# Variational Principles, Constraints, and Rotating Systems

• Jerrold E. Marsden
• Tudor S. Ratiu
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 17)

## 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.

## Keywords

Vector Field Variational Principle Rotating System Symplectic Form Geometric Phase
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