formulation of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by class ...
founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaMécanique analytique''. Lagrangian mechanics describes a mechanical system as a pair $(M,L)$ consisting of a configuration space $M$ and a smooth function $L$ within that space called a ''Lagrangian''. By convention, $L = T - V,$ where $T$ and $V$ are the
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ente ...
and
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from $L$ must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.

# Introduction

Suppose there exists a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects (bead, pendulum bob, etc.) as a particle, calculation of the motion of the particle using
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
would require solving for the time-varying constraint force required to keep the particle in the constrained motion (reaction force exerted by the wire on the bead, or tension in the pendulum rod). For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of ''independent'' generalized coordinates that completely characterize the possible motion of the particle. This choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment. For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
. For a system of ''N'' point particles with
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
es ''m''1, ''m''2, ..., ''mN'', each particle has a
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
, denoted r1, r2, ..., r''N''.
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
are often sufficient, so r1 = (''x''1, ''y''1, ''z''1), r2 = (''x''2, ''y''2, ''z''2) and so on. In three dimensional space, each position vector requires three
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
to uniquely define the location of a point, so there are 3''N'' coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is written r = (''x'', ''y'', ''z''). The
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...
of each particle is how fast the particle moves along its path of motion, and is the
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
of its position, thus $\mathbf_1 = \frac, \mathbf_2 = \frac,\ldots, \mathbf_N = \frac$ In Newtonian mechanics, the
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ve ...
are given by Newton's laws. The second law "net
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...
equals mass times
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...
",$\sum \mathbf = m\frac$applies to each particle. For an ''N'' particle system in 3 dimensions, there are 3''N'' second order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s in the positions of the particles to solve for.

## The Lagrangian

Instead of forces, Lagrangian mechanics uses the
energies In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of he ...
in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The ''non-relativistic'' Lagrangian for a system of particles can be defined by $L = T - V$ where $T = \frac \sum_^N m_k v^2_k$ is the total
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its ac ...
of the system, equalling the sum Σ of the kinetic energies of the particles, and ''V'' is the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
of the system. Kinetic energy is the energy of the system's motion, and ''vk''2 = v''k'' · v''k'' is the magnitude squared of velocity, equivalent to the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
of the velocity with itself. The kinetic energy is a function only of the velocities v''k'', not the positions r''k'' nor time ''t'', so ''T'' = ''T''(v1, v2, ...). The
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
of the system reflects the energy of interaction between the particles, i.e. how much energy any one particle will have due to all the others and other external influences. For
conservative force In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum ...
s (e.g. Newtonian gravity), it is a function of the position vectors of the particles only, so ''V'' = ''V''(r1, r2, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g.
electromagnetic potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
), the velocities will appear also, ''V'' = ''V''(r1, r2, ..., v1, v2, ...). If there is some external field or external driving force changing with time, the potential will change with time, so most generally ''V'' = ''V''(r1, r2, ..., v1, v2, ..., ''t''). The above form of ''L'' does not hold in relativistic Lagrangian mechanics, and must be replaced by a function consistent with special or general relativity. Also, for dissipative forces another function must be introduced alongside ''L''. One or more of the particles may each be subject to one or more
holonomic constraints In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that ...
; such a constraint is described by an equation of the form ''f''(r, ''t'') = 0. If the number of constraints in the system is ''C'', then each constraint has an equation, ''f''1(r, ''t'') = 0, ''f''2(r, ''t'') = 0, ..., ''fC''(r, ''t'') = 0, each of which could apply to any of the particles. If particle ''k'' is subject to constraint ''i'', then ''fi''(r''k'', ''t'') = 0. At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine the allowed paths the particles can move along, but not where they are or how fast they go at every instant of time. Nonholonomic constraints depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanics ''can only be applied to systems whose constraints, if any, are all holonomic''. Three examples of nonholonomic constraints are: when the constraint equations are nonintegrable, when the constraints have inequalities, or with complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to Newtonian mechanics, or use other methods. If ''T'' or ''V'' or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian ''L''(r1, r2, ... v1, v2, ... ''t'') is ''explicitly time-dependent''. If neither the potential nor the kinetic energy depend on time, then the Lagrangian ''L''(r1, r2, ... v1, v2, ...) is ''explicitly independent of time''. In either case, the Lagrangian will always have implicit time-dependence through the generalized coordinates. With these definitions, Lagrange's equations of the first kind are where ''k'' = 1, 2, ..., ''N'' labels the particles, there is a Lagrange multiplier ''λi'' for each constraint equation ''fi'', and $\frac \equiv \left(\frac,\frac,\frac\right)\,,\quad \frac \equiv \left(\frac,\frac,\frac\right)$ are each shorthands for a vector of
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s with respect to the indicated variables (not a derivative with respect to the entire vector).Sometimes in this context the variational derivative denoted and defined as $\frac \equiv \frac - \frac\frac$ is used. Throughout this article only partial and total derivatives are used. Each overdot is a shorthand for a
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
. This procedure does increase the number of equations to solve compared to Newton's laws, from 3''N'' to 3''N'' + ''C'', because there are 3''N'' coupled second order differential equations in the position coordinates and multipliers, plus ''C'' constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations. In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual
differentiation rules This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Elementary rules of differentiation Unless otherwise stated, all functions are functions of real numbers (R) that return real ...
(e.g. the partial derivative of ''L'' with respect to the ''z''-velocity component of particle 2, defined by $v_ = dz_2 / dt$, is just $\partial L / \partial v_$; no awkward
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
s or total derivatives need to be used to relate the velocity component to the corresponding coordinate ''z''2). In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number of ''independent'' coordinates is therefore ''n'' = 3''N'' − ''C''. We can transform each position vector to a common set of ''n''
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
, conveniently written as an ''n''-tuple q = (''q''1, ''q''2, ... ''qn''), by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time, $\mathbf_k = \mathbf_k(\mathbf, t) = (x_k(\mathbf, t), y_k(\mathbf, t), z_k(\mathbf, t), t)\,.$ The vector q is a point in the configuration space of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with re ...
of its position with respect to time, is $\dot_j = \frac \,, \quad \mathbf_k = \sum_^n \frac\dot_j +\frac\,.$ Given this v''k'', the kinetic energy ''in generalized coordinates'' depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so $T = T\left(\mathbf, \dot, t\right)$. With these definitions, the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovere ...
s, or Lagrange's equations of the second kind are mathematical results from the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, which can also be used in mechanics. Substituting in the Lagrangian ''L''(q, dq/d''t'', ''t''), gives the
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ve ...
of the system. The number of equations has decreased compared to Newtonian mechanics, from 3''N'' to ''n'' = 3''N'' − ''C'' coupled second order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for. Although the equations of motion include
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s, the results of the partial derivatives are still
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s in the position coordinates of the particles. The total time derivative denoted d/d''t'' often involves implicit differentiation. Both equations are linear in the Lagrangian, but will generally be nonlinear coupled equations in the coordinates.

# From Newtonian to Lagrangian mechanics

## Newton's laws

For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of ''N'' particles, all of these equations apply to each particle in the system). The equation of motion for a particle of mass ''m'' is
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
of 1687, in modern vector notation $\mathbf = m \mathbf \,,$ where a is its acceleration and F the resultant force acting ''on'' it. In three spatial dimensions, this is a system of three coupled second order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s to solve, since there are three components in this vector equation. The solution is the position vector r of the particle at time ''t'', subject to the
initial condition In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). Fo ...
s of r and v when ''t'' = 0. Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set of
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally i ...
ξ = (''ξ''1, ''ξ''2, ''ξ''3), the law in
tensor index notation In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
is the ''"Lagrangian form"'' $F^a = m \left( \frac + \Gamma^a _ \frac\frac \right) = g^ \left(\frac \frac - \frac\right) \,, \quad \dot^a \equiv \frac \,,$ where ''Fa'' is the ''a''th contravariant component of the resultant force acting on the particle, Γ''abc'' are the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dista ...
of the second kind, $T = \frac m g_ \frac \frac$ is the kinetic energy of the particle, and ''gbc'' the covariant components of the ''
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space a ...
'' of the curvilinear coordinate system. All the indices ''a'', ''b'', ''c'', each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates. It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle, F = 0, it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are ''
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connectio ...
s'', the curves of extremal length between two points in space (these may end up being minimal so the shortest paths, but that is not necessary). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation, and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces, F ≠ 0, the particle accelerates due to forces acting on it, and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4D curved spacetime, the above form of Newton's law also carries over to
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense. However, we still need to know the total resultant force F acting on the particle, which in turn requires the resultant non-constraint force N plus the resultant constraint force C, $\mathbf = \mathbf + \mathbf \,.$ The constraint forces can be complicated, since they will generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations. The constraint forces can either be eliminated from the equations of motion so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion.

## D'Alembert's principle

A fundamental result in
analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
is
D'Alembert's principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alembert ...
, introduced in 1708 by Jacques Bernoulli to understand static equilibrium, and developed by
D'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédi ...
in 1743 to solve dynamical problems. The principle asserts for ''N'' particles the virtual work, i.e. the work along a virtual displacement, δr''k'', is zero: $\sum_^N ( \mathbf _k + \mathbf _k - m_k \mathbf_k )\cdot \delta \mathbf_k = 0\,.$ The ''
virtual displacement In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very s ...
s'', δr''k'', are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system ''at an instant of time'', i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it.Here the virtual displacements are assumed reversible, it is possible for some systems to have non-reversible virtual displacements that violate this principle, see Udwadia–Kalaba equation.
Virtual work In mechanics, virtual work arises in the application of the '' principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different fo ...
is the work done along a virtual displacement for any force (constraint or non-constraint). Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero:In other words $\mathbf_k\cdot\delta\mathbf_k = 0$ for particle ''k'' subject to a constraint force, however $C_ \delta x_k \neq 0\,,\quad C_ \delta y_k \neq 0\,,\quad C_ \delta z_k \neq 0$ because of the constraint equations on the r''k'' coordinates. $\sum_^N \mathbf _k \cdot \delta \mathbf_k = 0\,,$ so that $\sum_^N (\mathbf _k - m_k \mathbf_k ) \cdot \delta \mathbf_k = 0\,.$ Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion. The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an arbitrary coordinate system since the displacements δr''k'' might be connected by a constraint equation, which prevents us from setting the ''N'' individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion.

## Equations of motion from D'Alembert's principle

If there are constraints on particle ''k'', then since the coordinates of the position r''k'' = (''xk'', ''yk'', ''zk'') are linked together by a constraint equation, so are those of the
virtual displacement In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very s ...
s ''δ''r''k'' = (''δxk'', ''δyk'', ''δzk''). Since the generalized coordinates are independent, we can avoid the complications with the ''δ''r''k'' by converting to virtual displacements in the generalized coordinates. These are related in the same form as a total differential, $\delta \mathbf_k = \sum_^n \frac \delta q_j \,.$ There is no partial time derivative with respect to time multiplied by a time increment, since this is a virtual displacement, one along the constraints in an ''instant'' of time. The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces N''k'' along the virtual displacements ''δ''r''k'', and can without loss of generality be converted into the generalized analogues by the definition of generalized forces $Q_j = \sum_^N \mathbf _k \cdot \frac \,,$ so that $\sum_^N \mathbf_k \cdot \delta \mathbf_k = \sum_^N \mathbf _k \cdot \sum_^n \frac \delta q_j = \sum_^n Q_j \delta q_j \,.$ This is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized coordinates and velocities can be found to give the desired result: $\sum_^N m_k \mathbf_k \cdot \frac = \frac\frac - \frac \,.$ Now D'Alembert's principle is in the generalized coordinates as required, and since these virtual displacements ''δqj'' are independent and nonzero, the coefficients can be equated to zero, resulting in Lagrange's equations or the generalized equations of motion, $Q_j = \frac\frac - \frac$ These equations are equivalent to Newton's laws ''for the non-constraint forces''. The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.

## Euler–Lagrange equations and Hamilton's principle

For a non-conservative force which depends on velocity, it ''may'' be possible to find a potential energy function ''V'' that depends on positions and velocities. If the generalized forces ''Qi'' can be derived from a potential ''V'' such that $Q_j = \frac\frac - \frac \,,$ equating to Lagrange's equations and defining the Lagrangian as ''L'' = ''T'' − ''V'' obtains Lagrange's equations of the second kind or the Euler–Lagrange equations of motion $\frac - \frac\frac = 0 \,.$ However, the Euler–Lagrange equations can only account for non-conservative forces ''if'' a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations. The Euler–Lagrange equations also follow from the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. The ''variation'' of the Lagrangian is $\delta L = \sum_^n \left(\frac \delta q_j + \frac \delta \dot_j \right) \,,\quad \delta \dot_j \equiv \delta\frac \equiv \frac \,,$ which has a form similar to the total differential of ''L'', but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
with respect to time can transfer the time derivative of ''δqj'' to the ∂''L''/∂(d''qj''/d''t''), in the process exchanging d(''δqj'')/d''t'' for ''δqj'', allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian, Now, if the condition ''δqj''(''t''1) = ''δqj''(''t''2) = 0 holds for all ''j'', the terms not integrated are zero. If in addition the entire time integral of ''δL'' is zero, then because the ''δqj'' are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of ''δqj'' must also be zero. Then we obtain the equations of motion. This can be summarized by
Hamilton's principle In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single functio ...
: $\int_^\delta L \, \mathrmt = 0 \,.$ The time integral of the Lagrangian is another quantity called the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 ...
, defined as $S = \int_^ L\,\mathrmt\,,$ which is a ''
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional s ...
''; it takes in the Lagrangian function for all times between ''t''1 and ''t''2 and returns a scalar value. Its dimensions are the same as angular_momentum_.html" ;"title="angular_momentum.html" ;"title="angular momentum">angular momentum ">angular_momentum.html" ;"title="angular momentum">angular momentum [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle is $\delta S = 0\,.$ Thus, instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is sometimes referred to as the '' principle of least action'', however the action functional need only be ''stationary'', not necessarily a maximum or a minimum value. Any variation of the functional gives an increase in the functional integral of the action. Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
to mechanical problems, such as the Brachistochrone problem solved by Jean Bernoulli in 1696, as well as
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathe ...
,
Daniel Bernoulli Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mec ...
, L'Hôpital around the same time, and Newton the following year. Newton himself was thinking along the lines of the variational calculus, but did not publish. These ideas in turn lead to the
variational principle In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those func ...
s of mechanics, of
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
, Maupertuis,
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries i ...
, Hamilton, and others. Hamilton's principle can be applied to nonholonomic constraints if the constraint equations can be put into a certain form, a linear combination of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation. This will not be given here.

## Lagrange multipliers and constraints

The Lagrangian ''L'' can be varied in the Cartesian r''k'' coordinates, for ''N'' particles, $\int_^ \sum_^N \left(\frac - \frac\frac \right)\cdot\delta \mathbf_k \, \mathrmt = 0 \,.$ Hamilton's principle is still valid even if the coordinates ''L'' is expressed in are not independent, here r''k'', but the constraints are still assumed to be holonomic. As always the end points are fixed ''δ''r''k''(''t''1) = ''δ''r''k''(''t''2) = 0 for all ''k''. What cannot be done is to simply equate the coefficients of δr''k'' to zero because the δr''k'' are not independent. Instead, the method of Lagrange multipliers can be used to include the constraints. Multiplying each constraint equation ''fi''(r''k'', ''t'') = 0 by a Lagrange multiplier ''λi'' for ''i'' = 1, 2, ..., ''C'', and adding the results to the original Lagrangian, gives the new Lagrangian $L' = L(\mathbf_1,\mathbf_2,\ldots,\dot_1,\dot_2,\ldots,t) + \sum_^C \lambda_i(t) f_i(\mathbf_k,t) \,.$ The Lagrange multipliers are arbitrary functions of time ''t'', but not functions of the coordinates r''k'', so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives $\int_^ \delta L' \mathrmt = \int_^ \sum_^N \left(\frac - \frac\frac + \sum_^C \lambda_i \frac\right)\cdot\delta \mathbf_k \, \mathrmt = 0 \,.$ The introduced multipliers can be found so that the coefficients of ''δ''r''k'' are zero, even though the r''k'' are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement $\frac - \frac\frac = 0 \quad \Rightarrow \quad \frac - \frac\frac + \sum_^C \lambda_i \frac = 0 \,,$ which are Lagrange's equations of the first kind. Also, the ''λi'' Euler-Lagrange equations for the new Lagrangian return the constraint equations $\frac - \frac\frac = 0 \quad \Rightarrow \quad f_i(\mathbf_k,t) = 0 \,.$ For the case of a conservative force given by the gradient of some potential energy ''V'', a function of the rk coordinates only, substituting the Lagrangian ''L'' = ''T'' − ''V'' gives $\underbrace_ + \underbrace_ + \sum_^C \lambda_i \frac = 0 \,,$ and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces are $\mathbf_k = \sum_^C \lambda_i \frac \,,$ thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers.

# Properties of the Lagrangian

## Non-uniqueness

The Lagrangian of a given system is not unique. A Lagrangian ''L'' can be multiplied by a nonzero constant ''a'' and shifted by an arbitrary constant ''b'', and the new Lagrangian ''L' = aL'' + ''b'' will describe the same motion as ''L''. If one restricts as above to trajectories $\mathbf$ over a given time interval

## Invariance under point transformations

Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates s according to a point transformation q = q(s, ''t''), the new Lagrangian ''L''′ is a function of the new coordinates :$L\left(\mathbf\left(\mathbf,t\right), \dot\left(\mathbf,\dot,t\right), t \right) = L\text{'}\left(\mathbf, \dot,t\right) \,,$ and by the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
for partial differentiation, Lagrange's equations are invariant under this transformation; :$\frac\frac = \frac \,.$ This may simplify the equations of motion.

## Cyclic coordinates and conserved momenta

An important property of the Lagrangian is that conserved quantities can easily be read off from it. The ''generalized momentum'' "canonically conjugate to" the coordinate ''qi'' is defined by :$p_i =\frac.$ If the Lagrangian ''L'' does ''not'' depend on some coordinate ''qi'', it follows immediately from the Euler–Lagrange equations that :$\dot_i = \frac\frac = \frac=0\,$ and integrating shows the corresponding generalized momentum equals a constant, a conserved quantity. This is a special case of
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
. Such coordinates are called "cyclic" or "ignorable". For example, a system may have a Lagrangian :$L\left(r,\theta,\dot,\dot,\dot,\dot,\dot,t\right)\,,$ where ''r'' and ''z'' are lengths along straight lines, ''s'' is an arc length along some curve, and ''θ'' and ''φ'' are angles. Notice ''z'', ''s'', and ''φ'' are all absent in the Lagrangian even though their velocities are not. Then the momenta :$p_z =\frac\,,\quad p_s =\frac\,,\quad p_\phi =\frac\,,$ are all conserved quantities. The units and nature of each generalized momentum will depend on the corresponding coordinate; in this case ''pz'' is a translational momentum in the ''z'' direction, ''ps'' is also a translational momentum along the curve ''s'' is measured, and ''pφ'' is an angular momentum in the plane the angle ''φ'' is measured in. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved.

## Energy

### Definition

Given a Lagrangian $L,$ the ''energy'' of the corresponding mechanical system is, by definition, :$E = \biggl\left(\sum^n_ _i\frac\biggr\right) - L.$

### Invariance under coordinate transformations

At every time instant $t,$ the energy is invariant under configuration space coordinate changes $\mathbf \to \mathbf$, i.e. :$E\left(\mathbf,\dot\mathbf,t\right) = E\left(\mathbf,\dot\mathbf,t\right).$ Besides this result, the proof below shows that, under such change of coordinates, the derivatives $\partial L/\partial _i$ change as coefficients of a linear form.

### Conservation

In Lagrangian mechanics, the system is closed if and only if its Lagrangian $L$ does not explicitly depend on time. The energy conservation law states that the energy $E$ of a closed system is an integral of motion. More precisely, let $\mathbf=\mathbf\left(t\right)$ be an ''extremal''. (In other words, $\mathbf$ satisfies the Euler-Lagrange equations). Taking the total time-derivative of $L$ along this extremal and using the EL equations leads to : If the Lagrangian $L$ does not explicitly depend on time, then $\partial L/\partial t = 0,$ so $E$ is, indeed, an integral of motion, meaning that :$E\left(\mathbf\left(t\right), \dot\mathbf\left(t\right), t\right) = \text.$ Hence, the energy is conserved.

### Kinetic and potential energies

It also follows that the kinetic energy is a homogenous function of degree 2 in the generalized velocities. If in addition the potential ''V'' is only a function of coordinates and independent of velocities, it follows by direct calculation, or use of Euler's theorem for homogenous functions, that :$\sum_^n \dot_i\frac = \sum_^n \dot_i\frac = 2T \,.$ Under all these circumstances, the constant :$E = T + V$ is the total energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy ''E'' is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates. In the case the velocity or kinetic energy or both depends on time, then the energy is ''not'' conserved.

## Mechanical similarity

If the potential energy is a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
of the coordinates and independent of time, and all position vectors are scaled by the same nonzero constant ''α'', r''k''′ = ''α''r''k'', so that :$V\left(\alpha\mathbf_1,\alpha\mathbf_2,\ldots, \alpha\mathbf_N\right)=\alpha^N V\left(\mathbf_1,\mathbf_2,\ldots, \mathbf_N\right)$ and time is scaled by a factor ''β'', ''t''′ = ''βt'', then the velocities v''k'' are scaled by a factor of ''α''/''β'' and the kinetic energy ''T'' by (''α''/''β'')2. The entire Lagrangian has been scaled by the same factor if :$\frac=\alpha^N \quad\Rightarrow\quad \beta = \alpha^\,.$ Since the lengths and times have been scaled, the trajectories of the particles in the system follow geometrically similar paths differing in size. The length ''l'' traversed in time ''t'' in the original trajectory corresponds to a new length ''l′'' traversed in time ''t′'' in the new trajectory, given by the ratios :$\frac= \left\left(\frac\right\right)^\,.$

## Interacting particles

For a given system, if two subsystems ''A'' and ''B'' are non-interacting, the Lagrangian ''L'' of the overall system is the sum of the Lagrangians ''LA'' and ''LB'' for the subsystems: :$L = L_A + L_B \,.$ If they do interact this is not possible. In some situations, it may be possible to separate the Lagrangian of the system ''L'' into the sum of non-interacting Lagrangians, plus another Lagrangian ''LAB'' containing information about the interaction, :$L = L_A + L_B + L_\,.$ This may be physically motivated by taking the non-interacting Lagrangians to be kinetic energies only, while the interaction Lagrangian is the system's total potential energy. Also, in the limiting case of negligible interaction, ''LAB'' tends to zero reducing to the non-interacting case above. The extension to more than two non-interacting subsystems is straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added.

# Examples

The following examples apply Lagrange's equations of the second kind to mechanical problems.

## Conservative force

A particle of mass ''m'' moves under the influence of a
conservative force In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum ...
derived from the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
∇ of a
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
, :$\mathbf = -\boldsymbol V\left(\mathbf\right)\,.$ If there are more particles, in accordance with the above results, the total kinetic energy is a sum over all the particle kinetic energies, and the potential is a function of all the coordinates.

### Cartesian coordinates

The Lagrangian of the particle can be written :$L\left(x,y,z, \dot, \dot,\dot\right) = \frac m \left(\dot^2 + \dot^2 + \dot^2\right) - V\left(x,y,z\right)\,.$ The equations of motion for the particle are found by applying the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovere ...
, for the ''x'' coordinate :$\frac \left\left( \frac \right\right) = \frac \,,$ with derivatives :$\frac = - \frac\,,\quad \frac = m \dot\,,\quad \frac \left\left( \frac \right\right) = m \ddot\,,$ hence :$m \ddot = - \frac\,,$ and similarly for the ''y'' and ''z'' coordinates. Collecting the equations in vector form we find :$m\ddot=-\boldsymbol V$ which is
Newton's second law of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
for a particle subject to a conservative force.

### Polar coordinates in 2D and 3D

Using the spherical coordinates (r, ''θ'', ''φ'') as commonly used in physics (ISO 80000-2:2019 convention), where ''r'' is the radial distance to origin, ''θ'' is polar angle (also known as colatitude, zenith angle, normal angle, or inclination angle), and ''φ'' is the azimuthal angle, the Lagrangian for a central potential is :$L = \frac\left(\dot^2+r^2\dot^2 +r^2\sin^2\theta \, \dot^2\right)-V\left(r\right)\,.$ So, in spherical coordinates, the Euler–Lagrange equations are :$m\ddot-mr\left(\dot^2+\sin^2\theta \, \dot^2\right)+\frac =0\,,$ :$\frac\left(mr^2\dot\right) -mr^2\sin\theta\cos\theta \, \dot^2=0\,,$ :$\frac\left(mr^2\sin^2\theta \, \dot\right)=0\,.$ The ''φ'' coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentum :$p_\varphi = \frac = mr^2\sin^2\theta \dot\,,$ in which ''r'', ''θ'' and ''dφ/dt'' can all vary with time, but only in such a way that ''pφ'' is constant. The Lagrangian in two-dimensional polar coordinates is recovered by fixing ''θ'' to the constant value ''π''/2.

## Pendulum on a movable support

Consider a pendulum of mass ''m'' and length ''ℓ'', which is attached to a support with mass ''M'', which can move along a line in the $x$-direction. Let $x$ be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle $\theta$ from the vertical. The coordinates and velocity components of the pendulum bob are :$\begin & x_\mathrm = x + \ell \sin \theta & \quad \Rightarrow \quad \dot_\mathrm = \dot + \ell \dot \cos \theta \\ & y_\mathrm = - \ell \cos\theta & \quad \Rightarrow \quad \dot_\mathrm = \ell \dot \sin \theta \,. \end$ The generalized coordinates can be taken to be $x$ and $\theta$. The kinetic energy of the system is then :$T = \frac M \dot^2 + \frac m \left\left( \dot_\mathrm^2 + \dot_\mathrm^2 \right\right)$ and the potential energy is :$V = m g y_\mathrm$ giving the Lagrangian : Since $x$ is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is :$p_x = \frac = \left(M + m\right) \dot x + m \ell \dot\theta \cos\theta \,$ and the Lagrange equation for the support coordinate $x$ is :$\left(M + m\right) \ddot x + m \ell \ddot\theta\cos\theta-m \ell \dot\theta ^2 \sin\theta = 0 \,.$ The Lagrange equation for the angle $\theta$ is : and simplifying :$\ddot\theta + \frac \cos\theta + \frac \sin\theta = 0.$ These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example, $\ddot x \to 0$ should give the equations of motion for a simple pendulum that is at rest in some
inertial frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
, while $\ddot\theta \to 0$ should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively.

## Two-body central force problem

Two bodies of masses and with position vectors and are in orbit about each other due to an attractive central potential . We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies and the location of the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...
. The Lagrangian is thenThe Lagrangian also can be written explicitly for a rotating frame. See Padmanabhan, 2000. :$L = \underbrace_ + \underbrace_$ where is the total mass, is the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the m ...
, and the potential of the radial force, which depends only on the magnitude of the separation . The Lagrangian splits into a ''center-of-mass'' term and a ''relative motion'' term . The Euler–Lagrange equation for is simply :$M\ddot = 0 \,,$ which states the center of mass moves in a straight line at constant velocity. Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates and take , :$L_\text=\frac \mu \left(\dot^2 +r^2 \dot^2 \right) - V\left(r\right) \,,$ so is a cyclic coordinate with the corresponding conserved (angular) momentum :$p_\theta = \frac = \mu r^2 \dot \theta = \ell \,.$ The radial coordinate and angular velocity can vary with time, but only in such a way that is constant. The Lagrange equation for is :$\mu r \dot \theta ^2 -\frac = \mu \ddot \,.$ This equation is identical to the radial equation obtained using Newton's laws in a ''co-rotating'' reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocity from this radial equation, :$\mu \ddot r = -\frac + \frac \,.$ which is the equation of motion for a one-dimensional problem in which a particle of mass is subjected to the inward central force and a second outward force, called in this context the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
:$F_ = \mu r \dot \theta ^2 = \frac \,.$ Of course, if one remains entirely within the one-dimensional formulation, enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated. If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of ''using polar coordinates''. As Hildebrand says: "Since such quantities are not true physical forces, they are often called ''inertia forces''. Their presence or absence depends, not upon the particular problem at hand, but ''upon the coordinate system chosen''." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the
centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous ...
for a curved motion. This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example, see for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in. Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an
inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as ''generalized'' inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal ''always'' with ''generalized'' forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently." It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.

## Electromagnetism

A test particle is a particle whose
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
and
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aq ...
are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s and
up quark The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up ...
s are more complex and have additional terms in their Lagrangians. The Lagrangian for a
charged particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
with
electrical charge Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described b ...
, interacting with an
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
, is the prototypical example of a velocity-dependent potential. The electric
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
and
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
are defined from the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
as follows: :$\mathbf = - \boldsymbol\phi - \frac \,, \quad \mathbf = \boldsymbol \times \mathbf \,.$ The Lagrangian of a massive charged test particle in an electromagnetic field :$L = \tfracm \dot^2 + q\, \dot \cdot \mathbf - q \phi \,,$ is called
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast ...
. Combined with
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovere ...
, it produces the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an el ...
law :$m \ddot = q \mathbf + q \dot \times \mathbf$ Under gauge transformation: :$\mathbf \rightarrow \mathbf+\boldsymbol f \,, \quad \phi \rightarrow \phi-\dot f \,,$ where is any scalar function of space and time, the aforementioned Lagrangian transforms like: :$L \rightarrow L+q\left\left(\dot\cdot\boldsymbol+\frac\right\right)f=L+q\frac \,,$ which still produces the same Lorentz force law. Note that the canonical momentum (conjugate to position ) is the kinetic momentum plus a contribution from the field (known as the potential momentum): :$\mathbf = \frac = m \dot + q \mathbf \,.$ This relation is also used in the
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast ...
prescription in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...
and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particl ...
. From this expression, we can see that the canonical momentum is not gauge invariant, and therefore not a measurable physical quantity; However, if is cyclic (i.e. Lagrangian is independent of position ), which happens if the and fields are uniform, then this canonical momentum given here is the conserved momentum, while the measurable physical kinetic momentum is not.

# Extensions to include non-conservative forces

Dissipation In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form ...
(i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom. In a more general formulation, the forces could be both conservative and
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
. If an appropriate transformation can be found from the Fi, Rayleigh suggests using a dissipation function, ''D'', of the following form: :$D = \frac \sum_^m \sum_^m C_ \dot_j \dot_k \,$ where ''Cjk'' are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. If ''D'' is defined this way, then :$Q_j = - \frac - \frac$ and :$\frac \left \left( \frac \right \right) - \frac + \frac = 0\,.$

# Other contexts and formulations

The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations.

## Alternative formulations of classical mechanics

A closely related formulation of classical mechanics is
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...
. The Hamiltonian is defined by :$H = \sum_^n \dot_i\frac - L \,$ and can be obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables canonically conjugate to the original variables. For example, given a set of generalized coordinates, the variables canonically conjugate are the generalized momenta. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is a particularly ubiquitous quantity in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...
(see
Hamiltonian (quantum mechanics) Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamilton ...
). Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.

## Momentum space formulation

The Euler–Lagrange equations can also be formulated in terms of the generalized momenta rather than generalized coordinates. Performing a Legendre transformation on the generalized coordinate Lagrangian ''L''(q, dq/d''t'', ''t'') obtains the generalized momenta Lagrangian ''L''′(p, dp/d''t'', ''t'') in terms of the original Lagrangian, as well the EL equations in terms of the generalized momenta. Both Lagrangians contain the same information, and either can be used to solve for the motion of the system. In practice generalized coordinates are more convenient to use and interpret than generalized momenta.

## Higher derivatives of generalized coordinates

There is no mathematical reason to restrict the derivatives of generalized coordinates to first order only. It is possible to derive modified EL equations for a Lagrangian containing higher order derivatives, see
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovere ...
for details. However, from the physical point-of-view there is an obstacle to include time derivatives higher than the first order, which is implied by Ostrogradsky's construction of a canonical formalism for nondegenerate higher derivative Lagrangians, see Ostrogradsky_instability

## Optics

Lagrangian mechanics can be applied to geometrical optics, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.

## Relativistic formulation

Lagrangian mechanics can be formulated in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The la ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects. In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian must be modified and is not simply the kinetic minus the potential energy of a particle. Also, it is not straightforward to handle multiparticle systems in a manifestly covariant way, it may be possible if a particular frame of reference is singled out.

## Quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...
,
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 ...
and quantum-mechanical
phase Phase or phases may refer to: Science * State of matter, or phase, one of the distinct forms in which matter can exist * Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathemati ...
are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s. In 1948,
Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
discovered the
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional ...
extending the principle of least action to
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...
for
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
and
photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they al ...
. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and
Fermat's principle Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
in
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultravio ...
.

## Classical field theory

In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In
classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantu ...
, the physical system is not a set of discrete particles, but rather a continuous field ''ϕ''(r, ''t'') defined over a region of 3D space. Associated with the field is a
Lagrangian density Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
:$\mathcal\left(\phi, \nabla \phi, \partial\phi/\partial t , \mathbf,t\right)$ defined in terms of the field and its space and time derivatives at a location r and time ''t''. Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in general, and the Lagrangian density has to be "reverse engineered"). The Lagrangian is then the
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
of the Lagrangian density over 3D space :$L\left(t\right) = \int \mathcal \, \mathrm^3 \mathbf$ where d3r is a 3D differential
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV = ...
. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may have explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian.

## Noether's theorem

The action principle, and the Lagrangian formalism, are tied closely to
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, which connects physical conserved quantities to continuous
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of a physical system. If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The la ...
or
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.

*
Canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of c ...
* Fundamental lemma of the calculus of variations *
Functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
*
Generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
*
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...
*
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrica ...
* Inverse problem for Lagrangian mechanics, the general topic of finding a Lagrangian for a system given the equations of motion. * Lagrangian and Eulerian specification of the flow field *
Lagrangian point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of ...
*
Lagrangian system In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of . In classical mechanics, many dynamical systems are Lagr ...
* Non-autonomous mechanics * Plateau's problem * Restricted three-body problem

# References

* * * * * * * * * * * * * *''The Principle of Least Action'', R. Feynman * * * * * * * * * * * * * * * * * *

* Gupta, Kiran Chandra, ''Classical mechanics of particles and rigid bodies'' (Wiley, 1988). * * Goldstein, Herbert, et al. ''
Classical Mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by class ...
''. 3rd ed., Pearson, 2002.