The main purpose of this course is to introduce the analytical formalism to describe the mechanical properties of single and multi-particle systems. We will extensively use various analytical techniques to perform such a description. In a sense, the course will provide the link between the mathematical techniques used in calculus and complex variable analysis, and behavior of physical systems. We will start from Lagrangian and Hamiltonian formalism; study the dynamics of some fundamental systems in the phase space in general, and a motion near the equilibrium point in particular. We will then continue to perturbation theory, and its limitations. We will see how the instabilities emerge from the resonance (nonlinear) response, and study the types of bifurcations, which will bring us to chaos. Finally, we will learn the ways to quantify and describe the chaotic motion.

Syllabus 
Homeworks
Physics Projects
  

lecture00 - Overview of relevant calculus topics

lecture01 - Lagrangian formulation

lecture02 - Constrains, symmetry, and conservation laws

lecture03 - Small oscillations

lecture04 - Forced and damped oscillations

lecture04 - supplement - Green's function formalism

lecture05 - Forced damped oscillations

lecture06 - Dynamics in the vicinity of fixed points

lecture07 - Hamilton description of motion

lecture08 - Canonical transformations and Liouville's theorem

lecture09 - Hamilton Jacobi equations and separation of variables

lecture10 - Adiabatic invariant and action angle variables

lecture11 - Time-dependent perturbation theory

lecture12 - Time-independent perturbation theory

lecture13 - Parametric resonance

lecture14 - Review lecture

lecture15 - Phase space dynamics in integrable systems

lecture16 - KAM

lecture17 - Lyapunov exponents and multiple dimensions

lecture18 - Maps, resonances, biffurcations

lecture19 - Rotational motion

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Math project:

(pdf) - due date: Oct 31 2006

Homeworks:

1. (due Jan. 18 2008) HF 1.14, HF 1.15 (b), HF 1.19, HF 1.20
2. (due Jan. 25 2008) HF 2.14, 2.16, 3.16 (a-c), 9.14
3. (due Feb.6 2008) HF 5.5, 5.8, 5.15, 6.14, 6.15
4. (due Feb. 20 2008) HF 6.1 (a-c,e), 6.4, 6.8, 6.10, 6.11
5. (due Mar. 07 2008) G 12.2, 12.3, 12.5, 12.7, 12.11
   Compare your solution to 12.2 to results of numerical integration
   mathematica code;

 

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Physics projects

For each project you have to:

• derive the equations of motion
• Solve equation of motion in the small-oscillations regime
• Use perturbation theory to find a correction due to weak nonlinearity
• Write a computer program which will simulate system behavior in a general case.
• Build Poincare surface of section, and consider its behavior as you change the parameters of the system.
• Calculate the Lyapunov' exponents for chaotic regime.
• Make a presentation.