In the previous chapter we addressed principles based on the conservation of vector quantities, specifically translational and angular momentum. In this chapter we will address principles rooted in the variation of scalar quantities like work and energy. We will begin with d'Alembert's Principle of Virtual Work, which is an extension of Bernoulli's static principle to dynamics. The principle is based on the notion of virtual displacement. We will refer to d'Alembert's Principle as a zeroth-order variational principle to denote that it is based on the variation of the zeroth-order derivative of displacement. This is in contrast to higher-order variations related to velocity and acceleration, which will be discussed in subsequent chapters.

Virtual Displacements

Virtual displacements refer to all displacements of a system that satisfy the scleronomic constraints of the system. Scleronomic constraints refer to constraints that are not explicitly dependent on time, as opposed to rheonomic constraints, which are explicitly dependent on time. In the case of virtual displacements, time is frozen or stationary.

D'Alembert's Principle of Virtual Work

PRINCIPLE 5.1 The virtual work of a system is stationary. That is,

δW = 0.

Additionally, the constraints of the system perform no virtual work:

δWc = 0.

This is known as d'Alembert's Principle.

It is noted that while d'Alembert'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 d'Alembert'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

D'Alembert's Principle for a single point mass with a discrete set of nf external forces, ﹛ f1, …, fnf ﹜, acting on it is expressed as

where δr represents the displacement variations. During these variations time is stationary. That is, δt = 0.