Variational Principles in Mechanics

Abstract

Differential variational principles in mechanics for mechanical systems with a finite number of degrees of freedom under constraints are obtained from the corresponding scalar motion equations of these systems, as written for the tangent space to the manifold of all positions of the system which it may occupy at a given time. The concept of a virtual (possible) displacement of a system under holonomic constraints is introduced to formulate the d’Alembert–Lagrange principle, while for the derivation of the Suslov–Jourdain principle we need the concept of the virtual velocity of a mechanical system subject to nonholonomic first-order constraints. We shall discuss the Chetaev-type constraints and the relationship between the generalized d’Alembert–Lagrange and the Suslov–Jourdain principles. To formulate the Gauss principle, we introduce the concept of a virtual acceleration of a system due to linear second-order nonholonomic constraints. The differential variational principles obtained in this chapter are used to derive the principal forms of motion equations of constrained mechanical systems. The integral variational Hamilton-Ostrogradskii and Lagrange principles, which reflect the extremal properties of the curves of motion under potential forces, are derived from the Hamilton principle of variable action. From this principle we shall also derive the Hamilton-Jacobi equation.