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We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's principle) and even General Relativity (we get Einstein's equation from the Einstein-Hilbert action). However, how do we explain this very principle, i.e., more mathematically, I want to ask the following:

If I am given a set of generalized positions and velocities, say, $\{q_{i}, \dot{q}_{i}\}$, which describes a classical system with known dynamics (equations of motion), then, how do I rigorously show that there always exists an action functional $A$, where $$A ~=~ \int L(q_{i}, \dot{q}_{i})dt,$$ such that $\delta A = 0$ gives the correct equations of motion and trajectory of the system?

I presume historically, the motivation came from Optics: i.e., light rays travel along a path where $S = \int_{A}^{B} n ds$ is minimized (or at least stationary). (Here, $ds$ is the differential element along the path). I don't mind some symplectic geometry talk if that is needed at all.

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2 Answers

I) Not all equations of motion (eom) are variational. A famous example is the self-dual five-form in type IIB superstring theory. In classical point mechanics, frictional forces typically lead to non-variational problems.

II) Consider for instance $n$ variable $q^i$ and $n$ eoms,

$$\tag{1} E_i~\approx~ 0, \qquad i~\in~\{1, \ldots, n\}. $$

A simplified version of OP's problem (v3) is the following:

Does there exist an action $$\tag{2} S[q] ~=~\int{\rm d}t~L$$ such that the Euler-Lagrange derivatives $$\tag{3} \frac{\delta S}{\delta q^i}~=~E_i $$ precisely become the given $E_i$-functions?

The above restricted problem is relatively easy to answer once and for all, because one may differentiate the known $E_i$-functions to arrive at a set of consistency conditions. Let us for simplicity assume that the functions $E_i=E_i(q)$ do not involve generalized velocities $\dot{q}^i$, accelerations $\ddot{q}^i$, and so forth. Then we may assume that the Lagrangian $L$ does not depend on time derivatives of $q^i$ as well. So the question becomes if

$$\tag{4} \frac{\partial L}{\partial q^i}~=~E_i ? $$

We can collect the information of the eoms in a one-form

$$\tag{5} E~:=~E_i ~{\rm d}q^i.$$

The question rewrites as

$$\tag{6} {\rm d}L~=~E? $$

Hence the Lagrangian $L$ exists if $E$ is an exact one-form.

III) However, the above discussion is in many ways oversimplified. The eoms (1) do not have a unique form! E.g. one may multiply the given $E_i$-functions with an invertible $q$-dependent matrix $A^i{}_j$ such that the eoms (1) equivalently read

$$\tag{7} \sum_{i=1}^n E_i A^i{}_j~\approx~ 0. $$

Or perhaps the system variables $q^i$ should be viewed as a subsystem of a larger system with more dynamical or auxiliary variables?

Ultimately, the main question is whether the eoms have an action principle or not; the particular form of the eoms (that the Euler-Lagrange equations spit out) is not important in this context.

This opens up a lot of possibilities, and it can be very difficult to systematically find an action principle; or conversely, to prove a no-go theorem that a given set of eoms is not variational.

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Obviously one can mathematically cook up equations of motion that would not arise from an action principle.

The original motivation for believing that Nature obeys a Law of Least Action was metaphysical, and then it turned out that in reality, one could only guarantee that the action was stationary, not necessarily minimal, which ruined the metaphysics... besides, one must be cautious about postulating that Nature has to do something which we have deduced based on philosophical motivations.

But ever since Hertz and Einstein, there has been another motivation. (Whether it will stand the test of time better than string theory, remains to be seen...) Gauss, Hertz, and after them, Klein (see Whittaker, Analytical Dynamics, p. 254ff. and Hertz, The Principles of Mechanics, http://www.archive.org/details/principlesofmech00hertuoft ) reformulated Newtonian Mechanics in terms of an abstract curved space on which all particles followed geodesics. The metric on the space was cooked up from the forces acting on the system, and all the laws of mechanics reduced to Hertz's principle of least curvature instead of least action. Now after Einstein we know that if we interpret gravity as the metric of space-time, then particles under the influence of gravity follow a geodesic. This is a generalisation of the very old principle of inertia: with Newton it was stated as, a particle not acted on by a force travels in a straight line, i.e., a geodesic in flat Newtonian space. Einstein re-formulated this as above stated. The quest for a (non-quantum) unified field theory was always motivated by this: define a geometry on space-time based on the forces of Nature so that all trajectories will be geodesics. The physical insight here is the same as that underlying the original law of inertia: natural, unconstrained motion is straight, i.e., geodesic. But geodesics always do obey some variational principle.

If we take Einstein's point of view seriously, and think it will survive when treated quantum-mechanically, then the answer to your question would be: if the set of trajectories arise as the set of geodesics from some metric on the relevant space, then there is a physically significant action principle which governs the dynamics.

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