1. SylabUZ
2. Faculty of Physics and Astronomy
3. winter term 2018/2019
4. Physics and Astronom - PhD studies
5. Lecture I-A

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Lecture I-A - course description

Aim of the course

The goal of the course is to acquaint the students with some of the selected methods of the dynamical systems that are described using ordinary differentia equations systems. The course will present basic notions of the theory of differential equations. Students will learn various types of equations and their solution methods. One of the most important goals will be to present the practical methods of the study of certain equation systems that appear in physics, astronomy and other applied sciences.

Prerequisites

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1. Knowledge of calculus and aglebra on the at the academic level for science or technical courses.

2. Knowledge of basic physics on the academic level.

3. Knowledge of theoretical mechanics.

Scope

Ordinary differential equations:

• phase curves and integral curves , first integral, phase portrait

• types of equations

• linear differentia equations

• equilibrium points and their classification; normalization

• stability

• numerical integration methods

• Lapunov’s exponents and deterministic chaos

Mechanics of material points and solids:

• Lanrange and Hamilton equations

• stability of equilibrium points in mechanicsl systems

• chaos in mechanical systems

• separatrix spliting amd Mielnikov method

Teaching methods

Traditional lecture using multimedia presentations.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Writen test.

Passing criterium – a positive grade from the test that involves questions/exercises of various degrees of difficulty.

Recommended reading

[1] Perko, Lawrence. 2001. Differential Equations and Dynamical Systems. Vol. 7. Texts in Applied Mathematics. New York, NY: Springer New York. http://link.springer.com/10.1007/978-1-4613-0003-8.

[2] Walter, Wolfgang. 1998. Ordinary Differential Equations. Vol. 182. Graduate Texts in Mathematics. New York, NY: Springer New York. http://link.springer.com/10.1007/978-1-4612-0601-9.

[3]. Schuster, Chaos deterministyczny, PWN, Warszawa 1993.

[4] Florian, Scheck. Mechanics: From Newton’s Laws to Deterministic Chaos. 3rd ed. New York, NY: Springer Verlag, 1999.

Further reading

[1] W.I. Arnold, Równania różniczkowe zwyczajne, PWN 1975.

[2] W. I. Arnold, Teoria Równań Różniczkowych, PWN, 1983.

Notes

Modified by dr Joanna Kalaga (last modification: 11-07-2018 13:07)