Jacobi-Poisson theorem | Poisson’s second theorem | Classical mechanics

Jacobi-Poisson theorem

Poisson’s second theorem

If u and v are any two constants of motion of any given system, then their Poisson bracket [u, v] are also a constant of motion.
If u is a constants of motion, then [u, H] + ∂u/∂t = 0 ⇒ [u, H] = - ∂u/∂t.
Given u and v are constant of motion

$\therefore \left [H,u \right ]=\frac{\partial u}{\partial t}and\left [H,v \right ]=\frac{\partial v}{\partial t}$

We have to prove [u, v] is also a constant of motion

$\therefore \left [H,\left [ u,v \right ] \right ]=\frac{\partial }{\partial t}[u,v]$

Proof

By Jacobi identity

This is mathematical form of Jacobi-Poisson’s theorem or Poisson's second theorem.
According to statement of Jacobi-Poisson theorem if u and v are any two constants of motion of any given system, then their Poisson bracket [u, v] are also a constant of motion.

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