Topics covered: CM, Autumn 2017

• Lecture 1 (Aug 1): Introduction
  • Course structure and modalities
• Lecture 2 (Aug 3): Newtonian mechanics
  • Diagnostic Test
  • Newton's laws: N1 -> assertion, N2 -> second order differential equation
  • Initial conditions and trajectories in (x, v) space: no force, constant force, (x, v)-dependent force
  • Conservation laws: trajectory interpretation, conservation of linear momentum, angular momentum and energy (for conservative force) for a single particle, from N2
  • N3 -> strong and weak versions, violation in EM
  • Conservation of linear momentum (with weak N3), angular momentum (with strong N3)
• Lecture 3 (Aug 17): Many-particle systems and constraints
  • Energy conservation (with conservative internal and external forces) for a system of particles
  • Internal potential energy, work done by internal forces
  • Counting degrees of freedom
  • Constraint forces and work done by them
• Drop Test
• Lecture 4 (Aug 22): Euler-Lagrange equations
  • Holonomic and non-holonomic constraints, rheonomous and scleronomous constraints
  • Generalized coordinates and generalized force
  • Virtual displacement and D'Alembrt's principle of vanishing virtual work
  • D'Alembert's principle leading to Lagrange's equation
  • Lagrange's equation for specific forces: [F = \grad Phi] (potential Phi), [F = f(v)] (Rayleigh's dissipation function),
  • Lagange's equation for specific forms of generalized force: [Q = - (\del U / \del q) + (d/dt) (\del U / \del qdot)] Generalized potential U. Electromagnetism as an example.
• Lecture 5 (Aug 24): Applications of E-L equation
  • Non-uniqueness of Lagrangian: [L' = L + (d/dt) F(q,t) ]
  • E-L equations in cartesian, polar and spherical polar coordinates, centripetal force, angular momentum and torque.
  • Examples: plenar / spherical pendulum, double pendulum, block sliding on inclined plane (with friction), flywheel, sliding ladder
• Assignment 1
• Lecture 6 (Aug 29): E-L equation from variational principle
  • Calculus of variations: extremization of [J =Int f(y, y', x) dx ] over paths, examples of least path, least time
  • Hamilton's principle: extremization of [I = Int L(q, qdot, t) dt] for [ L = T - U], leading to Lagrange's equation
  • Multiple interpretations of the same Lagrangian: example of motion of a particle and an electrical circuit
• Lecture 7 (Aug 31): Non-holonomic E-L equations (Lagrange multipliers)
  • Equivalence of [D'Alembert principle + N2] and [Hamilton's principle]
  • Problem of n (non-independent) coordinates and m non-holonomic constraints of the form [ Sum_k (a_lk) qkdot + (a_lt) =0]
  • Method of Lagrange multipliers to separate (n-m) independent coordinates and m constraint equations
  • Non-holonomic E-L equations, OK even for holonomic constraints
  • Forces of constraint from the extra equations of motion, example of a pendulum
• Lecture 8 (Sep 7): Example with non-holonomic constraints
  • Block sliding on an inclined plane: solving with different generalized coordinates, by substituting holonomic constraints. with Lagrange multipliers
  • An axle with two wheels, rolling without slipping on an inclined plane: Identifying generalized coordinates, constructing Lagrange multipliers, setting up E-L equations
• Lecture 9 (Sep 12): Generalization of kinetic energy, momentum, energy
  • Generalized kinetic energy: T2, T1, T0 terms
  • Generalized momentum: [p = del L / del qdot], distinction from kinematic momentum, example of electromagnetism
  • Generalized energy: Jacobi function [ h = p qdot - L]
  • Example of pulling a mass with a spring: difference between [h] and [T + V]
• Lecture 10 (Sep 14): Hamiltonian and EoM
  • Difference between change of coordinates and change of reference frame
  • Definition of Hamiltonian [H] and Hamilton's equations of motion
  • 2n first-order differential equations instead of n second-order equations (the E-L equations), phase space with coordinates (p,q)
  • Properties of Hamiltonian: conservation laws when [H] is independent of [p, q or t]
  • Existence of H: always possible for a "natural" kinetic energy term
  • Lagrangian -> Hamiltonian as a special case of Legendre transformations
  • Legendre transformations in thermodynamics: Internal energy -> Enthalpy, Free energy -> Gibbs free energy
• Assignment 2
• Lecture 11 (Sep 19): Two body problem with central potential
  • Solving CM problems using differential equations vs. solving them using constants of motion, in the phase space language
  • Reducing two-body problem to one-body problem (6 coordinates -> 3 coordinates): reduced mass and its coordinates
  • Conservation of direction of angular momentum [Lbar] -> planar motion
  • Conservation of magnitude of [Lbar] -> effective potential in 1-d coordinate [r], Qualitative understanding of trajectories and orbits
  • Conservation of energy, Integral equations for [r(t)] and [theta(t)]
  • Expressing the scenario in terms of Hamiltonian
• Lecture 12 (Sep 21): Central potential and orbits
  • Properties of orbits independent of the form of potential: Kepler's second law, symmetry about extreme points
  • Differential equation in [r] as a harmonic equation in [u = 1/r], potentials of the form [a r^(n+1)] that give "closed form" solutions
  • Closed orbits: Bertrand's theorem: only Kepler potential [k/r] and spring potential, precession of perihelion of planetary orbits
  • Kepler potential: the "epsilon" parameter [sqrt[1 + 2 E L^2/(m k^2)]] and energy, qualitative description of different conic sections as orbits
• Lecture 13 (Sep 24): Elliptical orbits
  • The equation [r = L^2/(mk) / (1 + epsilon cos(theta-theta0))] as an ellipse, epsilon as eccentricity, major axis [a = k / (2|E|)], minor axis [b = L / sqrt[2 m |E|]]
  • Kepler's third law: [T^2 \propto a^3], T independent of [b, L]
  • Runge Lenz /Laplace vector as a constant of motion, description of Kepler problem in terms of five conserved quantities
  • Virial theorem, application to Kepler potential, spring potential, and ideal gas law
• Lecture 14 (Sep 26): Scattering
  • Beam incident on a fixed target: parameters [Theta, s, psi]
  • Azimuthal symmetry, Differential cross section [dsigma/dTheta], need to determine [s(Theta)], possible multi-valued nature
  • Quantitative relation between [V(r)] and [Theta(s)]
  • Repulsive Coulomb potential: connection with Kepler potential, Rutherford cross section
• Lecture 15 (Sep 28): Scattering, Small oscillations
  • Qualitative behaviour of scattering angle [Theta(s)] and differential cross section [dsigma/dCosTheta] for specific potentials
  • Relating differential cross sections in Centre-of-Mass frame and lab frame
  • Examples of spring potential, with single and multiple springs
  • Formulation of the scenario of small oscillations about an equilibrium configuration
• Assignment 3
• Lecture 16 (Oct 3): Small oscillations: normal modes
  • Formulation with [L = (1/2) T_ij etadot_i etadot_j - (1/2) V_ij eta_i eta_j], E-L equations of motion as coupled differential equations
  • Solutions of the form [eta_i = a_i cos(omega t + delta)], Condition for existence: [Det(V - omega^2 T)=0]
  • Eigenvalues [omega_alpha] and eigenvectors [a_alpha], Orthonormalization of eigenvectors with respect to T_ij , i.e. [a(alpha)_i T_ij a(beta)_j = delta_(alpha,beta)].
  • Definition of Normal modes [chi_alpha] through [eta_i = a_(i,alpha) chi_alpha], Decoupling of differential equations in terms of the generalized coordinates [chi_alpha]
• Lecture 17 (Oct 5): Molecular vibrations, Forced oscillation, damping
  • Normal modes of vibration for a diatomic molecule and a linear triatomic molecule: Explicit calculations in 1D, Counting of degrees of freedom and number of normal modes in 3D
  • Forced oscillatons: with external oscillatory generalized force [Q_alpha = Q0_alpha cos(omega_ext t + delta_alpha)], Resonance
  • Oscillations with dissipation: addition of Rayleigh's dissipation function [ (1/2) F_ij etadot_i etadot_j], change in frequency and exponential decay
  • Combination of forced oscillations and damping: Lorentz / Breit-Wigner form of amplitudes of oscillations
• Lecture 18 (Oct 10): Non-oscillatory forces, anharmonic oscillations, perturbation series
  • Forced oscillations [d^2 chi_alpha / dt^2 + omega_alpha^2 chi_alpha = Q_alpha(t)], when the external generalized force [Q_alpha] does nor have an oscillatoty form. Total energy transmitted by the force to the system
  • Anharmonic oscillator by introducing third order terms in [eta, etadot]
  • Perturbative expansion of [chi_alpha] as [chi = chi(0) + lambda chi(1) + lambda^2 chi(2) + ...], Lambda as accounting parameter, Matching of coefficients of powers of alpha
  • Additional frequencies appearing at higher orders, need for perturbative expansion of [omega_alpha] as [omega = omega(0) + lambda omega(1) + lambda^2 omega(2) + ...]
• Lecture 19 (Oct 12): Perturbation series
  • Example of [L = (1/2) m xdot^2 - (1/2) m omega_o^2 x^2 - (1/3) alpha m x^3 - (1/4) beta m x^4] to explicitly calculate perturbative expansion
  • Conditions of ``no self-resonance'' for determining [omega(1)], Determination of [chi(1)] and higher order terms
  • The [omega(0) -> x(0) -> omega(1) -> x(1) -> omega(2) -> ...] route
  • Perturbative expansion as a result of two scales of "force"
  • Rapid-force oscillations: Different time scales for force and normal modes, solving for fast and slow osillations separately (separation of scales), effect of fast oscillations on slow ones
• Midterm Examination
• Assignment 4
• Lecture 20 (Oct 17): Rigid bodies and rotations
  • Six generalized coordinates for the rigid body: 3 for CM position, 3 for orientation
  • Representing orientation: direction cosines, orthogonal matrices [A], Euler angles [phi, theta, psi], active vs. passive transformations
  • Infinitesimal rotations, representations as [A = 1+epsilon], Relation between vectors in "body" frame vs. "space" frame
• Lecture 21 (Oct 24): Motion of particles in rigid body frame
  • Coriolis force and centrifugal force in the body frame, derivation starting from the Lagrangian, coriolis force in the northern vs southern hemisphere
  • Calculating coriolis force on a falling body using perturbation technique
  • Angular momentum for a rigid body, moment of inertia tensor, principle axes
  • Kinetic energy of a rigid body
• Lecture 22 (Oct 25): Motion of a rigid body
  • Angular velocities along the principle axes of a body in terms of the rates of change of Euler angles [phi, theta, psi]
  • E-L equations of motion for a rigid body in terms of its principle moments of inertia: non-linear equations of motion
  • Precession for a symmetric rigid body [I1 = I2]
  • Gyroscope effect: precession instead of falling
• Lecture 23 (Oct 31): Motion of a symmetric top
  Lagrangian in terms of three angular velocities, in turn in terms of the Euler angles
  • Two cyclic coordinates, two conserved (generalized) momenta, one conserved energy
  • Effective 1D problem with [udot^2 = a cubic polynomial] where [u=cos(theta)].
  • Precession and nutation in terms of roots of the cubic polynomial
• Assignment 5
• Lecture 24 (Nov 2): Canonical Transformations (CT)
  • Structure of classical mechanics: Lagrangian vs Hamiltonian framework
  • Canonical transformations: change of coordinates [(q,p,t) -> (Q,P,t)] and Hamiltonian [H(q,p,t) -> K(Q,P,t)] that retain the structure of Hamilton's equations of motion
  • Modified Hamilton's principle: new and old coordinates related by a Generating function [F] such that [p qdot - H = P Qdot - K + dF/dt], vanishing variation of [F] at the endpoints of the interval
  • Canonical transformations generated by [F = F1(q,Q,t)], giving [p = delF1/delq, P = - delF1/delQ]
  • Transformation of a harmonic oscillator problem into a problem with constant [P], Mapping the solutions in old and new coordinates
  • Generating function [F] as [F2(q,P,t)- Q P], [F3(p,Q,t)+ q p] and [F4(p,P,t) - Q P + q p]
  • [F1, F2, F3, F4] as Legendre Transforms of each other, Comparison with the thermodynamic quantities [U, F, H, G], Maxwell relations among these quantities
  • Examples of Canonical Transformations and generating functions: identity, [(q,p) <-> (P, -Q)], Infinitesimal translations, rotations and time evolutions, Generators for these operations
• Lecture 25 (Nov 7): CT and Poisson brackets (PB)
  • Examples of CT: Galilean transformations, uniform rotation, gauge transformations
  • Conservation of phase space volume
  • CT as a group
  • Poisson brackets: definition, basic properties, PB for coordinates and momenta
  • Poisson's theorem (time evolution of an observable), Jacobi-Poisson theorem (u, v constants of motion implies [u,v] constant of motion)
  • PB unchanged under CT
• Lecture 26 (Nov 9): Hamilton-Jacobi Theory
  • Choice of [F2] such that [K=0]
  • Hamilton-Jacobi equation, methods for solution
  • First and second integrals of motion [alpha_i and beta_i] to get [q_i(alpha, beta, t)] and hence [q_i(initial conditions, t)]
  • Determination of [p_i(initial conditions, t)] and [E - del_W/del_t]
  • Solving Harmonic oscillator and Kepler problem using HJ Theory
• Lecture 27 (Nov 14): Action-angle variables
  • Conditions for separability of [W]: multiply periodic system
  • Action variables for periodic, conservative, W-separable systems
  • Writing [E(J_i)] and hence frequency [nu_i = del_E/del_J_i], angle variables
  • Finding periodicity of Harmonic oscillator and periodicities in the Kepler problem, origin of orbit precession
• Assignment 6