Hamilton's principle
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In
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, Hamilton's principle is
William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
'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 function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the '' differential''
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. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
of the physical system. Although formulated originally for
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, Hamilton's principle also applies to classical
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
such as the
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
and gravitational
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
, and plays an important role in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
and criticality theories.


Mathematical formulation

Hamilton's principle states that the true evolution of a system described by
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. 397 ...
between two specified states and at two specified times and is a
stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...
(a point where the variation is zero) of the
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
functional \mathcal mathbf\ \stackrel\ \int_^ L(\mathbf(t),\dot(t),t)\, dt where L(\mathbf,\dot,t) is the Lagrangian
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
for the system. In other words, any ''first-order'' perturbation of the true evolution results in (at most) ''second-order'' changes in \mathcal. The action \mathcal is a functional, i.e., something that takes as its input a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
and returns a single number, a scalar. In terms of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, Hamilton's principle states that the true evolution of a physical system is a solution of the functional equation That is, the system takes a path in configuration space for which the action is stationary, with fixed boundary conditions at the beginning and the end of the path.


Euler–Lagrange equations derived from the action integral

Requiring that the true trajectory be a
stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...
of the action functional \mathcal is equivalent to a set of differential equations for (the Euler–Lagrange equations), which may be derived as follows. Let represent the true evolution of the system between two specified states and at two specified times and , and let be a small perturbation that is zero at the endpoints of the trajectory \boldsymbol\varepsilon(t_1) = \boldsymbol\varepsilon(t_2) \ \stackrel\ 0 To first order in the perturbation , the change in the action functional \delta\mathcal would be \delta \mathcal = \int_^\; \left L(\mathbf+\boldsymbol\varepsilon,\dot +\dot)- L(\mathbf,\dot) \rightt = \int_^\; \left( \boldsymbol\varepsilon \cdot \frac + \dot \cdot \frac \right)\,dt where we have expanded the Lagrangian ''L'' to first order in the perturbation . Applying
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 ...
to the last term results in \delta \mathcal = \left \boldsymbol\varepsilon \cdot \frac\right^ + \int_^\; \left( \boldsymbol\varepsilon \cdot \frac - \boldsymbol\varepsilon \cdot \frac \frac \right)\,dt The boundary conditions \boldsymbol\varepsilon(t_1) = \boldsymbol\varepsilon(t_2) \ \stackrel\ 0 causes the first term to vanish \delta \mathcal = \int_^\; \boldsymbol\varepsilon \cdot\left(\frac - \frac \frac \right)\,dt Hamilton's principle requires that this first-order change \delta \mathcal is zero for all possible perturbations , i.e., the true path is a
stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...
of the action functional \mathcal (either a minimum, maximum or saddle point). This requirement can be satisfied if and only if These equations are called the Euler–Lagrange equations for the variational problem.


Canonical momenta and constants of motion

The conjugate momentum for a generalized coordinate is defined by the equation p_k \ \overset\ \frac. An important special case of the Euler–Lagrange equation occurs when ''L'' does not contain a generalized coordinate explicitly, \frac=0 \quad \Rightarrow \quad \frac \frac = 0 \quad \Rightarrow \quad \frac = 0 \,, that is, the conjugate momentum is a ''constant of the motion''. In such cases, the coordinate is called a cyclic coordinate. For example, if we use polar coordinates , , to describe the planar motion of a particle, and if does not depend on , the conjugate momentum is the conserved angular momentum.


Example: Free particle in polar coordinates

Trivial examples help to appreciate the use of the action principle via the Euler–Lagrange equations. A free particle (mass ''m'' and velocity ''v'') in Euclidean space moves in a straight line. Using the Euler–Lagrange equations, this can be shown in
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy L = \frac mv^2= \fracm \left( \dot^2 + \dot^2 \right) in orthonormal (''x'',''y'') coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time, ''t''). Therefore, upon application of the Euler–Lagrange equations, \frac \left( \frac \right) - \frac = 0 \qquad \Rightarrow \qquad m\ddot = 0 And likewise for ''y''. Thus the Euler–Lagrange formulation can be used to derive Newton's laws. In polar coordinates the kinetic energy and hence the Lagrangian becomes L = \fracm \left( \dot^2 + r^2\dot^2 \right). The radial and components of the Euler–Lagrange equations become, respectively \frac \left( \frac \right) - \frac = 0 \qquad \Rightarrow \qquad \ddot - r\dot^2 = 0 \frac \left( \frac \right)-\frac = 0 \qquad \Rightarrow \qquad \ddot + \frac\dot\dot = 0. remembering that r is also dependent on time and the product rule is needed to compute the total time derivative \frac mr^2 \dot. The solution of these two equations is given by r = \sqrt \varphi = \tan^ \left( \frac \right) + d for a set of constants , , , determined by initial conditions. Thus, indeed, ''the solution is a straight line'' given in polar coordinates: is the velocity, is the distance of the closest approach to the origin, and is the angle of motion.


Applied to deformable bodies

Hamilton's principle is an important variational principle in elastodynamics. As opposed to a system composed of rigid bodies, deformable bodies have an infinite number of degrees of freedom and occupy continuous regions of space; consequently, the state of the system is described by using continuous functions of space and time. The extended Hamilton Principle for such bodies is given by \int_^ \left \delta W_e + \delta T - \delta U \rightt = 0 where is the kinetic energy, is the elastic energy, is the work done by external loads on the body, and , the initial and final times. If the system is conservative, the work done by external forces may be derived from a scalar potential . In this case, \delta \int_^ \left T - (U + V) \rightt = 0. This is called Hamilton's principle and it is invariant under coordinate transformations.


Comparison with Maupertuis' principle

Hamilton's principle and Maupertuis' principle are occasionally confused and both have been called the
principle of least action Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a Lagrangian describing the physical sy ...
. They differ in three important ways: * ''their definition of the
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
...'' Maupertuis' principle uses an integral over the
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. 397 ...
known as the abbreviated action or reduced action \mathcal_ \ \stackrel\ \int \mathbf \cdot d\mathbf where p = (''p''1, ''p''2, ..., ''pN'') are the conjugate momenta defined above. By contrast, Hamilton's principle uses \mathcal, the integral of the Lagrangian over time. *''the solution that they determine...'' Hamilton's principle determines the trajectory q(''t'') as a function of time, whereas Maupertuis' principle determines only the shape of the trajectory in the generalized coordinates. For example, Maupertuis' principle determines the shape of the ellipse on which a particle moves under the influence of an inverse-square central force such as
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, but does not describe ''per se'' how the particle moves along that trajectory. (However, this time parameterization may be determined from the trajectory itself in subsequent calculations using the
conservation of energy The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
). By contrast, Hamilton's principle directly specifies the motion along the ellipse as a function of time. *''...and the constraints on the variation.'' Maupertuis' principle requires that the two endpoint states ''q''1 and ''q''2 be given and that energy be conserved along every trajectory (same energy for each trajectory). This forces the endpoint times to be varied as well. By contrast, Hamilton's principle does not require the conservation of energy, but does require that the endpoint times ''t''1 and ''t''2 be specified as well as the endpoint states ''q''1 and ''q''2.


Action principle for fields


Classical field theory

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. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
for
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
, such as the
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
or
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
. The Einstein equation utilizes the ''
Einstein–Hilbert action The Einstein–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 action is given as :S = \int R \sqrt ...
'' as constrained by a
variational principle A variational principle is a mathematical procedure that renders a physical problem solvable by the calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the pr ...
. The path of a body in a gravitational field (i.e. free fall in space time, a so-called
geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...
) can be found using the action principle.


Quantum mechanics and quantum field theory

In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all imaginable paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, that gives the
probability amplitude In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity at a point in space represents a probability density at that point. Probability amplitu ...
s of the various outcomes. Although equivalent in classical mechanics with
Newton's laws Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
, 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. In particular, it is fully appreciated and best understood within
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
's
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...
of quantum mechanics is based on a stationary-action principle, using path integrals.
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, Electrical network, electr ...
can be derived as conditions of stationary action.


See also

*
Analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
* Configuration space *
Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
*
Phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
* Geodesics as Hamiltonian flows * Herglotz's variational principle


References

* W.R. Hamilton, "On a General Method in Dynamics.", ''Philosophical Transactions of the Royal Society'
Part II (1834) pp. 247–308Part I (1835) pp. 95–144
(''From the collectio
Sir William Rowan Hamilton (1805–1865): Mathematical Papers
edited by David R. Wilkins, School of Mathematics, Trinity College, Dublin 2, Ireland. (2000); also reviewed a
On a General Method in Dynamics
') * Goldstein H. (1980) ''Classical Mechanics'', 2nd ed., Addison Wesley, pp. 35–69. * Landau LD and Lifshitz EM (1976) ''Mechanics'', 3rd. ed., Pergamon Press. (hardcover) and (softcover), pp. 2–4. * Arnold VI. (1989) ''Mathematical Methods of Classical Mechanics'', 2nd ed., Springer Verlag, pp. 59–61. * Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013. * Bedford A.: Hamilton's Principle in Continuum Mechanics. Pitman, 1985. Springer 2001, ISBN 978-3-030-90305-3 ISBN 978-3-030-90306-0 (eBook), https://doi.org/10.1007/978-3-030-90306-0 {{Authority control Lagrangian mechanics Calculus of variations Principles William Rowan Hamilton