# Is it possible to have the principle of least action and multiple solutions?

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?

• Possible duplicates: physics.stackexchange.com/q/115208/2451 and physics.stackexchange.com/q/3928/2451 Oct 14 '14 at 13:30
• This is what I am asking. However the answers don't make much sense to me. So we can have multiple equations of motion for a single dynamical system? Oct 14 '14 at 14:10
• I included an example there. Oct 14 '14 at 14:53
• Consider the path of a light ray from one focus of a thin lens to the other expressed in least action terms... Oct 15 '14 at 2:11
• So if I understand you correctly you can have multiple different equations of motion. However these are all due to different boundary conditions. In the example that you give @Qmechanic this equation of motion would look different depending on the boundary conditions. Is that due to the fact that between these different boundaries there is a different extrema where dS=0? Oct 15 '14 at 10:37