The exam will cover chapters 6, 7, 9 to 11, 14, and 15, with particular emphasis on the material covered since the midterm exam, the last sections of chapters 7 and 10, 11, 14, and 15.

- Euler angles
- Eigenvalue problem (for coupled oscillators)
- Four-vectors: addition/subtraction, multiplication by scalar, scalar dot product, derivation
- Relativistic invariance of scalars

- Lagrange equations with multiple coordinates
- Euler angles
- Writing Lagrangian in terms of Euler angles
- Relation between body and space coordinates
- Motion of a spinning top
- Nutation
- Coupled oscillators
- Normal frequencies and normal modes
- Weak coupling limit
- Handling cases with many coupled oscillators
- Relativity
- Lorentz transformation
- time dilation, length contraction, velocity addition
- four-vectors, space-time, and metric
- invariant mass, proper time, four-velocity, and four-momentum
- relativistic collisions problems, conservation of energy and momentum
- Relativistic Dynamics: 3-force, modified Newton's second law
- Relativistic Lagrangian

Review any questions on these problems.

- angular velocity vector
- addition of angular velocities
- Rotating Coordinate Frames
- derivative of a vector: (d
**u**/dt)_{non-rotating}= (d**u**/dt)_{rotating}+**Ω**×**u** - Vector triple product, BAC-CAB rule
- Linear algebra
- Matrix representation of vectors and tensors
- Matrix multiplication
- Determinant of a matrix
- Eigenvalue problem: finding eigenvalues and eigenvectors
- Calculus of variations: the Euler-Lagrange equation.
- square of path length, ds², in cartesian, cylindrical, and spherical coordinates.
- shortcut relation: v² = ds²/dt²

- Linearly accelerating reference frame, fictitous force -m
**A**_{0} - Rotating reference frame, fictitous forces
- Centrifugal force -m
**Ω**×(**Ω**×**r**) - Coriolis force 2m
**v**×**Ω** - Transverse force
- Relation between ω and v for pure rolling
- Definitions of center of mass, and moment of inertia tensor
- Center of mass and moment of inertia tensor of composite objects
- Parallel axis theorem for the inertia tensor
- Conservation of linear and angular momentum for a rigid body
- General motion of a rigid body, 2 cases:
- motion of the CM, and rotation about the CM
- rotation about a fixed point
- Fixed axis rotation:
**Γ**= d**L**/dt and**L**=**I****ω** - General rotations
- moment of inertia tensor
- principal axis theorem
- finding principal axes, eigenvalue problem
- Euler's equations: I will give you the equations, but you must know how to use them.
- zero torque problems
- objects with axial symmetry
- Solving minimization / maximization problems with the Euler-Lagrange equation.
- Lagrangian Mechanics
- Determining the number of generalized coordinates
- Determining the Lagrangian, L = T - U, for those generalized coordinates
- Finding the motion with Lagrange's equation ∂L/∂q
_{i}- (d/dt)(∂L/∂q_{i}dot) = 0 - Recognizing a conserved quantity in the Lagrangian