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?
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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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