Things you should know (physics 321 final)
- Relativity:
- Physics 221/222 level relativity – spacetime Pythagorean theorem, slopes of world lines, lines of simultaneity, etc.
- Lorentz transformation of 4-vector components (positions, 4-momentum, 4-potential for electromagnetism, etc.)
- Relativistic form of Newton's second law.
- 4-velocity, 4-acceleration, relation to Minkowski force.
- Standard form of Lorentz force law. (Nothing more about electromagnetism in this chapter.)
- Planetary motion:
- Derivation of Kepler's laws.
- Geometry of ellipses and hyperbolas in context of Kepler's laws.
- Principle of virtual work:
- Definition and use of principle of virtual work. (No non-holonomic constraints.)
- Determination of degrees of freedom and constraints in physical systems.
- D'Alembert's principle:
- Extension of principle of virtual work to inertial forces – D'Alembert's principle.
- Application of D'Alembert's principle to simple physical systems.
- Moment of inertia for rotations about a fixed axis – parallel axis theorem.
- Use of non-Cartesian coordinates in D'Alembert's principle, e.g., angles.
- Motion of solid bodies in two dimensions.
- Static and dynamic friction in the context of D'Alembert's principle.
- Lagrange's equations:
- Understanding of generalized coordinates and use to reduce degrees of freedom due to constraints.
- Set up the Lagrangian for simple physical systems.
- Derive the Lagrange equations from the Lagrangian.
- Compute generalized momenta and determine which such momenta are conserved.
- Elimination of generalized velocities from governing equations that are associated with conserved generalized momenta.
- Incorporation of time-dependence and non-conservative forces.
- Oscillations:
- Understand physics 221/242 treatment of harmonic oscillatiors.
- Coupled oscillators.
- Linearization of governing equations to test for stability/instability of steady solutions.
- Solid bodies in 3-D:
- Inertia tensor – how to compute inertia tensor, meaning and computation of eigenvalues and principal axes, how to transform components from one reference frame to another.
- Angular momentum and kinetic energy of 3-D solid bodies.
- Rotation of solid bodies:
- Inertial forces in rotating reference frames (Coriolis and centrifugal).
- Application of Euler equations; axisymmetric and non-axisymmetric bodies.
- Tops using Lagrangian method. (This is pretty tough, so don't worry too much about this topic for the final.)
- Hamilton's mechanics:
- Quantum mechanical origin of Hamilton's principle.
- Precise statement of Hamilton's principle and derivation of Lagrange's equations from it.
- Obtain Hamiltonian from Lagrangian.
- Use of Hamilton's equations for simple problems.
- Generalization of Lagrangian to include the velocity-dependent forces of electromagnetism.