Hamilton’s principal function

The contours along which Hamilton s principal function S remains constant describe moving surfaces in Euclidean space. The gradient VS at each point of a moving surface is orthogonal to the surface, and the particle trajectories associated with S are given by the solutions to... 

The expression for the energy as a function of momenta and coordinates is called Hamilton s principal function or the classical Hamiltonian, and is denoted by Jf . 

The principal function S is the same quantity as the one defined by Hamilton s principle as in equation (2). It has the advantage of dealing with the scalars T and V only, whereas transformations of force components using Newton s laws deal with more complicated vector quantities. 

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