Disconnected extremums of an action

[ShayanJ]

If you want to find the extremums of an action, you just solve the associated Euler-Lagrange equations. But this procedure gives you only the continuous extremums. There may be extremums that consist of two separate pieces, like two separate surfaces. How would you find out whether such a solution exists for a given action or not?
Thanks

 

[jvicens]

for your post! You bring up an interesting point about finding extremums of an action. As you mentioned, solving the associated Euler-Lagrange equations will give us the continuous extremums, but there may be cases where the extremum consists of two separate pieces.

To determine if such a solution exists for a given action, one approach would be to use a variational principle known as the "Principle of Least Action." This principle states that the true path of a system is the one that minimizes the action, which is a mathematical quantity that describes the behavior of the system.

Using this principle, we can set up a variational problem where we minimize the action subject to certain constraints. These constraints can be used to ensure that the solution we find is continuous and does not consist of two separate pieces.

Another approach would be to use numerical methods to solve the Euler-Lagrange equations and then check for any discontinuities or jumps in the solution. If such discontinuities are present, it is an indication that the extremum may consist of two separate pieces.

In addition, it may also be helpful to analyze the physical system and its boundary conditions to see if they allow for such a solution. For example, in a physical system where the extremum represents the path of a particle, we can consider the forces acting on the particle and see if they allow for a discontinuous solution.

Overall, finding out whether a solution with two separate pieces exists for a given action requires a combination of mathematical analysis, physical understanding, and numerical methods. It is an interesting and challenging problem that requires careful consideration and investigation.