In physics, the equations of motion of a physical system can be derived by minimizing/maximizing an "action", i.e. a functional of the path of the system:
$$J(x)= \int_a^bL(t,x(t),x'(t))dt$$
where $x:\mathbb R\to X$ is the motion over time, for some state space $X$.
I assume that not all dynamical systems have the property that motion over time can be represented by the minimum of such a functional. Is there a name in mathematics for such systems?
A system of (differential) equations with a variational principle is called a variational system or sometimes a Lagrangian system$^1$.
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$^1$ Be aware that some systems in Lagrangian mechanics are non-variational, cf. e.g. this Phys.SE post.