1. komponenta

Lecture type	Total
Lectures	30
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE OBJECTIVE: The objective of this course is to familiarize students with the fundamental laws and methods of classical mechanics and dynamical systems, as well as to further develop their mathematical skills through concrete physical problems. Students will also be introduced to key methods for analyzing nonlinear dynamical systems of first and higher orders and introduced to the basics of chaos theory.

LEARNING OUTCOMES: After passing the Introduction to Dynamical Systems course, students will be able to:

Determine the equilibrium points of a system with any number of degrees of freedom, examine their stability, and linearize the equations of motion near stable equilibrium points.
Determine the normal coordinates of a system consisting of n coupled harmonic oscillators and sketch the trajectories of two uncoupled oscillators in the plane.
Derive the response of a harmonic oscillator under forced oscillation with and without damping and explain the phenomenon of resonance.
Demonstrate a thorough understanding of the Hamiltonian formulation of classical mechanics and the concept of phase space and sketch the phase portrait of a one-dimensional conservative system.
Explain the concept of action-angle variables, derive the generating function for the transformation to action-angle variables, and explain the relationship between system degeneracy and the number of global integrals of motion.
Demonstrate thorough knowledge of the fundamental aspects of dynamical systems physics covered in the course content and related models.
Classify and assess the behavior types of simpler dynamical systems using qualitative methods for analyzing the corresponding differential equation systems.
Present basic scenarios of entering chaos and explain them through the analysis of corresponding dynamical systems.
COURSE PROGRAM AND OUTLINE:

Linearization of equations of motion and stability analysis of systems with arbitrary degrees of freedom.
Determination of normal coordinates of the system. Lissajous trajectories.

Forced oscillation with damping.

Parametric resonance.

Phase space, Hamiltonian, and Hamilton's equations. Canonical transformations.

Introduction of action-angle variables using the example of trajectories in phase space with one degree of freedom.
The concept of canonical transformations. Systems with two or more degrees of freedom: motion on tori as a generalization of Lissajous curves. The concept of Poincare sections.

Overview of Hamilton's equations as an example of dynamical systems.
First and second-order dynamical systems. Examples of autonomous, non-autonomous, conservative, and dissipative systems.

Poincare sections as motivation for the transition to mappings.
May-Feigenbaum mapping. Lorenz model and Henon mapping.

Strange attractors. Fractal objects.

TEACHING METHODS: Lectures, exercises, independent assignments.

MONITORING AND ASSESSMENT: Regular attendance, homework, e-learning.

REQUIREMENTS FOR COURSE SIGNATURE: Students are required to regularly attend lectures and exercises (at least 70%) and actively participate in solving problems during exercises.

EXAMINATION METHOD: The exam consists of a written and an oral part.

1. H. Goldstein, C.P. Poole, J.L. Safko: Classical Mechanics 3rd Edition, Addison-Wesley Publishing Company, 2001.
   L.D. Landau, E.M. Lifschitz: Mechanics, Buttenworth-Heinemann, 2001.
   S.T. Strogatz: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering, Perseus Books, Reading 1994.

Enrollment :
Passed : General Physics 2
Passed : Linear Algebra 2
Passed : Mathematical Analysis 2

Mandatory course - Regular study - Bachelor of Geophysics