I'm wondering whether conservation laws are sufficient to imply equations of motions. Specifically:

1) In classical mechanics of point particles, are conservation of energy, conservation of momentum and conservation of angular momentum enough to imply the dynamics? Stated equivalently, given a particle trajectory in configuration space which is not a solution of the equation of motion, does that trajectory necessarily violate the conservation laws?

2) Same question about classical field theory.

3) Same question about point particle quantum mechanics.

4) Same question about quantum field theory.

I'm aware that my question isn't well defined, yet I'm very interested in your answers...


  • $\begingroup$ I am not sure to understand your question, but I may provide a funny example : The equation $\dot p + p = 0$, where p is the momentum, satisfyes space homogeneity, but not momentum conservation. The reason is that we cannot exhibit the Lagrangian. $\endgroup$
    – Trimok
    Jun 8 '13 at 11:13
  • $\begingroup$ I'm looking for the converse: an example of a trajectory which satisfies all conservation laws but not the equation of motion. Or a proof that there is no such example. $\endgroup$
    – Lior
    Jun 8 '13 at 11:31
  • $\begingroup$ How would you define your equation of motion which you would like to see violated whilst respecting the conservation laws? $\endgroup$
    – twistor59
    Jun 8 '13 at 11:44
  • $\begingroup$ I guess instead of using "equation of motion", what I really want to ask is whether there is a trajectory other than the physical one which satisfies the conservation laws, given that both trajectories have the same initial conditions. $\endgroup$
    – Lior
    Jun 8 '13 at 13:24

The conservation laws provide just a few equations so if there are more degrees of freedom you can find trajectories that obey all conservation laws but which do not obey the dynamics. E.g. two particles of the same mass with no forces acting on them or between them. If they travel on opposite sides of a circle at constant speed about their fixed centre of mass then they conserve energy, momentum and angular momentum, but the equations of motion would require them to travel in straight lines.

Same is true for field theories and quantum field theories.

  • $\begingroup$ Your answer does answer my question correctly, but now I understand that I need to restate my question to include all conserved quantities. For a given initial condition, is there a trajectory other than the physical one which has all of the exact same conserved quantities? In your example, each particle's linear momentum is different in the two trajectories. $\endgroup$
    – Lior
    Jun 8 '13 at 13:45
  • 1
    $\begingroup$ If there was a repulsive force acting between the particles then momentum and energy would only be conserved in total, not for each particle individually. So the answer is still no. $\endgroup$ Jun 8 '13 at 15:05

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