The connection between variational principle and differential equations

[Heirot]

It is very well known that the result of varying some functional gives a differential equation which solutions minimizes the given functional. What about the other way around? Can one find a functional that is minimized given a differential equation? Is there a procedure for this?

The reason I'm asking this is to see whether the principle of least action in physics is a true phyisical principle that asserts: only those diff. eqs. can appear in physics which result from variation of some functional "Action"? Or is the principle of least action a tautological one?

 

[gtoguy97]

Yes, it is possible to find a functional that is minimized given a differential equation. This is known as the principle of least action in physics, which states that the differential equations that appear in physical systems are those that result from the variation of some functional (known as the action). Thus, the principle of least action is not a tautological statement, but rather an assertion about the structure of physical systems. To find a functional that minimizes a given differential equation, one can use either the Hamiltonian or the Lagrangian method. In the former, the differential equation is transformed into a set of first-order differential equations. These equations are then used to construct a Hamiltonian function, which is then minimized to obtain the desired functional. The Lagrangian method, on the other hand, requires the transformation of the differential equation into a system of second-order differential equations. These equations are then used to construct a Lagrangian function, which is then minimized to obtain the desired functional. In both cases, the resulting functional is often referred to as the action. It is important to note, however, that the action is not necessarily the same as the energy of the system; instead, it is a measure of the difference between the initial and the final states of the system.