Ex 1.1: Degrees of Freedom & 1.2: Generalized Coordinates
This exercise is designed to get you thinking about the idea of configuration space. One goal of classical mechanics is to predict the future of a system, based on a description of the current state of the system. So how to describe the system?
One way would be to track, for every instant of time, a 3-dimensional position for every particle in a system, along with velocities in each direction, for a total of 6 numbers - or dimensions - per particle.
That's fine for particles moving in straight lines through space, not affecting each other. But remember, this is all an accounting trick that we're using to represent reality, not reality itself. If there is some more convenient way to track the state of the system, we're free to use that as well, provided we can recover the original positions and velocities after evolving the system.
Maybe two particles are attached to each other, so their positions in space are the same, or offset by a tiny amount. Then it's enough to track:
the position of one of the particles (3 numbers)
the angle and distance of the offset (2 numbers)
What seemed like a system that required 6 numbers actually required 5.
Theo Jansen's incredible Strandbeesten are built out of copies of Jansen's Linkage. Each of these legs has 11 pipes that flex and bend, but only one degree of freedom.
The first exercise gives us some practice thinking about the redundancy in different physical systems.
For each of the mechanical systems described below, give the number of degrees of freedom of the configuration space. (SICM, ex1)
Exercise 1.2 asks about the "generalized coordinates" of each system. What are the actual numbers that we want to track for each system, if not the inline_formula not implemented positions of each of inline_formula not implemented particles?
For each of the systems in exercise 1.1, specify a system of generalized coordinates that can be used to describe the behavior of the system.
Three juggling pins
The system has 18 degrees of freedom. Each pin requires 3 coordinates to specify its center of mass, and 3 angles for each pin. If you assume that each pin is symmetric about its central axis, then it doesn't matter how far around the pin has rotated and you can make do with 15 degrees of freedom. 3 positions and 2 angles for each.
Spherical Pendulum
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.
This system has only 2 degrees of freedom. One for the latitude of the pendulum, and one for the longitude.
Spherical Double Pendulum
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.
A point mass sliding without friction on a rigid curved wire.
1 degree of freedom; the distance along the wire.
Axisymmetric Top
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.
This system seems to have 2 degrees of freedom, for the angles off of vertical. It's like a spherical pendulum, but upside down. What I find strange about this answer is that the top does have a rotation speed, which is a measure of how far the top has rotated in time. How can we track this velocity if we don't track the top's spin angle? This may be wrong.
Non-Axisymmetric Top
A non-symmetric top has 3 degrees of freedom. 2 from before, and an additional angle to measure how far around the top has rotated.