Workshops - Application of mathematical methods to solving physical problems

elementary

MA1_W01, MA1_W03, MA1_W04, MA1_W08, MA1_W09, MA1_U12, MA1_U15, MA1_U16, MA1_U18, MA1_U20, MA1_U22, MA1_U25, MA1_U26 , MA1_U29, MA1_U36, MA1_K01, MA1_K02, MA1_K03, MA1_K04

Mathematical analysis, Probability calculus, Statistics

Objectives of the course: Knowledge and the ability to use mathematical methods to describe physical phenomena

Course program (30 h):

1. Monte Carlo method: how it was created and for what purpose, what it is about, examples of applications in physics;

2. Harmonic vibrations: theory, equations, equation derivation, damped vibrations, examples of vibrations (oscillator, string, pendulum), composite harmonic vibrations - Lissajous curves;

3. Geometric curves: Habenicht trifolium, Brocard trifolium, T. Fay butterfly, Moritz curve, Fermat spiral, Dürer spiral, Varignon spiral, cardioid, Pascal snails, curve patterns and drawings, applications;

4. Fractals: definitions, theory, examples of known fractals (eg Julia's set, Sierpinski's carpet, Mandelbrot beetle, Koch curve);

5. Spiral functions: definition, examples of their application in physics: error function, gamma function, Jacobi function, Riemann zeta function, Airy function and Bessel function;

6. Fourier series, Fourier transform - definition, the most important formulas in physics, Laplace transform - definition, the most important formulas in physics;

7. Mathematical description of RLC circuits as an example of the application of differential equations in physics;

8. Logistic curve, recursive curve y = cx (1-x), bifurcations, Feigenbaum constant, applications,

9. Elements of group theory: definitions, applications in physics, symmetry group, permutation group;

10. Cellular automata: definition, examples, Conway's game;

11. The most important distributions of discrete random variables (definition, function formula and distribution function, properties, distribution parameters, applications in physics, limit theorems, generation eg in Excel);

12. The most important distributions of continuous random variables (definition, function formula and distribution function, properties, distribution parameters, applications in physics, limit theorems, generation eg in Excel);

13. Orthogonal polynomials: examples, definitions, properties, applications;

14. The traveling salesman problem - looking for the shortest path after the Hamilton cycle.

Description prepared by: Paweł Pęczkowski - course coordinator

[1] Frederick W. Bayron, Robert W. Fuller, "Matematyka w fizyce klasycznej i kwantowej", PWN, Warszawa, 1989.

[2] William R. Bennett (Jr.), "Scientific and engineering problem - solving with the computer", Prentice-Hall, New Jersey, 1976.

[3] Iwo Białynicki-Birula, Iwona Białynicka Birula, "Modelowanie rzeczywistości, Od gry w życie Conwaya przez żuka Mandelbrota do maszyny Turinga", Prószyński i S-ka, Warszawa, 2002.

[4] Mery L. Boas, Mathematical methods in the physical sciences, 3-rd Edition, John Wiley & Sons, Inc., USA, 2006, (https://www.christs.cam.ac.uk/sites/www.christs.cam.ac.uk/files/inline-files/0a187866618ca3049030ec5014860ae8-original.pdf).

[5] John H. Conway, Richard K. Guy, "Księga liczb", WNT, Warszawa, 1999.

[6] Richard Courant, Herbert Robbins, "Co to jest matematyka?", Prószyński i S-ka, Warszawa, 1998.

[7] Jerzy Ginter, "Symetria w fizyce materii", Wydawnictwa Uniwersytetu Warszawskiego, Warszawa, 2017.

[8] Paweł Kowalczyk, Fizyka cząsteczek, PWN, Warszawa, 2000.

[9] Maciej Matyka, "Symulacje komputerowe w fizyce. Komputerowe symulowanie zjawisk fizycznych - teoria i praktyka", Wydawnictwo HELION, Gliwice, 2002.

[10] David Potter, "Metody obliczeniowe fizyki, fizyka komputerowa", PWN, Warszawa, 1981.

[11] Josep Sales, Francesc Benyuls ,"Niebezpieczne krzywe, Elipsy, hiperbole i inne geometryczne cuda", Świat jest matematyczny RBA, Warszawa, 2012.

[12] Dietrich Stauffer, H. Eugene Stanley, "Od Newtona do Mandelbrota. Wstęp do fizyki teoretycznej", WNT, Warszawa, 1996.

[13] Edgar T. Sokołow, "Centaur czyli jak matematyka pomaga fizyce", PWN, Warszawa, 1987.

[14] Maciej Zawacki, "Fizyka. Rozwiązywanie zadań w Excelu. Ćwiczenia praktyczne", Wydawnictwo HELION, Gliwice, 2002.

a) Knowledge. The student knows the mathematical methods used to solve physical problems. He knows computer programs that allow to use mathematical methods in practice.

b) Skills. The student is able to choose and use the appropriate methods and information techniques to solve a specific physical problem. He can use an existing computer program (or a spreadsheet), if necessary, modify (adapt) it to make it as effective and efficient as possible.

c) Social competences. The student is aware of the possibilities and advantages of applying mathematical methods and techniques in physics, but also is aware of the disadvantages and limitations of the usefulness of these methods.

1. Preparation and delivery of a paper consisting of a presentation on a previously assigned topic;

2. Active participation in classes;

3. Ability to perform tasks with the use of information technology and having the necessary knowledge to interpret the results obtained;

4. Ability to use mathematical methods to solve selected physical problems.

There are no apprenticeships.