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When we talk about integrability of classical systems in terms of Hamiltonian mechanics, it's all to do with counting independent conserved quantities.

Then when we move to the Hamilton-Jacobi formalism, suddenly everything is about separability of the Hamilton-Jacobi equation and Staeckel conditions. How do these two concepts relate to one-another? Does the existence of a certain number of conserved quantities imply separability of the Hamilton-Jacobi equation in some coordinate system?

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    $\begingroup$ see en.wikipedia.org/wiki/Action-angle_coordinates. The rough idea is that every conserved quantity can be assumed as a new coordinate. The new Hamiltonian is then independent on such coordinate, since it is constructed to be cyclic. If you have as many independent conserved quantities as the degrees of freedom you can replace all the variables in the way described above, so that the Hamiltonian depends on the time derivative of the conjugate variables alone. $\endgroup$
    – Phoenix87
    Commented Jan 30, 2015 at 14:57
  • $\begingroup$ Related: physics.stackexchange.com/q/291511/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Mar 2, 2018 at 18:26

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The answer to your question is yes, the existence of $n$ conserved quantities with $n$ degrees of freedom implies separability of HJ.

The massless HJ equation is $$g^{MN}\frac{\partial S}{\partial x^M}\frac{\partial S}{\partial x^N}=E.$$ It separates if there exists a new set of coordinates $Y^M$ such that $$ S(Y_1,...,Y_n)=\sum_{i=1}^n S_i(Y_i),$$ which implies existence of $n$ conserved quantities, because each term in the HJ equation depends on its own variable. The same procedure is used when we solve PDE. For example in $2D$ $$S=S_x(x)+S_y(y), \quad f(x)(\partial_x S_x(x))^2+f(y)(\partial_y S_y(y))^2=E.$$ The latter means that both terms in LHS are constants separately. So we have two independent conserved quantities.

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  • $\begingroup$ Ok great, together with Pheonix's answer that shows that if you have enough conserved quantities then HJE is seperable, and if HJE is seperable that implies our conserved quantities exist. I don't recognise your notation though, is your g the metric tensor? So your HJE is for a classical scalar field or something? I've only seen the analysis in classical mechanics so far $\endgroup$
    – Jojo
    Commented Mar 4, 2015 at 18:39
  • $\begingroup$ @Joe Yes, $g^{MN}$ is the metric tensor, and the equation is the most general HJE in a gravitational field. For more you can look here en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation $\endgroup$
    – Yuri
    Commented Mar 4, 2015 at 19:53

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