I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in $n$-dimensional space with n even, I am thinking in the phase space of a system with $f$-degrees of freedom that have dimensionality $2f,$ the aim is picking phase-space points that have the same total energy. Thanks in advance.


EDIT: OK, perhaps thinking in a phase space states with energy $E = T(p) +V(q),$ one can pick at random $q$ which energy is less or equal than $E,$ if the energy is less than $E$ the remaining momentum can be picked from the ball $T(p)=cte$. But this is not a satisfactory answer to the question I did. Is this approach correct? How to do it efficiently using a computer?



Sampling point from the surface of an n-dimensional ellipsoid with uniform distribution
If your ellipsoid is not too squashed, the method described in the first answer to [this MSE question](https://math.stackexchange.com/questions/973101/how-to-generate-points-uniformly-distributed-on-the-surface-of-an-ellipsoid) should work decently. If it *IS* very squashed, it won't, but an ellispoid with $a \gg b > c> \dots $ is essentially a product of a lower dimensional ellipsoid with an interval, so you can reduce the problem to a lower dimensional one (with some (slight) loss of uniformity.)


