Classical Mechanics

Units

1. 1

   Motion in phase space

   1 lesson

   1. Phase portraits and fixed pointsDraw one-dimensional motion as a flow in the (x, v) plane, read the energy contours of a potential, classify fixed points by the sign of U″, build the pendulum's phase portrait with its separatrix, and see a fixed-point pair collide as a parameter changes.
2. 2

   Linear and coupled oscillations

   2 lessons

   1. Small oscillations about equilibriumExpand a potential about a minimum to get ω₀² = U″(x₀)/m, solve the damped linear oscillator with a complex exponent, find the steady-state amplitude and phase of a driven oscillator with complex notation, and add the responses to several driving forces by superposition.
   2. Coupled oscillators and normal modesWrite the equations of two masses joined by three springs as a matrix equation, find the normal-mode frequencies and shapes as an eigenvalue problem, write any motion as a sum of modes, and explain beats between two modes.
3. 3

   The calculus of variations

   2 lessons

   1. Functionals and the Euler-Lagrange equationTreat the length of a path as a functional J[y] = ∫F(y, y′, x) dx, derive the Euler-Lagrange equation d/dx(∂F/∂y′) = ∂F/∂y by varying the path, show it gives a straight line as the shortest path, and use the first integral when F does not depend on x.
   2. The brachistochrone and other extremal pathsSolve the brachistochrone with the first integral to find a cycloid, derive Snell's law from Fermat's principle, find the catenary shape of a soap film between two rings, and handle a constraint with a Lagrange multiplier.
4. 4

   Lagrangian mechanics

   3 lessons

   1. Hamilton's principle and Lagrange's equationsDefine the Lagrangian L = T − U and the action S = ∫L dt, state Hamilton's principle that the true path makes S stationary, derive Lagrange's equations d/dt(∂L/∂q̇) = ∂L/∂q, and recover Newton's second law for a particle in Cartesian coordinates.
   2. Generalized coordinates and constraintsCount the degrees of freedom of a constrained system, choose generalized coordinates, write the Lagrangian of a bead on a rotating hoop and of a double pendulum, find the hoop's equilibria and bifurcation, and obtain a constraint force from a Lagrange multiplier.
   3. Symmetry and conservation lawsShow that a cyclic coordinate gives a conserved generalized momentum, state and prove Noether's theorem for a continuous symmetry, link translation, rotation and time translation to momentum, angular momentum and energy, and use the Jacobi function h, which equals the energy only when the constraints do not depend on time.
5. 5

   Central forces and the two-body problem

   3 lessons

   1. The two-body problem and reduced massSplit two bodies pulling on each other into a free centre of mass and one body of reduced mass μ moving in the force F(r), show that angular momentum is conserved and the motion planar, and read the radial motion from the effective potential with its centrifugal barrier.
   2. The Kepler problem and orbit equationTurn the radial equation of a central force into an orbit equation in u = 1/r, solve it for the inverse-square force to get the conics r = c/(1 + ε cos φ), tie eccentricity to energy, derive Kepler's three laws, and test when a circular orbit is stable for a power-law force.
   3. Scattering and cross sectionsDefine the impact parameter, scattering angle and differential cross section, derive dσ/dΩ = (b/sin θ)|db/dθ|, apply it to a hard sphere, and derive the Rutherford cross section from the Coulomb hyperbola.
6. 6

   Rotating frames and rigid bodies

   2 lessons

   1. Non-inertial and rotating framesDerive how time derivatives change in a rotating frame, write the centrifugal and Coriolis terms, and apply them to a puck on a turntable, a body falling on the rotating Earth, winds, and the Foucault pendulum.
   2. The inertia tensor and Euler's equationsWrite the angular momentum of a rigid body as L = Iω with the inertia tensor, find the principal axes and moments, derive Euler's equations for torque-free motion, solve the symmetric case, and show which principal axes are stable (the tennis-racket theorem).
7. 7

   Hamiltonian mechanics

   2 lessons

   1. The Hamiltonian and Hamilton's equationsDefine the canonical momentum, build H(q, p, t) = pq̇ − L by the Legendre transform, derive Hamilton's equations as first-order flow in phase space, show when H is the energy, and use them on the oscillator, the pendulum and a central force.
   2. Phase-space flow and Liouville's theoremShow that phase-space trajectories never cross, derive Liouville's theorem that the flow of Hamilton's equations conserves phase-space volume, use Poisson brackets to test constants of the motion, and see sensitive dependence in the driven pendulum.