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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 ...
, action is a scalar quantity describing how a
physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
has changed over time. Action is significant because 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 ...
of the system can be derived through the
principle of stationary action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
. In the simple case of a single particle moving with a constant velocity ( uniform linear motion), the action is the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
of the particle times the distance it moves, added up along its path; equivalently, action is twice the particle's
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 ...
times the duration for which it has that amount of energy. For more complicated systems, all such quantities are combined. More formally, action is a mathematical functional which takes the
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tr ...
(also called path or history) of the system as its argument and has a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
as its result. Generally, the action takes different values for different paths. Action has dimensions of
energy 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 ...
 × 
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
or
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
 × 
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
, and its SI unit is
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
-second (like the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
''h'').


Introduction

Hamilton's principle states that the differential equations of motion for ''any'' physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models. It applies not only to the
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 ...
of a single particle, but also to
classical field In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each poin ...
s such as the electromagnetic and
gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
fields. Hamilton's principle has also been extended 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, ...
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 particles and ...
—in particular 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 i ...
of quantum mechanics makes use of the concept—where a physical system randomly follows one of the possible paths, with the phase of the probability amplitude for each path being determined by the action for the path.Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,


Solution of differential equation

Empirical laws are frequently expressed as
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, which describe how physical quantities such as position and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
change continuously with
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
,
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
or a generalization thereof. Given the
initial In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph tha ...
and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called ''
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 ...
''.


Minimization of action integral

''Action'' is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the ''action is minimized'', or more generally, is stationary. In other words, the action satisfies a variational principle: the
principle of stationary action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
(see also below). The action is defined by an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
, and the classical equations of motion of a system can be derived by minimizing the value of that integral. This simple principle provides deep insights into physics, and is an important concept in modern
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 ...
.


History

''Action'' was defined in several now obsolete ways during the development of the concept.Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, * Gottfried Leibniz, Johann Bernoulli and
Pierre Louis Maupertuis Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
defined the action for light as the integral of its speed or inverse speed along its path length. *
Leonhard 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 ...
(and, possibly, Leibniz) defined action for a material particle as the integral of the particle's speed along its path through space. *
Pierre Louis Maupertuis Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
introduced several ''ad hoc'' and contradictory definitions of action within a single
article Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: ...
, defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.


Mathematical definition

Expressed in mathematical language, using 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
evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action. Several different definitions of "the action" are in common use in physics. The action is usually an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over time. However, when the action pertains to fields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system. The action is typically represented as an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over time, taken along the path of the system between the initial time and the final time of the development of the system: \mathcal = \int_^ L \, dt, where the integrand ''L'' is called the Lagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space. Action has the dimensions of nergynbsp;×  ime and its SI unit is
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
-second, which is identical to the unit of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
.


Action in classical physics

In
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, the term "action" has a number of meanings.


Action (functional)

Most commonly, the term is used for 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 ...
\mathcal which takes a function of time and (for fields) space as input and returns a scalar.The Road to Reality, Roger Penrose, Vintage books, 2007, T. W. B. Kibble, ''Classical Mechanics'', European Physics Series, McGraw-Hill (UK), 1973, In
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 ...
, the input function is the evolution q(''t'') of the system between two times ''t''1 and ''t''2, where q represents the generalized coordinates. The action \mathcal mathbf(t)/math> is defined as the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the Lagrangian ''L'' for an input evolution between the two times: \mathcal mathbf(t)= \int_^ L(\mathbf(t),\dot(t),t)\, dt, where the endpoints of the evolution are fixed and defined as \mathbf_ = \mathbf(t_) and \mathbf_ = \mathbf(t_). According to Hamilton's principle, the true evolution qtrue(''t'') is an evolution for which the action \mathcal mathbf(t)/math> is stationary (a minimum, maximum, or a
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...
). This principle results in the equations of motion in Lagrangian mechanics.


Abbreviated action (functional)

The abbreviated action is also 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 ...
. It is usually denoted as \mathcal_. Here the input function is the ''path'' followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path. The abbreviated action \mathcal_ is defined as the integral of the generalized momenta along a path in the generalized coordinates: \mathcal_0 = \int \mathbf \cdot d\mathbf = \int p_i \,dq_i. Spelled out concretely, this is \mathcal_0 = \int_^ \mathbf(t) \cdot \dot(t)\, dt = \int_^ p_i(t) \,\frac \,dt. According to
Maupertuis' principle In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of ''path'' and ''length''). It is a special case of ...
, the true path is a path for which the abbreviated action \mathcal_ is stationary.


Hamilton's principal function

Hamilton's principal function S=S(q,t;q_0,t_0) is obtained from the action functional \mathcal by fixing the initial time t_0 and the initial endpoint q_0, while allowing the upper time limit t and the second endpoint q to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, 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 classi ...
. Due to a similarity with 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 ...
, the Hamilton–Jacobi equation provides, arguably, the most direct link with
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, ...
.


Hamilton's characteristic function

When the total energy ''E'' is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables: S(q_1, \dots, q_N, t) = W(q_1, \dots, q_N) - E \cdot t, where the time-independent function ''W''(''q''1, ''q''2, ..., ''qN'') is called ''Hamilton's characteristic function''. The physical significance of this function is understood by taking its total time derivative \frac = \frac \dot q_i = p_i \dot q_i. This can be integrated to give W(q_1, \dots, q_N) = \int p_i\dot q_i \,dt = \int p_i \,dq_i, which is just the abbreviated action.


Other solutions of Hamilton–Jacobi equations

The Hamilton–Jacobi equations are often solved by additive separability; in some cases, the individual terms of the solution, e.g., ''Sk''(''qk''), are also called an "action".


Action of a generalized coordinate

This is a single variable ''Jk'' in the
action-angle coordinates In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solvin ...
, defined by integrating a single generalized momentum around a closed path 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 ...
, corresponding to rotating or oscillating motion: J_k = \oint p_k \,dq_k The variable ''Jk'' is called the "action" of the generalized coordinate ''qk''; the corresponding canonical variable conjugate to ''Jk'' is its "angle" ''wk'', for reasons described more fully under
action-angle coordinates In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solvin ...
. The integration is only over a single variable ''qk'' and, therefore, unlike the integrated dot product in the abbreviated action integral above. The ''Jk'' variable equals the change in ''Sk''(''qk'') as ''qk'' is varied around the closed path. For several physical systems of interest, Jk is either a constant or varies very slowly; hence, the variable ''Jk'' is often used in perturbation calculations and in determining adiabatic invariants.


Action for a Hamiltonian flow

See tautological one-form.


Euler–Lagrange equations

In Lagrangian mechanics, the requirement that the action integral be stationary under small perturbations is equivalent to a set of
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 (called the Euler–Lagrange equations) that may be obtained using 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 action principle


Classical fields

The action principle can be extended to obtain 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 ...
for fields, such as the
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 ...
or
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 ...
. The
Einstein equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the for ...
utilizes the ''
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the a ...
'' as constrained by a variational principle. The
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tr ...
(path in
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
) of a body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is a
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 ...
.


Conservation laws

Implications of symmetries in a physical situation can be found with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is Noether's theorem, which states that to every
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to ano ...
in a physical situation there corresponds 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 ...
(and conversely). This deep connection requires that the action principle be assumed.


Quantum mechanics and quantum field theory

In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, which gives the probability amplitudes of the various outcomes. Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly in Richard Feynman's
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 i ...
, where it arises out of destructive interference of quantum amplitudes.
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
can also be derived as conditions of stationary action.


Single relativistic particle

When relativistic effects are significant, the action of a point particle of mass ''m'' travelling a
world line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
''C'' parametrized by the proper time \tau is S = - m c^2 \int_ \, d \tau. If instead, the particle is parametrized by the coordinate time ''t'' of the particle and the coordinate time ranges from ''t''1 to ''t''2, then the action becomes S = \int_^ L \, dt, where the Lagrangian isL. D. Landau and E. M. Lifshitz (1971). ''The Classical Theory of Fields''. Addison-Wesley. Sec. 8. p. 24–25. L = -mc^2 \sqrt.


Modern extensions

The action principle can be generalized still further. For example, the action need not be an integral, because nonlocal actions are possible. The configuration space need not even be a functional space, given certain features such as noncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally.


See also

*
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 ...
*
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 w ...
* Functional integral * Hamiltonian mechanics * Lagrangian * Lagrangian mechanics *
Measure (physics) The measure in quantum physics is the integration measure used for performing a path integral. In quantum field theory, one must sum over all possible histories of a system. When summing over possible histories, which may be very similar to each ...
* Noether's theorem *
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 i ...
* Principle of least action *
Principle of maximum entropy The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition ...
* Some actions: **
Nambu–Goto action The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate string-like objects (for example, cosmic strings). It is the starting point of the analysis of zero-thickness (i ...
** Polyakov action **
Bagger–Lambert–Gustavsson action In theoretical physics, in the context of M-theory, the action for the '' N''=8 M2 branes in full is (with some indices hidden): : S = \intd\sigma^3 where is a generalisation of a Lie bracket which gives the group constants. The only known c ...
**
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the a ...


References


Sources and further reading

For an annotated bibliography, see Edwin F. Taylor wh
lists
among other things, the following books * ''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, . *
Cornelius Lanczos __NOTOC__ Cornelius (Cornel) Lanczos ( hu, Lánczos Kornél, ; born as Kornél Lőwy, until 1906: ''Löwy (Lőwy) Kornél''; February 2, 1893 – June 25, 1974) was a Hungarian-American and later Hungarian-Irish mathematician and physicist. Acco ...

The Variational Principles of Mechanics
(Dover Publications, New York, 1986). . ''The'' reference most quoted by all those who explore this field. * L. D. Landau and
E. M. Lifshitz Evgeny Mikhailovich Lifshitz (russian: Евге́ний Миха́йлович Ли́фшиц; February 21, 1915, Kharkiv, Russian Empire – October 29, 1985, Moscow, Russian SFSR) was a leading Soviet physicist and brother of the physicist ...
, Mechanics, Course of Theoretical Physics (Butterworth-Heinenann, 1976), 3rd ed., Vol. 1. . Begins with the principle of least action. * Thomas A. Moore "Least-Action Principle" in Macmillan Encyclopedia of Physics (Simon & Schuster Macmillan, 1996), Volume 2, , , pages 840–842. * Gerald Jay Sussman and Jack Wisdom
Structure and Interpretation of Classical Mechanics
(MIT Press, 2001). Begins with the principle of least action, uses modern mathematical notation, and checks the clarity and consistency of procedures by programming them in computer language. * Dare A. Wells, Lagrangian Dynamics, Schaum's Outline Series (McGraw-Hill, 1967) , A 350-page comprehensive "outline" of the subject. * Robert Weinstock, Calculus of Variations, with Applications to Physics and Engineering (Dover Publications, 1974). . An oldie but goodie, with the formalism carefully defined before use in physics and engineering. * Wolfgang Yourgrau and Stanley Mandelstam
Variational Principles in Dynamics and Quantum Theory
(Dover Publications, 1979). A nice treatment that does not avoid the philosophical implications of the theory and lauds the Feynman treatment of quantum mechanics that reduces to the principle of least action in the limit of large mass. * Edwin F. Taylor'


External links



Interactive explanation/webpage {{Authority control Lagrangian mechanics Hamiltonian mechanics Calculus of variations Dynamics (mechanics)