the action functional (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 An important special case of the Euler–Lagrange equation occurs when L does not contain a generalized coordinate explicitly, 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 as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy 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, 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 The radial and components of the Euler–Lagrange equations become, respectively remembering that r is also dependent on time and the product rule is needed to compute the total time derivative . The solution of these two equations is given by 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 where is the kinetic energy, is the elastic energy, is the work done by external loads on