In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
and
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations 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 classi ...
. It was developed by many scientists and mathematicians during the 18th century and onward, after
Newtonian mechanics. Since Newtonian mechanics considers
vector quantities of motion, particularly
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 ...
s,
momenta
Momenta is an autonomous driving company headquartered in Beijing, China that aims to build the 'Brains' for autonomous vehicles.
In December 2021, Momenta and BYD established a 100 million yuan ($15.7 million) joint venture to deploy autonomous ...
,
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 a ...
s, of the constituents of the system, an alternative name for the mechanics governed by
Newton's laws and
Euler's laws is ''vectorial mechanics''.
By contrast, analytical mechanics uses ''
scalar'' properties of motion representing the system as a whole—usually its 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 acce ...
and
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 ...
—not Newton's vectorial forces of individual particles.
A scalar is a quantity, whereas a vector is represented by quantity and direction. 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 (V ...
are derived from the scalar quantity by some underlying principle about the scalar's
variation.
Analytical mechanics takes advantage of a system's ''constraints'' to solve problems. The constraints limit the
degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as
generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-
conservative forces or dissipative forces like
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
, in which case one may revert to Newtonian mechanics.
Two dominant branches of analytical mechanics are
Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in
configuration space) and
Hamiltonian mechanics (using coordinates and corresponding momenta in
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
). Both formulations are equivalent by a
Legendre transformation on the generalized coordinates, velocities and momenta, therefore both contain the same information for describing the dynamics of a system. There are other formulations such as
Hamilton–Jacobi theory,
Routhian mechanics alt=
In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the ...
, and
Appell's equation of motion
In classical mechanics, Appell's equation of motion (aka the Gibbs–Appell equation of motion) is an alternative general formulation of classical mechanics described by Josiah Willard Gibbs in 1879 and Paul Émile Appell in 1900.
Statement
...
. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the
principle of least action. One result is
Noether's theorem, a statement which connects
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
s to their associated
symmetries.
Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. In fact the same principles and formalisms can be used in
relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
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 ...
, and with some modifications,
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.
Analytical mechanics is used widely, from fundamental physics to
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, particularly
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
.
The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics.
Subject of analytical mechanics
The most obvious goal of mechanical theory is to solve mechanical problems which arise in physics or astronomy. Starting from a physical concept, such as a mechanism or a star system, a mathematical concept, or
model, is developed in the form of a differential equation or equations and then an attempt is made to solve them.
The vectorial approach to mechanics, as founded by Newton, is based on the Newton's laws which describe motion with the help of
vector quantities such as
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 a ...
,
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 i ...
,
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 ...
. These quantities characterise the
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
of a body which is idealised as a
"mass point" or a "
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
" understood as a single point to which a mass is attached. Newton's method was successful and was applied to a wide range of physical problems, starting from the motion of a particle in the
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
of
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
and then extended to the motion of planets under the action of the sun. In this approach, Newton's laws describe the motion by a differential equation and then the problem is reduced to the solving of that equation.
When the particle is a part of a system of particles, such as a
solid body or a
fluid, in which particles do not move freely but interact with each other, the Newton's approach is still applicable under proper precautions such as isolating each single particle from the others, and determining all the forces acting on it: those acting on the system as a whole as well as the forces of interaction of each particle with all other particles in the system. Such analysis can become cumbersome even in relatively simple systems. As a rule, interaction forces are unknown or hard to determine making it necessary to introduce new postulates. Newton thought that
his third law "action equals reaction" would take care of all complications. This is not the case even for such simple system as
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s of a solid body. In more complicated systems, the vectorial approach cannot give an adequate description.
The analytical approach to the problem of motion views the particle not as an isolated unit but as a part of a
mechanical system understood as an assembly of particles that interact with each other. As the whole system comes into consideration, the single particle loses its significance; the dynamical problem involves the entire system without breaking it in parts. This significantly simplifies the calculation because in the vectorial approach the forces have to be determined individually for each particle while in the analytical approach it is enough to know one single function which contains implicitly all the forces acting on and in the system. Such simplification is often done using certain kinematical conditions which are stated a priori; they are pre-existing and are due to the action of some strong forces. However, the analytical treatment does not require the knowledge of these forces and takes these kinematic conditions for granted. Considering how much simpler are these conditions in comparison with the multitude of forces that maintain them, the superiority of the analytical approach over the vectorial one becomes apparent.
Still, the equations of motion of a complicated mechanical system require a great number of separate differential equations which cannot be derived without some unifying basis from which they follow. This basis are the
variational principles: behind each set of equations there is a principle that expresses the meaning of the entire set. Given a fundamental and universal quantity called
'action', the principle that this action be stationary under small variation of some other mechanical quantity generates the required set of differential equations. The statement of the principle does not require any special
coordinate system
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 ...
, and all results are expressed in
generalized coordinates. This means that the analytical equations of motion do not change upon a
coordinate transformation, an
invariance property that is lacking in the vectorial equations of motion.
It is not altogether clear what is meant by 'solving' a set of differential equations. A problem is regarded as solved when the particles coordinates at time ''t'' are expressed as simple functions of ''t'' and of parameters defining the initial positions and velocities. However, 'simple function' is not a
well-defined concept: nowadays, a
function ''f''(''t'') is not regarded as a formal expression in ''t'' (
elementary function) as in the time of Newton but most generally as a quantity determined by ''t'', and it is not possible to draw a sharp line between 'simple' and 'not simple' functions. If one speaks merely of 'functions', then every mechanical problem is solved as soon as it has been well stated in differential equations, because given the initial conditions and ''t'' determine the coordinates at ''t''. This is a fact especially at present with the modern methods of
computer modelling which provide arithmetical solutions to mechanical problems to any desired degree of accuracy, the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s being replaced by
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s.
Still, though lacking precise definitions, it is obvious that the
two-body problem has a simple solution, whereas the
three-body problem has not. The two-body problem is solved by formulas involving parameters; their values can be changed to study the class of all solutions, that is, the
mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additiona ...
of the problem. Moreover, an accurate mental or drawn picture can be made for the motion of two bodies, and it can be as real and accurate as the real bodies moving and interacting. In the three-body problem, parameters can also be assigned specific values; however, the solution at these assigned values or a collection of such solutions does not reveal the mathematical structure of the problem. As in many other problems, the mathematical structure can be elucidated only by examining the differential equations themselves.
Analytical mechanics aims at even more: not at understanding the mathematical structure of a single mechanical problem, but that of a class of problems so wide that they encompass most of mechanics. It concentrates on systems to which Lagrangian or Hamiltonian equations of motion are applicable and that include a very wide range of problems indeed.
Development of analytical mechanics has two objectives: (i) increase the range of solvable problems by developing standard techniques with a wide range of applicability, and (ii) understand the mathematical structure of mechanics. In the long run, however, (ii) can help (i) more than a concentration on specific problems for which methods have already been designed.
Intrinsic motion
Generalized coordinates and constraints
In
Newtonian mechanics, one customarily uses all three
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 ...
, or other 3D
coordinate system
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 refer to a body's
position during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion. These are known as ''generalized coordinates'', denoted ''q
i'' (''i'' = 1, 2, 3...).
Difference between curvillinear and generalized coordinates
Generalized coordinates incorporate constraints on the system. There is one generalized coordinate ''q
i'' for each
degree of freedom (for convenience labelled by an index ''i'' = 1, 2...''N''), i.e. each way the system can change its
configuration; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of ''curvilinear'' coordinates equals the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of the position space in question (usually 3 for 3d space), while the number of ''generalized'' coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:
[''Analytical Mechanics'', L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ]
For a system with ''N'' degrees of freedom, the generalized coordinates can be collected into an ''N''-
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
:
and the
time derivative (here denoted by an overdot) of this tuple give the ''generalized velocities'':
D'Alembert's principle
The foundation which the subject is built on is ''D'Alembert's principle''.
This principle states that infinitesimal ''
virtual work'' done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful - since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is:
where
are the
generalized forces (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and are the generalized coordinates. This leads to the generalized form of
Newton's laws in the language of analytical mechanics:
where ''T'' 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 acce ...
of the system, and the notation
is a useful shorthand (see
matrix calculus for this notation).
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 d ...
If the curvilinear coordinate system is defined by the standard
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 ...
, and if the position vector can be written in terms of the generalized coordinates and time in the form:
and this relation holds for all times , then are called ''Holonomic constraints''. Vector is explicitly dependent on in cases when the constraints vary with time, not just because of . For time-independent situations, the constraints are also called
scleronomic
A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. ...
, for time-dependent cases they are called
rheonomic.
Lagrangian mechanics
Lagrangian and
Euler–Lagrange equations
The introduction of generalized coordinates and the fundamental Lagrangian function:
:
where ''T'' 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 acce ...
and ''V'' is the total
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 entire system, then either following 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 ...
or using the above formula - lead to the
Euler–Lagrange equations;
:
which are a set of ''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, one for each ''q
i''(''t'').
This formulation identifies the actual path followed by the motion as a selection of the path over which the
time integral of
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 acce ...
is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit.
Configuration space
The Lagrangian formulation uses the configuration space of the system, the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all possible generalized coordinates:
:
where
is ''N''-dimensional
real space (see also
set-builder notation
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Defining ...
). The particular solution to the Euler–Lagrange equations is called a ''(configuration) path or trajectory'', i.e. one particular q(''t'') subject to the required
initial conditions. The general solutions form a set of possible configurations as functions of time:
:
The configuration space can be defined more generally, and indeed more deeply, in terms of
topological manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s and the
tangent bundle.
Hamiltonian mechanics
Hamiltonian and Hamilton's equations
The
Legendre transformation of the Lagrangian replaces the generalized coordinates and velocities (q, q̇) with (q, p); the generalized coordinates and the ''
generalized momenta'' conjugate to the generalized coordinates:
:
and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):
:
where • denotes 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 ...
, also leading to
Hamilton's equations:
:
which are now a set of 2''N'' first-order ordinary differential equations, one for each ''q
i''(''t'') and ''p
i''(''t''). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:
:
which is often considered one of Hamilton's equations of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's second law:
:
Generalized
momentum space
Analogous to the configuration space, the set of all momenta is the ''momentum space'' (technically in this context; ''generalized momentum space''):
:
"Momentum space" also refers to "k-space"; the set of all
wave vectors (given by
De Broglie relations) as used in quantum mechanics and theory of
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s: this is not referred to in this context.
Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
The set of all positions and momenta form the ''phase space'';
:
that is, the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
× of the configuration space and generalized momentum space.
A particular solution to Hamilton's equations is called a ''
phase path'', a particular curve (q(''t''),p(''t'')) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is the ''
phase portrait'':
:
;The
Poisson bracket
All dynamical variables can be derived from position r, momentum p, and time ''t'', and written as a function of these: ''A'' = ''A''(q, p, ''t''). If ''A''(q, p, ''t'') and ''B''(q, p, ''t'') are two scalar valued dynamical variables, the ''Poisson bracket'' is defined by the generalized coordinates and momenta:
:
Calculating the
total derivative of one of these, say ''A'', and substituting Hamilton's equations into the result leads to the time evolution of ''A'':
:
This equation in ''A'' is closely related to the equation of motion in the
Heisenberg picture of
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, ...
, in which classical dynamical variables become
quantum operators (indicated by hats (^)), and the Poisson bracket is replaced by the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
of operators via Dirac's
canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.
Historically, this was not quit ...
:
:
Properties of the Lagrangian and Hamiltonian functions
Following are overlapping properties between the Lagrangian and Hamiltonian functions.
* All the individual generalized coordinates ''q
i''(''t''), velocities ''q̇
i''(''t'') and momenta ''p
i''(''t'') for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time ''t'' as a variable in addition to the q(''t''), p(''t''), not simply as a parameter through q(''t'') and p(''t''), which would mean explicit time-independence.
* The Lagrangian is invariant under addition of the ''
total''
time derivative of any function of q and ''t'', that is:
so each Lagrangian ''L'' and ''L describe ''exactly the same motion''. In other words, the Lagrangian of a system is not unique.
* Analogously, the Hamiltonian is invariant under addition of the ''
partial
Partial may refer to:
Mathematics
*Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...
'' time derivative of any function of q, p and ''t'', that is:
(''K'' is a frequently used letter in this case). This property is used in
canonical transformations
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian mechanics, Hami ...
(see below).
*If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are
constants of the motion, i.e. are
conserved, this immediately follows from Lagrange's equations:
Such coordinates are "
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates.
*If the Lagrangian is time-independent the Hamiltonian is also time-independent (i.e. both are constant in time).
*If the kinetic 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 ''d ...
of degree 2 of the generalized velocities, ''and'' the Lagrangian is explicitly time-independent, then:
where ''λ'' is a constant, then the Hamiltonian will be the ''total conserved energy'', equal to the total kinetic and potential energies of the system:
This is the basis for the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, inserting
quantum operators directly obtains it.
Principle of least action
Action is another quantity in analytical mechanics defined as 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 sy ...
of the Lagrangian:
:
A general way to find the equations of motion from the action is the ''
principle of least action'':
:
where the departure ''t''
1 and arrival ''t''
2 times are fixed.
[ The term "path" or "trajectory" refers to the ]time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
of the system as a path through configuration space , in other words q(''t'') tracing out a path in . The path for which action is least is the path taken by the system.
From this principle, ''all'' 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 (V ...
in classical mechanics can be derived. This approach can be extended to fields rather than a system of particles (see below), and underlies the path integral formulation of 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, ...
,[''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ][Quantum Field Theory, D. McMahon, Mc Graw Hill (US), 2008, ] and is used for calculating 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 connecti ...
motion in 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 ...
.
Hamiltonian-Jacobi mechanics
;Canonical transformations
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian mechanics, Hami ...
The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function of p, q, and ''t'') allows the Hamiltonian in one set of coordinates q and momenta p to be transformed into a new set Q = Q(q, p, ''t'') and P = P(q, p, ''t''), in four possible ways:
:
With the restriction on P and Q such that the transformed Hamiltonian system is:
:
the above transformations are called ''canonical transformations'', each function ''Gn'' is called a generating function of the "''n''th kind" or "type-''n''". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem.
The choice of Q and P is completely arbitrary, but not every choice leads to a canonical transformation. One simple criterion for a transformation q → Q and p → P to be canonical is the Poisson bracket be unity,
:
for all ''i'' = 1, 2,...''N''. If this does not hold then the transformation is not canonical.
;The Hamilton–Jacobi equation
By setting the canonically transformed Hamiltonian ''K'' = 0, and the type-2 generating function equal to Hamilton's principal function (also the action ) plus an arbitrary constant ''C'':
:
the generalized momenta become:
:
and P is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:
:
where ''H'' is the Hamiltonian as before:
:
Another related function is Hamilton's characteristic function
:
used to solve the HJE by additive separation of variables for a time-independent Hamiltonian ''H''.
The study of the solutions of the Hamilton–Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology. In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curves of Hamiltonian vector fields.
Routhian mechanics
Routhian mechanics alt=
In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the ...
is a hybrid formulation of Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates. If the Lagrangian of a system has ''s'' cyclic coordinates q = ''q''1, ''q''2, ... ''qs'' with conjugate momenta p = ''p''1, ''p''2, ... ''ps'', with the rest of the coordinates non-cyclic and denoted ζ = ''ζ''1, ''ζ''1, ..., ''ζN − s'', they can be removed by introducing the ''Routhian'':
:
which leads to a set of 2''s'' Hamiltonian equations for the cyclic coordinates q,
:
and ''N'' − ''s'' Lagrangian equations in the non cyclic coordinates ζ.
:
Set up in this way, although the Routhian has the form of the Hamiltonian, it can be thought of a Lagrangian with ''N'' − ''s'' degrees of freedom.
The coordinates q do not have to be cyclic, the partition between which coordinates enter the Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian equations of motion.
Appellian mechanics
Appell's equation of motion
In classical mechanics, Appell's equation of motion (aka the Gibbs–Appell equation of motion) is an alternative general formulation of classical mechanics described by Josiah Willard Gibbs in 1879 and Paul Émile Appell in 1900.
Statement
...
involve generalized accelerations, the second time derivatives of the generalized coordinates:
:
as well as generalized forces mentioned above in D'Alembert's principle. The equations are
:
where
:
is the acceleration of the ''k'' particle, the second time derivative of its position vector. Each acceleration a''k'' is expressed in terms of the generalized accelerations ''αr'', likewise each rk are expressed in terms the generalized coordinates ''qr''.
Extensions to classical field theory
; Lagrangian field theory
Generalized coordinates apply to discrete particles. For ''N'' scalar fields ''φi''(r, ''t'') where ''i'' = 1, 2, ... ''N'', the Lagrangian density is a function of these fields and their space and time derivatives, and possibly the space and time coordinates themselves:
and the Euler–Lagrange equations have an analogue for fields:
where ''∂μ'' denotes the 4-gradient and the summation convention has been used. For ''N'' scalar fields, these Lagrangian field equations are a set of ''N'' second order partial differential equations in the fields, which in general will be coupled and nonlinear.
This scalar field formulation can be extended to vector fields, tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
s, and spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
s.
The Lagrangian is the volume integral of the Lagrangian density:[Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ]
Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such as Newtonian gravity
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
, classical electromagnetism, 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 ...
, and quantum field theory. It is a question of determining the correct Lagrangian density to generate the correct field equation.
;Hamiltonian field theory
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory ...
The corresponding "momentum" field densities conjugate to the ''N'' scalar fields ''φi''(r, ''t'') are:
where in this context the overdot denotes a partial time derivative, not a total time derivative. The Hamiltonian density is defined by analogy with mechanics:
The equations of motion are:
where the variational derivative
must be used instead of merely partial derivatives. For ''N'' fields, these Hamiltonian field equations are a set of 2''N'' first order partial differential equations, which in general will be coupled and nonlinear.
Again, the volume integral of the Hamiltonian density is the Hamiltonian
Symmetry, conservation, and Noether's theorem
; Symmetry transformations in classical space and time
Each transformation can be described by an operator (i.e. function acting on the position r or momentum p variables to change them). The following are the cases when the operator does not change r or p, i.e. symmetries.
where ''R''(n̂, θ) is the rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\ ...
about an axis defined by the unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
n̂ and angle θ.
; Noether's theorem
Noether's theorem states that a continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
symmetry transformation of the action corresponds to a conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
, i.e. the action (and hence the Lagrangian) doesn't change under a transformation parameterized by a parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
''s'':
the Lagrangian describes the same motion independent of ''s'', which can be length, angle of rotation, or time. The corresponding momenta to ''q'' will be conserved.
See also
* Lagrangian mechanics
* Hamiltonian mechanics
* Theoretical mechanics
*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 classi ...
* Dynamics
* Nazariy Mexanika
* Hamilton–Jacobi equation
* Hamilton's principle
*Kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
* Kinetics (physics)
* Non-autonomous mechanics
* Udwadia–Kalaba equation
References and notes
{{DEFAULTSORT:Analytical Mechanics
Mathematical physics
Theoretical physics
Dynamical systems