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Heinrich Hertz was not the first to introduce non-holonomic constraints into mechanics, but his careful distinction between holonomic and non-holonomic constraints in his book Principles of Mechanics influenced the general history of that discipline. This chapter provides a brief history of non-holonomic constraints and Hertz's place in it. The immediate reaction to Hertz's book is first discussed, and a general overview of the very messy history of repeated independent mistakes, rejections, and rescues is then presented. Hertz derived various integral principles such as the principle of least action and Hamilton's principle for holonomic conservative systems, but concluded that these principles are invalid for non-holonomic systems. This had devastating effects on the energetic image because it meant that its fundamental law, Hamilton's principle, was, in fact, incorrect for a wide variety of mechanical systems. This was one of Hertz's major reasons for rejecting that image. The situation called for an immediate rescue operation. The operation was led by the mathematician Otto Hölder.

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

This chapter builds on the previous two chapters to tackle constrained systems, using Lagrangian mechanics and constrained variations. The first section deals with holonomic constraint equations using Lagrange multipliers; these can be used to reduce the number of coordinates until a linearly independent minimal set is obtained that describes a constraint surface within configuration space, so that Lagrange equations can be set up and solved. Motion is understood to be confined to a constraint submanifold. The variational formulation of non-holonomic constraints is then discussed to derive the vakonomic formulation. These erroneous equations are then compared to the central Lagrange equation, and the precise nature of the variations used in each formulation is investigated. The vakonomic equations are then presented in their Suslov form (Suslov–vakonomic form) in an attempt to reconcile the two approaches. In addition, the structure of biological membranes is framed as a constrained optimisation problem.

Chapter 2 provides a comprehensive review of the classical mechanics required when building vehicle models. Both vector-based methods and the variational approach to classical mechanics are reviewed, and efforts to highlight the links between the two approaches have been made. A wide range of illustrative examples with a particular focus on non-holonomic systems are studied, including Chaplygin’s sleigh, rolling balls, and rolling discs. Equilibria and stability, and the connection between timereversal symmetry and dissipation, are also briefly discussed.