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Classical Hamiltonian system
Set
| definiendum | $\langle \mathcal M, H\rangle \in \mathrm{it} $ |
| postulate | $ \mathcal M$ … smooth manifold |
| range | $ \Gamma_{\mathcal M}\equiv \mathcal M\times T^*\mathcal M $ |
| postulate | $ H:\Gamma_{\mathcal M} \times \mathbb R \to \mathbb R $ |
| postulate | $ H $ … differentiable in $\Gamma_{\mathcal M}$ |
Discussion
The Hamiltonian function is related to the Lagrangian function via Legendre transformation.
Volume in statistical physics
It's worth noting that $\hbar$ translates energy to frequency (or time) and $c$ further translates time to length. This way one can write down models involving a volume parameter $V$, defining a characteristic energy $\frac{(\hbar c)^3}{V}$. This may then e.g. be embedded via (unitless!) expressions as complicated as
$\frac{V}{(\hbar c)^3}\int {\mathrm d}E\, f(\frac{V}{(\hbar c)^3}E)$.
Such a volume $V$ may actually also not just be a parameter of the theory, but fixed by the space ${\mathcal M}$ - e.g. the integral over the space, if such a number exists.
This e.g. happens in Classical density of states and is how $V$ enters $U=\langle{H}\rangle$ and eventually $p := -\frac{\partial U}{\partial V} = \frac{N}{V}\cdot k_B T$.
Reference
Wikipedia: Hamiltonian mechanics,
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