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# 6. Integrable Dynamics

Marguerite Martyn, 1914

Figure 6.1: The point-particle idealization of a girl on a swing is the mathematical pendulum of Figures 1.2 and 1.3.

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.

The PDF file of the chapter is available here.

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