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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 …
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Classroom Contents
Chaos in Mechanical Systems, Routh Procedure, Ignorable Coordinates and Symmetries - Noether's Theorem
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- 1 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 …
- 2 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 fr…
- 3 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
- 4 We discuss the connection between ignorable coordinates, constants of motion and physical symmetries of a system (related to Noether's theorem).
- 5 We work through several examples, starting with a point mass in a gravity field.
- 6 Example: cart-pendulum system
- 7 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).
- 8 Example: spinning top
- 9 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.
- 10 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 th…
