Topics covered: CM, Autumn 2019

• Lecture 1 (Aug 7): Introduction
  • Course structure and modalities
  • Diagnostic Test
• Lecture 2 (Aug 13): Newtonian mechanics
  • 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
  • System of particles: conservation of linear momentum (with weak N3), angular momentum (with strong N3)
• Lecture 3 (Aug 14): Euler-Lagrange equations
  • Counting degrees of freedom
  • Constraint forces and work done by them, Holonomic and non-holonomic constraints
  • Generalized coordinates, velocities and forces
  • Virtual displacement and D'Alembert's principle of vanishing virtual work --> Euler-Lagrange (E-L) equation
  • E-L equation for specific forces: [F = \grad Phi] (potential Phi), [F = f(v)] (Rayleigh's dissipation function),
  • E-L equation with specific form of generalized force: [Q = - (\del U / \del q) + (d/dt) (\del U / \del qdot)] Generalized potential U. Electromagnetism as an example.
• Lecture 4 (Aug 20): 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
  • Multiple interpretations of the same Lagrangian: example of motion of a particle and an electrical circuit
• Lecture 5 (Aug 21): E-L equation from variational principle, Lagrange multipliers
  • 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
  • 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
• Assignment 1
• Lecture 6 (Aug 27): Non-holonomic E-L equations (Lagrange multipliers)
  • Examples with holonomic as well as non-holonomic constraints: Pendulum, Block on inclined plane, Rolling cylinder
  • An example that needs non-holonomic methods: Axle with two wheels, rolling without slipping on an inclined plane
• Lecture 7 (Aug 28): Generalization of kinetic energy, momentum, energy, Hamiltonian
  • 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], Change of coordinates vs change of reference frame
  • Definition of Hamiltonian [H] and Hamilton's equations of motion
• Lecture 8 (Sep 3): Hamiltonian and EoM
  • Solving problems with Hamilton's equation of motion: Pendulum, Block sliding down an inclined plane, Electromagnetism
  • Properties of Hamiltonian: conservation laws when [H] is independent of [p, q or t], Conditions for existence of H
  • Lagrangian -> Hamiltonian as a special case of Legendre transformations
  • Legendre transformations in thermodynamics: Internal energy -> Enthalpy, Free energy -> Gibbs free energy
  • The concept of Routhian
• Assignment 2
• Lecture 9 (Sep 4): Two body problem with central potential
  • 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)]
• Lecture 10 (Sep 11): 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 and conic section orbits: [r = L^2/(mk) / (1 + epsilon cos(theta-theta0))]
  • Elliptical orbits, major and minor axis, eccentricity [sqrt[1 + 2 E L^2/(m k^2)]]
  • Kepler's third law: [T^2 \propto a^3], T independent of [b, L]
• Lecture 11 (Sep 17): Runge-Lenz vector; scattering
  • Runge Lenz / Laplace vector as a constant of motion, description of Kepler problem in terms of five conserved quantities
  • Formulation of scattering in terms of a beam incident on a fixed target: parameters [Theta, s, psi]
  • Azimuthal symmetry, Differential cross section [dsigma/dcosTheta], need to determine [s(Theta)], possible multi-valued nature
  • Quantitative relation between [V(r)] and [Theta(s)]
• Lecture 12 (Sep 18): Scattering
  • Repulsive Coulomb potential: connection with Kepler potential, Rutherford cross section
  • 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
• Assignment 3
• Lecture 13 (Sep 24): Virial theorem; Small oscillations: formalism
  • Virial theorem, application to Kepler potential, spring potential, and ideal gas law
  • Examples of spring potential, with single and multiple springs
  • Formulation of the scenario of small oscillations about an equilibrium configuration
  • [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
• Lecture 14 (Sep 25): Normal modes, Molecular vibrations
  • [omega^2_alpha] as eigenvalues of [T^(-1) V], Condition for existence: [Det(V - omega^2 T)=0]
  • 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]
  • Normal modes of vibration for a diatomic molecule and a linear triatomic molecule: Explicit calculations in 1D
• Lecture 15 (Oct 1): Tutorial
• Lecture 16 (Oct 2): Midterm Exam
• Lecture 17 (Oct 9): Forced oscillation, damping, perturbation
  • 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
  • Anharmonic oscillator with higher order terms in [eta, etadot], need for perturbative expansion
  • Perturbative expansion of [chi_alpha] as [chi = chi(0) + lambda chi(1) + lambda^2 chi(2) + ...]
  • Additional frequencies appearing at higher orders, need for perturbative expansion of [omega_alpha] as [omega = omega(0) + lambda omega(1) + lambda^2 omega(2) + ...]
  • Lambda as accounting parameter, Matching of coefficients of powers of lambda
• Lecture 18 (Oct 15): Perturbation techique: anharmonic oscillator; Non-oscillatory forces
  • Example of [L = (1/2) m xdot^2 - (1/2) m omega_0^2 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
  • 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.
• Assignment 4
• Lecture 19 (Oct 16): 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
  • Euler force, Coriolis force and centrifugal force in the body frame
  • Lagrangian using quantities in rotating frame, Generalized potential U(r,v)
  • Calculating coriolis force on a falling body using perturbation technique
  • Foucault's pendulum using perturbation expansion
• Lecture 20 (Oct 22): Motion of particles in rigid body frame
  • Angular momentum for a rigid body, moment of inertia tensor, principle axes
  • Kinetic energy of a rigid body
  • Euler equations of motion for a rigid body in terms of its principle moments of inertia: non-linear equations of motion
• Lecture 21 (Oct 23): Motion of a rigid body
  • Unstable rotations of a rigid body
  • Precession for a symmetric rigid body [I1 = I2]
  • Gyroscope effect: precession instead of falling
• Lecture 22 (Oct 29): Lagrangian, E-L, Symmetric top
• Angular velocities along the principle axes of a body in terms of the rates of change of Euler angles [phi, theta, psi]
  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
• Lecture 23 (Oct 30): Midterm and Project discussion
• Assignment 5
• Lecture 24 (Nov 5): Canonical Transformations (CT)
  • Structure of classical mechanics: 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
• Lecture 25 (Nov 6): CT and Poisson brackets (PB)
  • Examples of Canonical Transformations and generating functions: identity, [(q,p) <-> (P, -Q)], Infinitesimal translations, rotations and time evolutions, Galilean transformations, uniform rotation, gauge transformations
  • Generators for the above transformations
  • Conservation of phase space volume
  • Poisson brackets: definition, basic properties, PB for coordinates and momenta
  • Poisson's theorem (time evolution of an observable)
  • PB unchanged under CT
• Lecture 26 (Nov 12): CT and Hamilton-Jacobi method
  • Equivalent statements for CT: Generating functions, Phase space volume conservation, Maxwell relations, Invariant PB
  • Jacobi-Poisson theorem (u, v constants of motion implies [u,v] constant of motion)
  • Hamilton-Jacobi idea: Choice of [F2] such that [K=0]
  • Hamilton-Jacobi equation, methods for solution
  • Obtaining first and second integrals of motion [alpha_i and beta_i]
• Lecture 27 (Nov 13): Hamilton-Jacobi theorem
  • Jacobi theorem: [beta_i = P_i] constants of motion
  • [alpha_i] = [Q_i] constants depending on initial conditions
  • Obtaining [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]
  • Time-independent Hamiltonian, separation of [W]
  • Solving Harmonic oscillator and Kepler problem using HJ Theory
• Lecture 28 (Nov 19): Trajectories in phase space
  • Periodic and ergodic trajectories, definition of chaos
  • Conditions for separability of [W]: multiply periodic system
  • Action variables [J_i] for periodic, conservative, W-separable systems
  • Writing [E(J_i)] and hence frequency [nu_i = del_E/del_J_i], angle variables [Theta_i]
  • Finding periodicity of Harmonic oscillator and periodicities in the Kepler problem, origin of orbit precession
  • Action variables as adiabatic invariants
• Lecture 29 (Nov 13): Continuous media
  • Example of an infinite chain of masses and springs, Lagrangian and equations of motion in the limit of small springs
  • Transformation to continuous variables: [eta_i \(t) -> eta(x,t)], Lagrangian density and equation of motion -> wave equation
  • Variational principle to get E-L equations for continuous media
  • Symmetry in space and time, new notation [X_{,nu}] for derivatives
  • Stress-energy tensor and conservation laws
  • [Only rough introduction from this point onwards]
  • Hamiltonian and Hamilton's equations of motion
  • Noether's theorem
• Assignment 6