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6.946

Classical Mechanics: A Computational Approach

Graduate · Historical course

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
Sources