First-Order Variational Principles

Abstract

In the previous chapter we addressed zeroth-order variational principles rooted in the fundamental principle of d'Alembert. In this chapter we will focus on the first-order variation of displacement. We will begin with Jourdain's Principle of Virtual Power (Jourdain 1909). The principle is based on the notion of virtual velocity. The derivation of the equations of motion for particles and rigid bodies using Jourdain's Principle closely parallels the derivation of the equations of motion using d'Alembert's Principle in the previous chapter. While Jourdain's Principle is analogous to d'Alembert's Principle, it can be seen as an independent principle of analytical dynamics. Therefore, the material in the previous chapter is not a prerequisite to this chapter, and this chapter will be treated in a stand-alone manner. Virtual Velocities Virtual velocities refer to all velocities of a system that satisfy the scleronomic constraints of the system. In the case of virtual velocities, time and position are frozen or stationary. Jourdain's Principle of Virtual Power PRINCIPLE 6.1 The virtual power of a system is stationary. That is, δP = 0. Additionally, the constraints of the system generate no virtual power, δP c = 0. This is known as Jourdain's Principle . As with d'Alembert's Principle, it is noted that while Jourdain's Principle can be seen as providing an alternate statement of Newton's second law, for interacting bodies, a law of action and reaction (Newton's third law) is still needed. Therefore, when we use Jourdain's Principle to derive the equations of motion for systems of particles/bodies, we will invoke the law of action and reaction. A Single Particle Jourdain's Principle for a single point mass with a discrete set of n f external forces, ﹛ f 1 , …, f nf ﹜, acting on it is expressed as where δ v represents the velocity variations. During these variations, time and position are stationary. That is, δt = 0 and δ r = 0.