# PHY6200 W07

## Final Review

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.

## Review of Material from First Half of Semester for Final Exam

### Mathematical Techniques

1. Euler angles
2. Eigenvalue problem (for coupled oscillators)
3. Solid angle
4. Four-vectors: addition/subtraction, multiplication by scalar, scalar dot product, derivation
5. Relativistic invariance of scalars

### Physics Topics

1. Lagrange equations with multiple coordinates
2. Euler angles
3. Writing Lagrangian in terms of Euler angles
1. Relation between body and space coordinates
2. Motion of a spinning top
3. Nutation
4. Coupled oscillators
5. Normal frequencies and normal modes
6. Weak coupling limit
7. Handling cases with many coupled oscillators
8. Collisions
1. definitions of impact parameter, (differential) cross section, and scattering angle
2. solid angle
9. Relativity
1. Lorentz transformation
2. time dilation, length contraction, velocity addition
3. four-vectors, space-time, and metric
4. invariant mass, proper time, four-velocity, and four-momentum
5. relativistic collisions problems, conservation of energy and momentum
10. Relativistic Dynamics: 3-force, modified Newton's second law
11. Relativistic Lagrangian

### Problem Sets 1 to 13 with Emphasis on 8 to 13

Review any questions on these problems.

## Review of Material from First Half of Semester for Final Exam

### Mathematical Techniques

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

### Physics Topics

1. Linearly accelerating reference frame, fictitous force -mA0
2. Rotating reference frame, fictitous forces
1. Centrifugal force -mΩ×(Ω×r)
2. Coriolis force 2mv×Ω
3. Transverse force
3. Relation between ω and v for pure rolling
4. Definitions of center of mass, and moment of inertia tensor
5. Center of mass and moment of inertia tensor of composite objects
6. Parallel axis theorem for the inertia tensor
7. Conservation of linear and angular momentum for a rigid body
8. General motion of a rigid body, 2 cases:
1. motion of the CM, and rotation about the CM
2. rotation about a fixed point
9. Fixed axis rotation: Γ = dL/dt and L = Iω
10. General rotations
1. moment of inertia tensor
2. principal axis theorem
3. finding principal axes, eigenvalue problem
4. Euler's equations: I will give you the equations, but you must know how to use them.
5. zero torque problems
6. objects with axial symmetry
11. Solving minimization / maximization problems with the Euler-Lagrange equation.
12. Lagrangian Mechanics
1. Determining the number of generalized coordinates
2. Determining the Lagrangian, L = T - U, for those generalized coordinates
3. Finding the motion with Lagrange's equation ∂L/∂qi - (d/dt)(∂L/∂qidot) = 0
4. Recognizing a conserved quantity in the Lagrangian