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12.620

Classical Mechanics: A Computational Approach

3-3-6 units · Graduate

Classical mechanics in a computational framework, Lagrangian formulation, action, variational principles, and Hamilton's principle. Conserved quantities, Hamiltonian formulation, surfaces of section, chaos, and Liouville's theorem. Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems. Invariant curves and cantori. Nonlinear resonances, resonance overlap and transition to chaos. Symplectic integration. Adiabatic invariants. Applications to simple physical systems and solar system dynamics. Extensive use of computation to capture methods, for simulation, and for symbolic analysis. Programming experience required.

Fall 2008

We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: the Lagrangian formulation; action, variational principles, and equations of motion; Hamilton's principle; conserved quantities; rigid bodies and tops; Hamiltonian formulation and canonical equations; surfaces of section; chaos; canonical transformations and generating functions; Liouville's theorem and Poincaré integral invariants; Poincaré-Birkhoff and KAM theorems; invariant curves and cantori; nonlinear resonances; resonance overlap and transition to chaos; properties of chaotic motion. Ideas will be illustrated and supported with physical examples. We will make extensive use of computing to capture methods, for simulation, and for symbolic analysis.

Readings

  • Buy at MIT Press Sussman, Gerald Jay, and Jack Wisdom. Structure and Interpretation of Classical Mechanics . Cambridge, MA: MIT Press, 2001. ISBN: 9780262194556. ↗
    Listed reading
  • All readings are taken from the course textbook:
    Listed reading
  • SES #: 1 · TOPICS: Mechanics is more than equations of motion · READINGS: Notation appendix, Scheme appendix, sections 1 through 1.4
    Reading assignment
  • SES #: 3 · TOPICS: Lagrange equations · READINGS: Section 1.5
    Reading assignment
  • SES #: 4 · TOPICS: Hamilton’s principle · READINGS: Sections 1.6 up to 1.6.2
    Reading assignment
  • SES #: 5 · TOPICS: Coordinate transformations and rigid constraints · READINGS: Sections 1.6.2, 1.6.3
    Reading assignment
  • SES #: 6 · TOPICS: Total-time derivatives and the Euler-Lagrange operator · READINGS: Section 1.6.4
    Reading assignment
  • SES #: 7 · TOPICS: State and evolution: chaos · READINGS: Section 1.7
    Reading assignment
  • SES #: 8 · TOPICS: Conserved quantities · READINGS: Section 1.8
    Reading assignment
  • SES #: 9 · TOPICS: Kinematics of rigid bodies, moments of inertia · READINGS: Sections 2.1, 2.2, 2.3, 2.4, 2.5
    Reading assignment
  • SES #: 10 · TOPICS: Generalized coordinates for rigid bodies · READINGS: Sections 2.6, 2.7
    Reading assignment
  • SES #: 11 · TOPICS: Motion of a free rigid body · READINGS: Sections 2.8, 2.9
    Reading assignment
  • SES #: 12 · TOPICS: Axisymmetric top · READINGS: Section 2.10
    Reading assignment
  • SES #: 13 · TOPICS: Spin-orbit coupling · READINGS: Sections 2.11, 2.12
    Reading assignment
  • SES #: 14 · TOPICS: Hamilton’s equations · READINGS: Sections 3.1 up to 3.1.1
    Reading assignment
  • SES #: 15 · TOPICS: Legendre transformation, Hamiltonian actian · READINGS: Sections 3.1.1, 3.1.2, 3.1.3
    Reading assignment
  • SES #: 16 · TOPICS: Phase space reduction, Poisson brackets · READINGS: Sections 3.2, 3.3, 3.4
    Reading assignment
  • SES #: 17 · TOPICS: Evolution and surfaces of section · READINGS: Sections 3.5, 3.6 through 3.6.2
    Reading assignment
  • SES #: 18 · TOPICS: Autonomous systems: Henon and Heiles · READINGS: Sections 3.6.3, 3.6.4
    Reading assignment
  • SES #: 19 · TOPICS: Exponential divergence, solar system · READINGS: Section 3.7
    Reading assignment
  • SES #: 20 · TOPICS: Liouville theorem, Poincare recurrence · READINGS: Sections 3.8, 3.9
    Reading assignment
  • SES #: 23 · TOPICS: Linear stability · READINGS: Sections 4.1, 4.2
    Reading assignment
  • SES #: 24 · TOPICS: Homoclinic tangle · READINGS: Section 4.3
    Reading assignment
  • SES #: 25 · TOPICS: Integrable systems · READINGS: Section 4.4
    Reading assignment
  • SES #: 26 · TOPICS: Poincare-Birkhoff theorem · READINGS: Section 4.5
    Reading assignment
  • SES #: 27 · TOPICS: Invariant curves, KAM theorem · READINGS: Section 4.6
    Reading assignment
  • SES #: 28 · TOPICS: Canonical transformations, point transforms, symplectic conditions · READINGS: Sections 5.1, 5.2, 5.3, 5.4
    Reading assignment
  • SES #: 29 · TOPICS: Mixed-variable generating functions · READINGS: Sections 5.6, 5.6.4
    Reading assignment
  • SES #: 30 · TOPICS: Time evolution is canonical · READINGS: Section 5.7
    Reading assignment
  • SES #: 31 · TOPICS: Hamilton-Jacobi equation · READINGS: Section 5.8
    Reading assignment
  • SES #: 32 · TOPICS: Lie transforms and Lie series · READINGS: Sections 5.9, 5.10
    Reading assignment
  • SES #: 33 · TOPICS: Perturbation theory with Lie series · READINGS: Sections 6.1 up to 6.2.1
    Reading assignment
  • SES #: 34 · TOPICS: Small denominators and secular terms, pendulum to higher order and many degrees of freedom · READINGS: Sections 6.2.1, 6.2.2, 6.3
    Reading assignment
  • SES #: 35 · TOPICS: Nonlinear resonances, reading the Hamiltonian, resonance overlap · READINGS: Sections 6.4 up to 6.4.4
    Reading assignment
  • SES #: 36 · TOPICS: Second-order resonances, stability of the vertical equilibrium · READINGS: Sections 6.4.4, 6.4.5
    Reading assignment

Prerequisites & course information

Same subject as 6.5160J, 8.351[J] Prereq: Physics I (GIR) , 18.03 , and permission of instructor G (Fall) Not offered regularly; consult department 3-3-6 units

Sources