# Differences

This shows you the differences between two versions of the page.

Both sides previous revision Previous revision | |||

classical_hamiltonian_system [2015/08/16 15:29] nikolaj |
classical_hamiltonian_system [2015/08/16 15:30] (current) nikolaj |
||
---|---|---|---|

Line 11: | Line 11: | ||

=== Discussion === | === Discussion === | ||

The Hamiltonian function is related to the Lagrangian function via Legendre transformation. | 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 === | === Reference === |