Variational Principles, Constraints, and Rotating Systems

Abstract

This chapter deals with two related topics: constrained Lagrangian (and Hamiltonian) systems and rotating systems. Constrained systems are illustrated by a particle constrained to move on a sphere. Such constraints that involve conditions on the configuration variables are called “holonomic.”¹ For rotating systems, one needs to distinguish systems that are viewed from rotating coordinate systems (passively rotating systems) and systems that themselves are rotated (actively rotating systems—such as a Foucault pendulum and weather systems rotating with the Earth). We begin with a more detailed look at variational principles, and then we turn to a version of the Lagrange multiplier theorem that will be useful for our analysis of constraints.