Algebraic Equivalence of the Lagrangian and of the Hamiltonian Forms of the Equations of Motion

Abstract

The problem of deriving the Hamiltonian form of the classical equations of motion from the Lagrangian form, and vice versa, is treated algebraically utilizing the explicit dependence of the kinetic energy on the generalized velocities. A method of constructing the Hamiltonian function and the canonical equations of motion from the dependence of the kinetic energy on the generalized velocities is given. The relationship between the Hamiltonian of a system and the energy of a system is obtained without the use of Euler's theorem, and is given for the general case, that is for the case in which the equations transforming from the Cartesian coordinates to the generalized coordinates explicitly involve the time. Some identities involving second partials of the Hamiltonian function and the Lagrangian function are obtained.