AMCS 255: Advanced Computational Physics (Spring 2022)

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.

Goals and Objectives

The course is problem oriented aiming to enable the students to develop accurate solutions for practically relevant problems based on solid theoretical foundations and mathematically precise modeling. It prepares the students to study and understand advanced literature and state of the art publications on topics related to computational physics.

Required Knowledge

The course will assume basic knowledge (calculus and linear algebra) such as taught in undergraduate mathematics courses or in AMCS 101, 131, and 151.

Assignments and Evaluation

There will be a problem set assigned each week. This homework track is mostly theoretical, but it will include a final project and smaller programming tasks along the way. A final project will consist of writing a physically-based simulation. The students may collaborate on the assignments and the final project provided each student writes up his or her own solutions and clearly lists the names of all the students in the group (grading policy: 25% homework assignments and 75% final project).

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 and C. Lubich

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

Springer, 2010

• L. D. Landau and E. M. Lifshitz

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

Butterworth-Heinemann, 1982

• R. H. Landau, M. J. Paez, and C. C. Bordeianu

Computational Physics: Problem Solving with Computers

Wiley, 2007

Instructor

Prof. Dr. Dominik L. Michels

Assistant

Dr. Dmitry A. Lyakhov

Class Schedule

10:15 AM – 11:45 AM | Tue Thu | 2021-01-25 – 2021-05-17 | Bldg 1, R 2107