Poisson Brackets of Harmonic Oscillator

Theorem
Let $P$ be a classical harmonic oscillator.

Let the real-valued function $\map x t$ be the position of $P$, where $t$ is time.

Then $P$ has the following Poisson brackets.


 * $\sqbrk {x, p} = 1$


 * $\displaystyle \sqbrk {x, H} = \frac p m$


 * $\sqbrk {p, H} = - k x$

Theorem
The standard Lagrangian of $P$ is:


 * $L = \frac 1 2 \paren{m {\dot x}^2 - k x^2}$.

The canonical momentum is:


 * $p = \dfrac {\partial L}{\partial \dot x} = m \dot x$

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


 * $\displaystyle H = \frac {p^2}{2m} + \frac k 2 x^2$

Then:

$\displaystyle \sqbrk {x, p} = \dfrac{\partial x}{\partial x} \dfrac{\partial p}{\partial p} - \dfrac{\partial p}{\partial x} \dfrac{\partial x}{\partial p} = 1$

$\displaystyle \sqbrk {x, H} = \dfrac{\partial x}{\partial x} \dfrac{\partial \paren{\frac {p^2} {2m} +\frac {k x^2} 2} }{\partial p} -\dfrac{\partial \paren{\frac {p^2} {2m} +\frac {k x^2} 2} }{\partial x} \dfrac{\partial x}{\partial p} = \frac p m$

$\displaystyle \sqbrk {p, H} = \dfrac{\partial p}{\partial x} \dfrac{\partial \paren{\frac {p^2} {2m} +\frac {k x^2} 2} }{\partial p} -\dfrac{\partial \paren{\frac {p^2} {2m} +\frac {k x^2} 2} }{\partial x} \dfrac{\partial p}{\partial p} = - k x$