PHY5210 W15

Final Review

The exam will cover chapters 6, 7, 9 to 11, and 13 to 16, with particular emphasis on the material covered since the midterm exam, chapters 11, and 13 to 16.

Review of Material after Second Midterm for Final Exam

Mathematical Techniques

  1. Eigenvalue problem (for coupled oscillators)
  2. Solid angle
  3. Four-vectors: addition/subtraction, multiplication by scalar, scalar dot product, derivation
  4. Relativistic invariance of scalars and transformatoin of 4--vectors
  5. Generalization of Lagrange's equations to the Lagrange denity
  6. Convective derivative

Physics Topics

  1. Coupled oscillators
    1. Normal frequencies and normal modes
    2. Weak coupling limit
    3. Handling cases with many coupled oscillators
    4. Normal coordinates
  2. Hamiltonian Mechanics
    1. H = ∑ p qdot - L
    2. Hamilton's equations
    3. Ignorable coordinates
    4. Phase space orbits
  3. Collisions
    1. definitions of impact parameter, (differential) cross section, and scattering angle
    2. solid angle
    3. Rutherford scattering
  4. Special 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
    6. Relativistic Dynamics: 3-force, modified Newton's second law
    7. Relativistic Lagrangian
  5. Continuum Mechanics
    1. Lagrange density for a string
    2. Waves on a string
    3. Stress and strain, elastic modulus
    4. Strain tensor
    5. Hydrodynamics, Bernoulli's principle

Problem Sets 1 to 12 with Emphasis on 8 to 12

Review any questions on these problems.

Review of Material from First Half of Semester for Final Exam

Mathematical Techniques

  1. Note that it is expected that you can handle vector algebra (addition, scalar multiplication, dot and cross products, componenets, unit vectors, and cartesian, cylindirical and spherical coordinates) and differentiation and integration involving vectors.
  2. angular velocity vector
  3. addition of angular velocities
  4. Rotating Coordinate Frames
    1. derivative of a vector: (du/dt)non-rotating = (du/dt)rotating + Ω×u
    2. Vector triple product, BAC-CAB rule
  5. Linear algebra
    1. Matrix representation of vectors and tensors
    2. Matrix multiplication
    3. Determinant of a matrix
    4. Eigenvalue problem: finding eigenvalues and eigenvectors
  6. Calculus of variations: the Euler-Lagrange equation.
  7. Full and partial derivatives.
  8. Determining v² from time derivative of (x(q), y(q), z(q)).
  9. square of path length, ds², in cartesian, cylindrical, and spherical coordinates.
  10. shortcut relation: v² = ds²/dt²
  11. Kronecker delta and summation notation.
  12. Taking discrete sums to integrals.
  13. Volume, surface, and line integrals (for mass, CM, and inertia tensor).

Physics Topics

  1. Noninertial Reference Frames
    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
  2. Rotations of Rigid Bodies
    1. Relation between ω and v for pure rolling
    2. Definitions of center of mass, and moment of inertia tensor
    3. Center of mass and moment of inertia tensor of composite objects
    4. Parallel axis theorem for the inertia tensor
    5. Conservation of linear and angular momentum for a rigid body
    6. General motion of a rigid body, 2 cases:
      1. motion of the CM, and rotation about the CM
      2. rotation about a fixed point
    7. Fixed axis rotation: Γ = dL/dt and L = Iω
    8. 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
  3. Solving minimization / maximization problems with the Euler-Lagrange equation.
  4. 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
© 2015 Robert Harr