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
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.23$: The Hamilton-Jacobi Equation. Jacobi's Theorem