Classical field theory

Remarks

The field equations form a system of $k$ partial differential equations of order at most two for $\varphi$.

Example: for $n = 2$, $k = 1$ and $\mathcal{L} = \frac{1}{2} z_0^2 - \frac{c^2}{2} z_1^2$ with $c > 0$, one has $\partial \mathcal{L} / \partial y = 0$, $\partial \mathcal{L} / \partial z_0 = z_0$ and $\partial \mathcal{L} / \partial z_1 = -c^2 z_1$. The field equation is $\partial_0^2 \varphi - c^2 \partial_1^2 \varphi = 0$, the one-dimensional wave equation.

Non-example: the expression $\varphi \, \partial_0^2 \varphi$ depends on a second derivative of the field, so it is not a first-order Lagrangian density in the sense above. Field theories with such higher-order densities exist, but their Euler–Lagrange equations have extra terms.

In the geometric setting, the fields are sections of a fiber bundle $Y \to X$ over a spacetime manifold $X$. The Lagrangian density is then defined on the first jet bundle of $Y$. Field theories can also be given by Hamiltonian formulations, or directly by field equations without a Lagrangian.