3

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?

Qmechanic
  • 201,751
Alexander Morton
  • 41
  • 1
    Possible duplicates: https://physics.stackexchange.com/q/115208/2451 and https://physics.stackexchange.com/q/3928/2451 – Qmechanic 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? – Alexander Morton Oct 14 '14 at 14:10
  • I included an example there. – Qmechanic 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... – dmckee --- ex-moderator kitten Oct 15 '14 at 02:11
  • 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? – Alexander Morton Oct 15 '14 at 10:37
  • No, the EOM and BC are the same. Still, the the system (EOM,BC) has infinitely many solutions. – Qmechanic Oct 15 '14 at 10:41
  • How do you know though for every possible Lagrangian that there is never two different possible EOM? I know physically I can explain this but is it built into the math somewhere? – Alexander Morton Oct 15 '14 at 10:44
  • Mathematically, it follows from the defining properties of a functional derivative that the Euler-Lagrange eqs. are unique. – Qmechanic Oct 15 '14 at 10:47
  • Sorry maybe I am missing something but what defining property? – Alexander Morton Oct 15 '14 at 10:58

1 Answers1

2

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.

Frederic Brünner
  • 15,722
  • I should have explained this better. I understand that the final differential equation can have multiple solutions. However is it possible to construct a lagrangian that will give you multiple different equations of motion? – Alexander Morton Oct 14 '14 at 13:56
  • @AlexanderMorton: No, the Euler-Lagrange equations are unique for a given Lagrangian. There might, however be an ambiguity if one encounters boundary terms. The latter can vanish for different choices of boundary conditions. An example can be found in string theory, where the choice of boundary conditions determines whether you are dealing with an open or a closed string. – Frederic Brünner Oct 14 '14 at 14:08