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Say we construct the Lagrangian for a system and minimise the action, what happens if this is not unique? In other words the action is minimised by two distinct (not infinitesimally separated) paths. Is there something else that governs the evolution of the system, or is there always a unique solution to the Euler-Lagrange equations?

Qmechanic
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Charlie
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    Possible duplicates: https://physics.stackexchange.com/q/203493/2451 , https://physics.stackexchange.com/q/203493/2451 and links therein. – Qmechanic Mar 31 '20 at 16:39

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Such a scenario is metastable. It has several time evolutions possible. The single time evolution for the system is not a defined thing. These scenarios are easy to construct. A simple example would be a roller coaster at the top of its highest hill, with zero velocity as an initial state, and its final state at the bottom of the coaster. It could either travel forwards or backwards. All we know is that it traveled from the initial point to the final point. Either path can minimize action.

In practice, this cannot possibly ever occur because we cannot construct such a system. If we did, we would soon find that its time evolution couples to everything. Whether mercury is in retrograde or not could start to affect the system because the gravity from mercury would affect the Lagrangian in one direction or the other.

These sorts of events are symmetry-breaking events, and are of great interest.

Cort Ammon
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  • What do you mean by "couples to everything"? I understand that Lagrangians are used in field theories that I haven't met yet, but what would happen if I constructed a simple classical system in which the Lagrangian has "degenerate" solutions in this way? – Charlie Mar 31 '20 at 16:32
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    I mean the change in potential energy w.r.t. position is always perturbed by every object in your light cone. Gravity, electrostatics, all sorts of "environmental" forces will perturb the real system, wrecking your perfect symmetry. If you are just forming a mathematically perfect system, all you can say in such a degenerate system is "it took one path or the other." – Cort Ammon Mar 31 '20 at 16:35
  • Solved, thank you. – Charlie Mar 31 '20 at 16:42