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​​​​CS 290C: Advanced Computational Physics (Spring 2017)

Overview

This course covers a selection of advanced topics related to computational physics. Based on prior knowledge in calculus and linear algebra, the following topics are considered: Lagrangian formalism, symmetries and conservation laws, stability and bifurcation, multi-body problems and rigid bodies, linear and nonlinear oscillations, Hamiltonian formalism, canonical transformations and invariances, Liouville's theorem, discrete Lagrangian and Hamiltonian formalisms, Hamilton Jacobi theory, transition to quantum mechanics, relativity, fields.

Syllabus

  1. Euler-Lagrange Equations of Second Kind

    Phase Space, Generalized Coordinates, Constraints, Calculus of Variations and Euler-Lagrange Equations of Second Kind.

  2. Lagrangian Formalism

    Lagrangian Formalism, Particle Systems, Friction and Dissipation.

  3. Symmetries and Conservation Laws

    Generalized Momenta, Cyclic Coordinates, Noether's Theorem, Conservation of Energy.

  4. Stability and Bifurcation

    Chaotic and Nonchaotic Dynamics, Sensitivity to Initial Conditions and Deterministic Chaos, Lyapunov Stability, Lyapunov's First Method, Lyapunov's Second Method, Bifurcations, Attractors.

  5. Euler-Lagrange Equations of First Kind

    Lagrange Multiplier, Euler-Lagrange Equations of First Kind, SHAKE and RATTLE.

  6. Multi-body Problems and Rigid Bodies

    Central Force, Two-body Problem, Effective Potential, Multi-body Problems, Center of Mass Theorem, Angular Momentum Theorem, Euler Angles, Lagrangian Equations of the Rigid Body.

  7. Linear and Nonlinear Oscillations

    Oscillators with a Single Degree of Freedom, Transition to the Continuum, Linear and Non-linear Forces, Calculation of Perturbations, Harmonic Balance, Enforced Non-linear Oscillation, Self- and Parameter-excited Oscillation.

  8. Hamiltonian Formalism

    Legendre Transformation, Hamiltonian mechanics, Poisson Brackets.

  9. Canonical Transformations and Invariances

    Point Transformations, Canonical Transformations, Generators, Canonical Invariances of Poisson Brackets, Canonical Invariances of the Phase Volume.

  10. Liouville's Theorem

    Phase Space Trajectories, Foundations of Statistical Mechanics, Liouville's Theorem and its Consequences.

  11. Discrete Lagrangian and Hamiltonian Formalisms

    Symplectic Transformations, Symplecticity and Variational Integrators.

  12. Hamilton Jacobi Theory

    Hamilton-Jacobi Formalism, Principal Function, Integrability, Level Set Method.

  13. Transition to Quantum Mechanics

    Quantum Objects, Copenhagen Interpretation, Time-independent Schrödinger Equation, Time-dependent Schrödinger Equation, Single Configuration Ansatz, Time-dependent Self-consistent Field System, Ehrenfest’s Molecular Dynamics.

  14. Relativity

    Space and Time, Galileo's principle, Einstein's Postulates, Lorentz Transformation, Time Dilation and Length Contraction, Minkowski Diagrams, Doppler Effect, Spacetime and Four-vectors, Relativistic Momentum, Mass and Energy, Photons.

  15. Fields

    Classical View on Gravitation, Electrostatics, Magnetostatics, Electrodynamics, Maxwell's Equations, Gravitation in General Relativity, Quantum Fields.

Literature

  • J.-L. Basdevant

    Variational Principles in Physics

    Springer, 2007

  • E. Hairer, C. Lubich

    Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations

    Springer, 2010

  • L.D. Landau, E.M. Lifshitz

    Mechanics, Third Edition, Course of Theoretical Physics, Volume 1

    Butterworth-Heinemann, 1982

  • R.H. Landau, M.J. Páez, C.C. Bordeianu

    Computational Physics: Problem Solving with Computers

    Wiley, 2007

  • The Feynman Lectures on Physics

Instructor

Prof. Dr. Dominik L. Michels, M.Sc.

Assistant

Dr. Dmitry A. Lyakhov, M.Sc.

Class Schedule

04:00 PM – 05:30 PM | Sun Wed | 2017-01-22 – 2017-05-17 | Bldg 1, R 2107