Analytical mechanics and field theory: derivation of equations from energy conservation

Abstract

Equations of motion in mechanics and field equations in field theory are conventionally derived using the least action principle. This paper presents a nonvariational derivation of Hamilton's and Lagrange's equations. The derivation starts by specifying the system energy as a function of generalized coordinates and velocities and then introduces generalized momenta in such a way that the energy remains unchanged under variations of any degree of freedom. This immediately leads to Hamilton's equations with an as yet undefined Hamiltonian. The explicit dependence of generalized momenta on the coordinates and velocities is determined by first finding the Lagrangian from the known energy function. We discuss electrodynamics as an illustrative example. The proposed approach provides new insight into the nature of canonical momenta and offers a way to find the Lagrangian from the known energy of the system.