The Hamilton-Jacobi equation and its complementary form

The Hamilton-Jacobi equation is briefly reviewed and a complementary form of this equation is derived. Both equations involve the Hamiltonian function but, whereas in the Hamilton-Jacobi equation it is the kinetic energy of the Hamiltonian that is expressed in terms of the derivatives of the generating function, in the complementary Hamilton-Jacobi equation it is the potential energy of the Hamiltonian that is expressed in terms of the derivatives of the generating function. Three elementary problems are analyzed to illustrate the underlying theory.