$$ V(E) = \int_{H\leq E} d\mu = \int_\Gamma d\mu\, \Theta\bigl(E-H(q,p)\bigr) . $$

To compute the volume within the equinergetic surface in the microcanonical ensamble, we use the formula above, where $\mu$ is an infinitesimal volume of the phase space. The only way I have learned to compute this, is when the hamiltonian is of the form, $H=\sum p^2/(2m)$, because here the integral over the momenta will be the volume of a sphere of $N$ dimension with radius $\sqrt{E}$, but what if the hamiltonian is a function of the positions, like the hamiltonian of a classical harmonic oscillator, how can I compute this integral?

  • 1
    $\begingroup$ I'm not quite clear about what conceptual problem is troubling you here. If you are just interested in classical harmonic oscillators, the problem is a straightforward extension of the one that you say you understand: the positional degrees of freedom are quadratic, so you end up with $N$ more dimensions, and the volume of an ellipsoid rather than a sphere, but a coordinate transformation will convert it into a sphere. If you are interested in the general case, then an analytical formula is probably not available. $\endgroup$
    – user197851
    Jan 25, 2019 at 18:53
  • $\begingroup$ PS please use MathJax (like LaTeX) to format mathematical expressions, not images. I've edited your question, if you are happy with the new formula, please delete the image. $\endgroup$
    – user197851
    Jan 25, 2019 at 18:59


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