In a system consisting of a single particle in a potential, the classical action ∫L dt is the number of phase waves that pass through the moving particle, as the particle moves from its initial to its final position. Thus the Lagrangian can be cast into the form L=p(vg−vp), where vg and vp are the group and phase velocities and p is the momentum.

Good treatments are found in Edwin C. Kemble, The Fundamental Principles of Quantum Mechanics (McGraw–Hill, New York, 1937), pp. 35–51;
Wolfgang Yourgrau and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory (Dover, New York, 1968), pp. 45–64, 116–126.
As with all treatments of phase waves for the nonrelativistic Lagrangian, this is somewhat fortuitous, because the true frequency of the de Broglie waves differs from H/ℏ by the enormous contribution mc2/ℏ due to the rest mass. However, when the rest mass energy is included, the result is only an effective constant increment of mc2 to the potential energy U, with no consequences for the validity of the theorem.
For example, Jerry B. Marion and Stephen T. Thornton, Classical Dynamics of Particles and Systems (Harcourt Brace Jovanovich, San Diego, 1988), 3rd ed., p. 539;
Herbert Goldstein, Classical Mechanics (Addison–Wesley, Reading, MA, 1980), 2nd ed., p. 321. In contrast, a very clear and complete derivation is given by Wolfgang Rindler, Introduction to Special Relativity (Clarendon, Oxford, 1991), pp. 93–96.
Max Born and Emil Wolf, Principles of Optics (Pergamon, Oxford, 1980), 6th ed., p. 18.
The fact that k points in the direction of p, even when a vector potential A is present, is, of course, due to Louis de Broglie. See Ondes et Mouvements (Gauthier-Villars, Paris, 1926), pp. 32–35.
For a more detailed justification, see
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Matter waves in a gravitational field: An index of refraction for massive particles in general relativity
Am. J. Phys.
We are, of course, free to choose the coordinates as we please. Choosing the pi as coordinates is one standard way, among several, of passing from Euler–Lagrange to Hamiltonian equations of motion. For a detailed discussion, including the fact that this procedure has no consequences for the underlying variational principle, see Goldstein (Ref. 3), pp. 362–365.
This is usually a convenient way to regard vp. For example, for gravity surface waves on water, the phase velocity depends upon the wave number k and the depth d of the water, vp=[(g/k)tanh(kd)]1/2. See for example, William C. Elmore and Mark A. Heald, Physics of Waves (McGraw–Hill, New York, 1969), p. 187.
For an especially clear and simple proof, see T. T. Taylor, Mechanics: Classical and Quantum (Pergamon, Oxford, 1976), pp. 78–82.
An explicit calculation, showing that the theorem holds in an arbitrary metric, has been made by Paul M. Alsing (personal communication).
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