# Is there a trajectory which is not a solution of the equation of motion but satisfies all conservation laws?

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...

Thanks!

• 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. Jun 8 '13 at 11:13
• 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.
– Lior
Jun 8 '13 at 11:31
• How would you define your equation of motion which you would like to see violated whilst respecting the conservation laws? Jun 8 '13 at 11:44
• 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.
– Lior
Jun 8 '13 at 13:24