Many-Body Dynamics: Conserved Hamiltonian?

[Gear300]

For a system of more than 2 particles each interacting with each other under a potential (conservative force), does the dynamics of anyone particle exhibit a conserved (energy) quantity --- in other words, is it possible to write down a conserved Hamiltonian for each particle? The reason I ask is that the net force on anyone particle is not necessarily symmetrical.

 

[Bill_K]

You cannot write down individual conserved Hamiltonians for interacting particles even if there are only two particles, so I must not be understanding your question.

 

[Gear300]

No, you seem to have understood it.

Though, doesn't it depend on the frame of reference? For instance, in the 2-body problem, if you were to take the frame of reference such that one particle lies at the origin and use a reduced mass description, couldn't you come up with a conserved Hamiltonian?

 

[kanato]

You can't take an inertial frame of reference where one particle lies at the origin. Since the other particle will exert a force on it, a frame of reference tacked onto one particle will be non-inertial, and then questions like "what is a conservation law" get really confusing. If you use a reduced mass description in a two body problem, you have one Hamiltonian for the center of mass, and one Hamiltonian for the relative distance between the particles. These are both conservative, but neither of of them belongs to one particle or the other.

 

[Gear300]

You can't take an inertial frame of reference where one particle lies at the origin. Since the other particle will exert a force on it, a frame of reference tacked onto one particle will be non-inertial, and then questions like "what is a conservation law" get really confusing. If you use a reduced mass description in a two body problem, you have one Hamiltonian for the center of mass, and one Hamiltonian for the relative distance between the particles. These are both conservative, but neither of of them belongs to one particle or the other.

If the acceleration of one particle was taken relative to another and it was found that the effective or apparent force in this frame behaved such that a scalar potential existed, then could you not still be able to make a statement on a conservation law despite being in a non-inertial frame?

 

[espen180]

You would, but it would not be as pretty as in an inertial frame. For example, in an inertial frame you have a symmetric momentum conservation law. However, in an accelerating frame you would have a nonsymmetric momentum conservation law. Especially when the acceleration of the frame is time-dependent, things will likely become very messy.