In Chapter 5 we considered objects that consist of a mass points with fixed relative positions, like a flying and spinning ping-pong ball. Rather than providing a description of each individual mass element, we established equations of motion for their center of mass and the orientation of the body in space. From the perspective of theoretical mechanics the fixing of relative positions is a constraint to their motion, just as the ropes of a swing enforces a motion on a one-dimensional circular track, rather than in two dimensions. The deflection angle θ of the pendulum, and the center of mass and orientation of the ball are examples of generalized coordinates that automatically take into account the constraints.

In this chapter we discuss how to set up generalized coordinates and how to find the associated equations of motion. The discussion will be driven by examples. The examples will be derived from the realm of integrable dynamics. These are systems where conservation laws can be used to break down the dynamics into separate problems that can be interpreted as motion with a single degree of freedom.

At the end of the chapter you know why coins run away rolling on their edge, and how the speed of a steam engine was controlled by a mechanical device. Systems where the dynamics is not integrable will subsequently be addressed in Chapter 7.

• 6.1 Motivation and Outline: How to deal with constraint motion?
• 6.2 Lagrange formalism
• 6.3 Dynamics with one degree of freedom
• 6.4 Several degrees of freedom and conservation laws
• 6.5 Dynamics of 2-particle systems
• 6.6 Worked problems: spinning top and running wheel
• 6.7 Problems

The PDF file of the chapter is available here.

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