Hamilton's principle
Hamilton's principle states that the true evolution of a system described by generalized coordinates between two specified states and at two specified times and is a stationary point (a point where the variation is zero) of the action functional
formula_1
where formula_2 is the Lagrangian function for the system. In other words, any "first-order" perturbation of the true evolution results in (at most) "second-order" changes in formula_3. The action formula_3 is a functional, i.e., something that takes as its input a function and returns a single number, a scalar. In terms of functional analysis, Hamilton's principle states that the true evolution of a physical system is a solution of the functional equation
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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. See also more rigorous derivation Euler–Lagrange equation
Requiring that the true trajectory be a stationary point of the action functional formula_3 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
formula_6
To first order in the perturbation , the change in the action functional formula_7 would be
formula_8
where we have expanded the Lagrangian "L" to first order in the perturbation . Applying integration by parts to the last term results in
formula_9
The boundary conditions formula_10 causes the first term to vanish
formula_11
Hamilton's principle requires that this first-order change formula_12 is zero for all possible perturbations , i.e., the true path is a stationary point of the action functional formula_3 (either a minimum, maximum or saddle point). This requirement can be satisfied if and only if
 - \frac{d}{dt}\frac{\partial L}{\partial \dot{\mathbf{q}}} = 0</math> These equations are called the Euler–Lagrange equations for the variational problem. The conjugate momentum for a generalized coordinate is defined by the equation
formula_14
An important special case of the Euler–Lagrange equation occurs when "L" does not contain a generalized coordinate explicitly,
formula_15
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. 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 as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy 
formula_16
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,
formula_17
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
formula_18
The radial and components of the Euler–Lagrange equations become, respectively
formula_19
formula_20
remembering that r is also dependent on time and the product rule is needed to compute the total time derivative formula_21. The solution of these two equations is given by
formula_22
formula_23
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. 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
formula_24
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,
formula_25
This is called Hamilton's principle and it is invariant under coordinate transformations. Hamilton's principle and Maupertuis' principle are occasionally confused and both have been called (incorrectly) the principle of least action. They differ in three important ways: The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravity. The Einstein equation utilizes the "Einstein–Hilbert action" as constrained by a variational principle. The path of a body in a gravitational field (i.e. free fall in space time, a so-called geodesic) can be found using the action principle. In quantum mechanics, 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 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. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman's path integral formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations can be derived as conditions of stationary action.
Mathematical formulation
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single 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 of the physical system.