Editing Analytical mechanics (section)

==Classical field theory==
===[[Lagrangian field theory]]===

Generalized coordinates apply to discrete particles. For ''N'' [[scalar field]]s ''φ<sub>i</sub>''('''r''', ''t'') where ''i'' = 1, 2, ... ''N'', the '''[[Lagrangian density]]''' is a function of these fields and their space and time derivatives, and possibly the space and time coordinates themselves:
<math display="block">\mathcal{L} = \mathcal{L}(\phi_1, \phi_2, \dots, \nabla\phi_1, \nabla\phi_2, \dots, \partial_t \phi_1, \partial_t \phi_2, \ldots, \mathbf{r}, t)\,.</math>
and the Euler–Lagrange equations have an analogue for fields:
<math display="block">\partial_\mu \left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi_i)}\right) = \frac{\partial \mathcal{L}}{\partial \phi_i}\,,</math>
where ''∂<sub>μ</sub>'' denotes the [[4-gradient]] and the [[summation convention]] has been used. For ''N'' scalar fields, these Lagrangian field equations are a set of ''N'' second order partial differential equations in the fields, which in general will be coupled and nonlinear.

This scalar field formulation can be extended to [[vector field]]s, [[tensor field]]s, and [[spinor field]]s.

The Lagrangian is the [[volume integral]] of the Lagrangian density:<ref name="autogenerated3"/><ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, {{ISBN|0-7167-0344-0}}</ref>
<math display="block">L = \int_\mathcal{V} \mathcal{L} \, dV \,.</math>

Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such as [[Newton's law of universal gravitation|Newtonian gravity]], [[classical electromagnetism]], [[general relativity]], and [[quantum field theory]]. It is a question of determining the correct Lagrangian density to generate the correct field equation.

===[[Hamiltonian field theory]]===

The corresponding "momentum" field densities conjugate to the ''N'' scalar fields ''φ<sub>i</sub>''('''r''', ''t'') are:<ref name="autogenerated3"/>
<math display="block">\pi_i(\mathbf{r},t) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}_i}\,\quad\dot{\phi}_i\equiv \frac{\partial \phi_i}{\partial t}</math>
where in this context the overdot denotes a partial time derivative, not a total time derivative. The '''Hamiltonian density''' <math>\mathcal{H}</math> is defined by analogy with mechanics:
<math display="block">\mathcal{H}(\phi_1, \phi_2,\ldots, \pi_1, \pi_2, \ldots,\mathbf{r},t) = \sum_{i=1}^N \dot{\phi}_i(\mathbf{r},t)\pi_i(\mathbf{r},t) - \mathcal{L}\,.</math>

The equations of motion are:
<math display="block">\dot{\phi}_i = +\frac{\delta\mathcal{H}}{\delta \pi_i}\,,\quad \dot{\pi}_i = - \frac{\delta\mathcal{H}}{\delta \phi_i} \,, </math>
where the [[variational derivative]]
<math display="block">\frac{\delta}{\delta \phi_i} = \frac{\partial}{\partial \phi_i} - \partial_\mu \frac{\partial }{\partial (\partial_\mu \phi_i)} </math>
must be used instead of merely partial derivatives. For ''N'' fields, these Hamiltonian field equations are a set of 2''N'' first order partial differential equations, which in general will be coupled and nonlinear.

Again, the volume integral of the Hamiltonian density is the Hamiltonian
<math display="block">H = \int_\mathcal{V} \mathcal{H} \, dV \,.</math>