Editing Calculus of variations (section)

=== Mechanics ===
{{main|Action (physics)}}
In classical mechanics, the action, <math>S,</math> is defined as the time integral of the Lagrangian, <math>L.</math>  The Lagrangian is the difference of energies,
<math display="block">L = T - U, </math>
where <math>T</math> is the [[kinetic energy]] of a mechanical system and <math>U</math> its [[potential energy]]. [[Hamilton's principle]] (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral
<math display="block">S = \int_{t_0}^{t_1} L(x, \dot x, t) \, dt</math>
is stationary with respect to variations in the path <math>x(t).</math>
The Euler–Lagrange equations for this system are known as Lagrange's equations:
<math display="block">\frac{d}{dt} \frac{\partial L}{\partial \dot x} = \frac{\partial L}{\partial x}, </math>
and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta <math>P</math> are defined by
<math display="block">p = \frac{\partial L}{\partial \dot x}. </math>
For example, if
<math display="block">T = \frac{1}{2} m \dot x^2, </math>
then <math display="block">p = m \dot x. </math>
[[Hamiltonian mechanics]] results if the conjugate momenta are introduced in place of <math>\dot x</math> by a Legendre transformation of the Lagrangian <math>L</math> into the Hamiltonian <math>H</math> defined by
<math display="block">H(x, p, t) = p \,\dot x - L(x,\dot x, t).</math>
The Hamiltonian is the total energy of the system: <math>H = T + U.</math>
Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of <math>X.</math> This function is a solution of the [[Hamilton–Jacobi equation]]:
<math display="block">\frac{\partial \psi}{\partial t} + H\left(x,\frac{\partial \psi}{\partial x},t\right) = 0.</math>