Can partial vanishing of Poisson bracket determine local constants of motion?

In summary, If {f,H}=0 only on a subset of the phase space, then f will only be a constant of motion for the trajectories contained in that subset. However, these local constants may not have significant physical meaning if considering the analytic continuation.
  • #1
giova7_89
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I don't know if this is the right place to post this, but my question is: if i have an Hamiltonian defined on the whole phase space and a function f which is also defined on the whole phase space and doesn't depend explicitly on time, i know that if its poisson bracket with the Hamiltonian vanishes everywhere, f is a constant of the motion. But what happens if this poisson bracket doesn't vanish everywhere, but only on a subset of the phase space? This subset could be for example the one i get from the equation {f,H}=0

Thanks!
 
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  • #2
Welcome giova7_89,
If the locus of {f,H}=0 contains a trajectory, f will indeed be constant for that trajectory. If one considers the analytic continuation, one must say that such local constants have little physical significance.
 

FAQ: Can partial vanishing of Poisson bracket determine local constants of motion?

What are local constants of motion?

Local constants of motion are physical quantities that are conserved within a small region or locality. They are derived from the laws of physics and are independent of time, meaning they remain the same over time within a specific region.

How are local constants of motion different from global constants of motion?

Local constants of motion are specific to a small region, while global constants of motion apply to an entire system. Local constants of motion can vary from one region to another, but global constants of motion are constant throughout the system.

What is the significance of local constants of motion?

Local constants of motion play a crucial role in understanding and predicting the behavior of a system. They provide information about the dynamics and stability of a system, and can also be used to simplify complex systems by reducing the number of variables.

How are local constants of motion determined?

Local constants of motion are determined through mathematical equations and analysis of the laws of physics governing a system. They are often derived from the principles of energy conservation, momentum conservation, and angular momentum conservation.

Can local constants of motion change?

Local constants of motion are usually considered to be constant, but they can change in certain situations. For example, if external forces are applied to a system, the local constants of motion may change. Additionally, local constants of motion may change if the laws of physics governing the system are altered.

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