Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

Hamilton’s variation principle can be shown to correspond to Fermat’s Principle for a wave propagation in configuration space (q-space), and the Hamilton-Jacobi equation expresses Huygens’ Principle for this wave propagation. Unfortunately this powerful and momentous conception of Hamilton, is deprived, in most modern reproductions, of its beautiful raiment as a superfluous accessory, in favour of a more colourless representation of the analytical correspondence. E. Schrödinger, Quantisation as a Problem of eigenvalues (Part II), Annalen der Physik, 1926