This is the draft lecture sequence from 1999.
The actual sequence used in 1999 and 2000 can be seen by looking at the
- The energy method.
-
Lagrangian and
Hamiltonian dynamics (introduced without using calculus of variations).
Illustrate with numerous examples.
- Examples of the energy method.
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Compound pendulum. Rolling hoop with mass on the perimeter.
Slipping ladders. Moment of inertia. Conservation of angular momentum.
Other conserved quantities: energy, phase space volume.
Use perturbation methods to estimate error of pendulum clock.
Show Huygen's method for making isochronous pendulum and relate
to dynamics of the rolling hoop.
Particles connected to each other by strings and springs.
This leads to the central topic:
- Normal modes.
-
Modes of molecules.
Stability analysis using normal modes.
Square matrices as linear operators and in quadratic forms.
Eigenvectors. Transformation of linear operators and of
quadratic forms.
Perturbation theory: coupling of normal modes.
Weak coupling of nonlinear oscillators.
Taking the continuum limit of systems of masses and springs,
we come to the wave equation and elasticity.
- Elasticity.
- Definitions of strain and stress as tensors.
Young's modulus, Poisson ratio, Shear and bulk modulus.
Perturbation of wave equation on a wire by the stiffness of the wire,
for example, harmonics of piano strings.
Microscopic view of elastic behaviour. Rebound
of elastic ball or rod from hard surface.
- Central forces.
-
Kepler's laws.
Planetary orbits.
Perturbations of circular orbits.
Scattering. Contrast between cross sections of
hard spheres and inverse square potentials. Predicting cross sections
using dimensional analysis.
Orbits resulting from other force laws. General relativity
as a perturbation.
Gravitational slingshot.
Tides.
- Rotating frames and fictitious forces.
-
Centrifugal and Coriolis forces. Central force problems
re-expressed as one-dimensional problems.
Nearly circular orbits revisited.
The three-body system, the Lagrange points and Trojan asteroids.
- Rigid bodies, especially the gyroscope.
-
Precession of gyroscope subjected to a torque. Examples:
the earth; the earth-moon system; NMR; levitron.
Free precession of a rigid body will be mentioned briefly.
Euler's equations and tennis racquet theorem if time permits.
- Interesting dynamical systems.
-
Possibilities include:
The driven inverted pendulum.
Different ways of driving a playground swing.
Harrison's clocks. (How to make a clock immune to linear acceleration,
centrifugal forces, and temperature variations. How to make a driving
mechanism that does not affect the period of the oscillator.)
Chaotic systems.
The relationship between a periodically driven dynamical system
and static equilibrium points of masses and springs in a periodic
potential.