Poisson Brackets of Classical Particle in Radial Potential on Plane

Theorem
Let $P$ be a classical particle embedded in a 2-dimensional Euclidean manifold.

Let the real-valued functions $\map r t$, $\map \theta t$ denote the position of $P$ in polar coordinates, where $t$ is time.

Suppose, the potential energy of $P$ depends only on $r$.

Then $P$ has the following Poisson brackets:

Proof
The standard Lagrangian of $P$ in polar coordinates is:


 * $L = \dfrac 1 2 m \paren { {\dot r}^2 + r^2 {\dot \theta}^2 } - \map U r$

The canonical momenta are:


 * $p_r = \dfrac {\partial L} {\partial \dot r} = m \dot r$


 * $p_\theta = \dfrac {\partial L} {\partial \dot \theta} = m r^2 \dot \theta$

The Hamiltonian associated to $L$ in canonical coordinates reads:


 * $H = \dfrac {p_r^2} {2 m} + \dfrac 1 2 \dfrac {p_\theta^2} {m r^2} + \map U r$

Then: