Hamilton’s Principle and Action Integrals

Here we show that for systems of particles subject to holonomic constraints and forces derivable from a generalized potential, Lagrange’s equations can also be obtained by finding the stationary values of a functional, called the “action.” By doing so, we convert the problem of finding the equations of motion to a problem in the calculus of variations. We also introduce the Hamiltonian for a system of particles, which can be obtained from the Lagrangian by performing a Legendre transformation. This process leads to Hamilton’s equations of motion, which are 1st-order differential equations for the generalized coordinates and momenta. The Hamiltonian formulation is particularly well-suited for illustrating the intimate connection between continuous symmetries of the system and conserved quantities.