Theorem: Poisson Brackets of Classical Particle in Radial Potential on Plane

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:
{{begin-eqn}}
{{eqn | l = \sqbrk {r, p_r} 
      | r = 1
}}
{{eqn | l = \sqbrk {\theta, p_\theta}
      | r = 1
}}
{{eqn | l = \sqbrk {r, H}
      | r = \dfrac {p_r} m
}}
{{eqn | l = \sqbrk {\theta, H} 
      | r = \dfrac {p_\theta} {m r^2}
}}
{{eqn | l = \sqbrk {p_r, H} 
      | r = -\dfrac {\partial U} {\partial r}
}}
{{eqn | l = \sqbrk {p_\theta, H} 
      | r = 0
}}
{{end-eqn}}

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:
{{begin-eqn}}
{{eqn | l = \sqbrk {r, p_r}
      | r = \paren {\dfrac {\partial r} {\partial r} \dfrac {\partial p_r} {\partial p_r} - \dfrac {\partial p_r} {\partial r} \dfrac {\partial r} {\partial p_r} } + \paren {\dfrac {\partial r} {\partial \theta} \dfrac {\partial p_r} {\partial p_\theta} - \dfrac {\partial p_r} {\partial \theta} \dfrac {\partial r} {\partial p_\theta} }
      | c = 
}}
{{eqn | r = 1
      | c = 
}}
{{eqn | l = \sqbrk {\theta, p_\theta}
      | r = \paren {\dfrac{\partial \theta} {\partial r} \dfrac {\partial p_\theta} {\partial p_r} - \dfrac {\partial p_\theta} {\partial r} \dfrac {\partial \theta} {\partial p_r} } + \paren {\dfrac {\partial \theta} {\partial \theta} \dfrac {\partial p_\theta} {\partial p_\theta} - \dfrac {\partial p_\theta} {\partial \theta} \dfrac {\partial \theta} {\partial p_\theta} }
      | c = 
}}
{{eqn | r = 1
      | c = 
}}
{{eqn | l = \sqbrk {r, H}
      | r = \paren {\dfrac {\partial r} {\partial r} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} { m r^2 } + \map U r} } {\partial p_r} - \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2}  + \map U r} } {\partial r} \dfrac {\partial r} {\partial p_r} } + \paren {\dfrac {\partial r} {\partial \theta} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial p_\theta} - \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2}  + \map U r} } {\partial \theta} \dfrac {\partial r} {\partial p_\theta} }
      | c = 
}}
{{eqn | r = \dfrac {p_r} m
      | c = 
}}
{{eqn | l = \sqbrk {p_r, H}
      | r = \paren {\dfrac {\partial p_r} {\partial r} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial p_r} - \dfrac {\partial \paren {\frac {p_r^2}{2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial r} \dfrac {\partial p_r} {\partial p_r} } + \paren {\dfrac {\partial p_r} {\partial \theta} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2}  + \map U r} } {\partial p_\theta} - \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2}  + \map U r} } {\partial \theta} \dfrac {\partial p_r} {\partial p_\theta} }
      | c = 
}}
{{eqn | r = -\dfrac {\partial U} {\partial r}
      | c = 
}}
{{eqn | l = \sqbrk {\theta, H}
      | r = \paren {\dfrac {\partial \theta} {\partial r} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial p_r} - \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial r} \dfrac {\partial \theta} {\partial p_r} } + \paren {\dfrac {\partial \theta} {\partial \theta} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial p_\theta} - \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial \theta} \dfrac {\partial \theta} {\partial p_\theta} }
      | c = 
}}
{{eqn | r = \dfrac {p_\theta} {m r^2}
      | c = 
}}
{{eqn | l = \sqbrk {p_\theta, H}
      | r = \paren {\dfrac {\partial p_\theta} {\partial r} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial p_r} - \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial r} \dfrac {\partial p_\theta} {\partial p_r} } + \paren {\dfrac {\partial p_\theta} {\partial \theta} \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial p_\theta} - \dfrac {\partial \paren {\frac {p_r^2} {2 m} + \frac 1 2 \frac {p_\theta^2} {m r^2} + \map U r} } {\partial \theta} \dfrac {\partial p_\theta} {\partial p_\theta} }
      | c = 
}}
{{eqn | r = 0
      | c = 
}}
{{end-eqn}}
{{qed}}