ABSTRACT
A well-known formulation of classical mechanics of system of particles is based on solving the Hamilton-Jacobi equation
∂S
∂t + H
( qi ,
∂S
∂qi , t
) = 0, (19.1)
where S(qi , αi , t) is Hamilton’s principal function, H is the Hamiltonian, qi are the configuration space variables, αi are constants of integration, and t denotes time. If the Hamiltonian is independent of t and is a constant of motion equal to the total energy E, then S and Hamilton’s characteristic function W are related by
S = W − Et . (19.2) Since W is independent of time, the surfaces with W = constant have fixed locations in configuration space. However, the S = constant surfaces move in time and may be considered as wavefronts propagating in this space [1]. For a single particle one can show that the wave velocity at any point is given by
u = E|∇W | = E
p = E
mv . (19.3)
T&F Cat # K11224, Chapter 19, Page 302, 15-9-2010
This shows that the velocity of a point on this surface is inversely proportional to the spatial velocity of the particle. Also one can show that the trajectories of the particle must always be normal to the surfaces of constant S. The momentum of the particle is obtained as [1]
p = ∇W . (19.4) It is not clear whether this wave picture was known to classical physicists. However,
in hindsight, we may now observe that there exists a correspondence between classical mechanics and geometrical optics. That is, the particle trajectories orthogonal to surfaces of constant S in classical mechanics, as we have discussed above, are similar to light rays traveling orthogonal to Huygens’ wavefronts.
