conservation laws - Separability of Hamilton Jacobi Equation

When we talk about integrability of classical systems in terms of Hamiltonian or Lagrangian 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?

Answer

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.