Chaos in Mechanical Systems, Routh Procedure, Ignorable Coordinates and Symmetries - Noether's Theorem

We start with a brief discussion of chaotic dynamics in mechanical system and how it relates to the number of constants of motion in a system, then spend the rest of the time discussing the general Routhian method, and its relationship to physical symmetries of a system. For example, the spherical pendulum and double pendulum are both 2 degrees of freedom, with a 4-dimensional ODE phase space, but the double pendulum is chaotic and the spherical pendulum is not..
We discuss the general Routhian procedure for n degrees of freedom with n-k of them 'ignorable' (also called 'cyclic variables'). This is a method of effectively reducing the number of degrees of freedom from n to k. Each ignorable coordinate is related to a conserved 'generalized momentum'. .
Routh-Jacobi constant: Under certain circumstances, there is an energy-like conserved quantity, which we might call the Routh-Jacobi integral or constant of motion.
We discuss the connection between ignorable coordinates, constants of motion and physical symmetries of a system (related to Noether's theorem)..
We work through several examples, starting with a point mass in a gravity field..
Example: cart-pendulum system.
Example: 2-mass baton or dumbbell system sliding along the ground right after it released from the wall (e.g., a ladder sliding off a wall)..
Example: spinning top.
Word of caution: sometimes the choice of generalized coordinates for describing your system conceals the existence of symmetries. Orbital motion in 2D is given as an example..
We end with the tantalizing possibility that one can find coordinates for a system in which *all* coordinates are ignorable, which is related to Hamiltonian mechanics, action-angle variables, and the Hamilton-Jacobi approach, which is described starting here:.