Classical Dynamics

by Jorge V. José and Eugene Saletan

Chapter Eight:

Introduction

1. rigidity and kinematics: definition, the angular velocity vector omega
2. kinetic energy and angular momentum
3. dynamics: space and body systems, dynamical equations, example: the gyrocompass, motion of the angular momentum J, fixed points and stability, the Poinsot construction

The Lagrangian and Hamiltonian formulations

1. the configuration manifold Q_R: inertial, space, and body systems, the dimension of Q_R, the structure of Q_R
2. the Lagrangian: kinetic energy, the constraints
3. the Euler-Langrange equations: derivation, the angular velocity matrix omega
4. the Hamiltonian formalism
5. equivalence to Euler's equations: antisymmetric matrix-vector correspondence, the torque, the angular velocity pseudovector and kinematics, transformations of velocities, Hamilton's canonical equations
6. discussion

Euler angles and spinning tops

1. Euler angles: definition, R in terms of the Euler angles, angular velocities, discussion
2. geometric phase for a rigid body
3. spinning tops: the Lagrangian and Hamiltonian, the motion of the top, nutation and precession, quadratic potential; the Neumann problem

Cayley-Klein parameters

1. 2x2 matrix representations of 3-vectors and rotations: 3-vectors, rotations
2. the Pauli matrices and CK parameters: definitions, finding R_U , axis and angle in terms of the CK parameters
3. relation between SU(2) and SO(3)

Problems

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Contact:
CIRCS, Dana Research Center,
Physics Department ,
Northeastern University,
Boston, MA 02115