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:


 * $\sqbrk {r, p_r} = 1$


 * $\displaystyle \sqbrk {\theta, p_\theta} = 1$


 * $\displaystyle \sqbrk {r, H} = \frac {p_r} m$


 * $\displaystyle \sqbrk {\theta, H} = \frac {p_\theta} {m r^2}$


 * $\displaystyle \sqbrk {p_r, H} = - \dfrac {\partial U}{\partial r}$


 * $\sqbrk {p_\theta, H} = 0$

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


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

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:


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

Then:

$\displaystyle \sqbrk {r, p_r} = \paren {\dfrac{\partial r}{\partial r} \dfrac{\partial p_r}{\partial p_r} - \dfrac{\partial p_r}{\partial r} \dfrac{\partial r}{\partial p_r} } + \paren {\dfrac{\partial r}{\partial \theta} \dfrac{\partial p_r}{\partial p_\theta} - \dfrac{\partial p_r}{\partial \theta} \dfrac{\partial r}{\partial p_\theta} } = 1$

$\displaystyle \sqbrk {\theta, p_\theta} = \paren{ \dfrac{\partial \theta}{\partial r} \dfrac{\partial p_\theta}{\partial p_r} - \dfrac{\partial p_\theta}{\partial r} \dfrac{\partial \theta}{\partial p_r} } + \paren{ \dfrac{\partial \theta}{\partial \theta} \dfrac{\partial p_\theta}{\partial p_\theta} - \dfrac{\partial p_\theta}{\partial \theta} \dfrac{\partial \theta}{\partial p_\theta} } = 1$

$\displaystyle \sqbrk {r, H} = \paren {\dfrac{\partial r}{\partial r} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 } + \map U r} }{\partial p_r} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial r} \dfrac{\partial r}{\partial p_r} } + \paren {\dfrac{\partial r}{\partial \theta} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial p_\theta} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial \theta} \dfrac{\partial r}{\partial p_\theta} } = \frac {p_r} m$

$\displaystyle \sqbrk {p_r, H} = \paren {\dfrac{\partial p_r}{\partial r} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 } + \map U r} }{\partial p_r} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial r} \dfrac{\partial p_r}{\partial p_r} } + \paren {\dfrac{\partial p_r}{\partial \theta} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial p_\theta} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial \theta} \dfrac{\partial p_r}{\partial p_\theta} } = - \dfrac {\partial U}{\partial r}$

$\displaystyle \sqbrk {\theta, H} = \paren {\dfrac{\partial \theta}{\partial r} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 } + \map U r} }{\partial p_r} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial r} \dfrac{\partial \theta}{\partial p_r} } + \paren {\dfrac{\partial \theta}{\partial \theta} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial p_\theta} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial \theta} \dfrac{\partial \theta}{\partial p_\theta} } = \frac {p_\theta} {m r^2}$

$\displaystyle \sqbrk {p_\theta, H} = \paren{ \dfrac{\partial p_\theta}{\partial r} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 } + \map U r} }{\partial p_r} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial r} \dfrac{\partial p_\theta}{\partial p_r} } + \paren{ \dfrac{\partial p_\theta}{\partial \theta} \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial p_\theta} - \dfrac{\partial \paren{\frac {p_r^2}{2m} + \frac 1 2 \frac {p_\theta^2} { m r^2 }  + \map U r} }{\partial \theta} \dfrac{\partial p_\theta}{\partial p_\theta} } = 0$