In physics, we often describe the dynamic properties of fields using variational principles like defining an action or a Lagrangian. A field however is simply some function of space $\phi(x)$ so I wonder what kind of properties the dynamics must follow to lend itself to description by an action principle?

For example, can all dynamics that are continuous be described by some specific action?

Just want to get an idea of the restrictions.


1 Answer 1


Since you have tagged it as classical field theory, I am assuming you are asking what properties $\phi(x)$ must have to admit equations of motion which may be derived from an action principle.

The property $\phi(x)$ must have to be amenable by an action principle is that it must satisfy equations of motion which are Euler-Lagrange equations for some action.

Unfortunately, I do not know of any papers on what properties $\phi(x)$ must have, but there is a classical theorem on what properties the equations must have, in classical mechanics. If one has $u : [0,T] \to \mathbb R^n$ which is differentiable, and satisfying $\ddot u = f^i(u, \dot u)$, we require that for the existence of Lagrangian, the existence of a non-singular symmetric matrix $M_{ij}(u,\dot u)$ satisfying:

  1. $$(M \Phi) = (M\Phi)^T$$
  2. $$\forall_{i,j}\quad \dot M_{ij} = \frac12 \frac{\partial f^k}{\partial \dot u^i} M_{kj} + \frac12 \frac{\partial f^k}{\partial \dot u^j} M_{ki} $$
  3. $$\forall_{i,j,k} \quad \frac{\partial M_{ij}}{\partial \dot u^k} = \frac{\partial M_{ik}}{\partial \dot u^j}$$

where $\Phi^i_j = \frac12 \partial_t \partial_{\dot u^j} f^i - \partial_{u^j}f^i - \frac14 (\partial_{\dot u^k} f^i)(\partial_{\dot u^j} f^k)$. These are the Helmholtz conditions. I do not know of a generalisation to field theory, but this work has been expanded on in papers such as here, here and the citations of this paper. Another way to view these conditions is as those for $0+1$-dimensional field theory.


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