Poisson Brackets of Harmonic Oscillator

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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:

\(\ds \sqbrk {x, p}\) \(=\) \(\ds 1\)
\(\ds \sqbrk {x, H}\) \(=\) \(\ds \dfrac p m\)
\(\ds \sqbrk {p, H}\) \(=\) \(\ds -k x\)


Proof

The standard Lagrangian of $P$ is:

$L = \dfrac 1 2 \paren {m {\dot x}^2 - k x^2}$

The canonical momentum is:

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

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

$H = \dfrac {p^2} {2 m} + \dfrac k 2 x^2$


Then:

\(\ds \sqbrk {x, p}\) \(=\) \(\ds \dfrac {\partial x} {\partial x} \dfrac {\partial p} {\partial p} - \dfrac {\partial p} {\partial x} \dfrac {\partial x} {\partial p}\) \(\ds = 1\)
\(\ds \sqbrk {x, H}\) \(=\) \(\ds \dfrac {\partial x} {\partial x} \dfrac {\map \partial {\frac {p^2} {2 m} + \frac {k x^2} 2} } {\partial p} - \dfrac {\map \partial {\frac {p^2} {2 m} + \frac {k x^2} 2} } {\partial x} \dfrac {\partial x} {\partial p}\) \(\ds = \dfrac p m\)
\(\ds \sqbrk {p, H}\) \(=\) \(\ds \dfrac {\partial p} {\partial x} \dfrac {\map \partial {\frac {p^2} {2 m} + \frac {k x^2} 2} } {\partial p} - \dfrac {\map \partial {\frac {p^2} {2 m} + \frac {k x^2} 2} } {\partial x} \dfrac {\partial p} {\partial p}\) \(\ds = -k x\)

$\blacksquare$


Sources

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