SICM Exercise 1.01

Solution to exercise 1.1 of Structure and Interpretation of Classical Mechanics by Gerald Jay Sussman and Jack Wisdom.

Degrees of freedom

🛈 Note

For each of the mechanical systems described below, give the number of degrees of freedom of the configuration space.

1. Three juggling pins.
2. A spherical pendulum, consisting of a point mass (the pendulum bob) hanging from a rigid massless rod attached to a fixed support point. The pendulum bob may move in any direction subject to the constraint imposed by the rigid rod. The point mass is subject to the uniform force gravity.
3. A spherical double pendulum, consisting of one point mass hanging from a rigid massless rod attached to a second point mass hanging from a second massless rod attached to a fixed support point . The point masses are subject to the uniform force of gravity.
4. A point mass sliding without friction on a rigid curved wire.
5. A top consisting of a rigid axisymmetric body with one point on the symmetry axis of the body attached to a fixed support , subject to a uniform gravitational force.
6. The same as e, but not axisymmetric.

1. Six parameters are needed to describe the position of each juggling pin, so 18 parameters are needed for 3 juggling pins.
2. Two parameters are needed to describe the orientation of the pendulum.
3. Four parameters are needed. Two for the system described in the previous task and two parameters extra to specify the orientation of the second pendulum with respect to the first pendulum.
4. One parameter, which describes the distance from the beginning of the wire.
5. Two parameters are needed to describe the orientation of the top. Since the top is axisymmetric the rotation around the axis can be ignored.
6. Three parameters. Two from the previous task plus an extra parameter for the rotation around the axis.