This is possibly a silly question but when we derive the equations of motion of a particle using the principle of least action. We must assume that there is a single minimum (for a fixed choice of boundary conditions), right? What happens if we have two minimums? How do we decide what trajectory that particle took? Is it just a case of never creating a Lagrangian with that form?
There seems to be a slight confusion about the meaning of solution: The principle of least action leads to the equation of motion (Euler-Lagrange equation), which correspond to a minimum of the action functional. These equations can have multiple solutions, so there is no contradiction in the formalism. There can multiple solutions that minimize the energy, but only one equation determined from the principle of least action.
As I have pointed out in the comment section below, the Euler-Lagrange equations are obtained in a unique way from the principle of least action. Ambiguities can only arise when one takes into account boundary conditions.