Analytical mechanics

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Lagrangian mechanics: Configurational space. Holonomic and nonholonomic constraints. Principle of virtual work. D’Alember principle. Lagrangian equations. Constant of motion. Cyclic variables, Jacobi energy function, Emmy-Noether theorem. Variational principles. Small oscillations. Generalized potential.
Hamiltonian mechanics: Legendre transformation. Hamiltonian function, canonical system. Poisson bracket, differentiation along solution of the canonical system, integrals of motion, Poisson theorem. Canonical transformation, symplectic matrix, symplectic condition. Generating functions. Hamilton-Jacobi equation.

Readings

V. I. Arnold: Mathematical Methods of Classical Mechanics, 2nd edition, Springer, New York, 1997.
H. Goldstein, C. P. Poole, J. L. Safko: Classical Mechanics, 3rd edition, Addison-Wesley, Reading, 2002.
A. Fasano, S. Marmi, Analytical Mechanics: An Introduction, Oxford University Press, Oxford, 2006
J. V. José, E. J. Saletan: Classical Dynamics : A Contemporary Approach, Cambridge Univ. Press, Cambridge, 1998.

Objectives and competences

The goal is to obtain basic knowledge of principles of analytical mechanics. Mastering them enables problem solving of dynamical problems and to understand the role of mathematics in mechanics

Intended learning outcomes

Knowledge and understanding: Knowledge and understanding of basic prnciples and methods of analytical mechanics.
Application: Application of the learnt methods in solving dynamical real word problems. First step for further graduate level study of methods of classical and relativistic mechanics.
Reflection: Crossbreeding of different mathematical subjects within a single course and their application.
Transferable skills: Students develop abilities to clearly and logically formulate problems. They learn to critically assess modeling by analyzing their predictions and comparing them with real examples.

Learning and teaching methods

Lectures, exercises, seminar, homeworks, consultations

Assessment

Regular homework assignments:
Oral presentation of homework

Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

George Mejak:
MEJAK, George. On extension of functions with zero trace on a part of boundary. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 1993, let. 175, str. 305-314. [COBISS-SI-ID 5828441]
MEJAK, George. Finite element solution of a model free surface problem by the optimal shape design approach. International journal for numerical methods in engineering, ISSN 0029-5981. [Print ed.], 1997, vol. 40, str. 1525-1550. [COBISS-SI-ID 9983833]
MEJAK, George. Eshebly tensors for a finite spherical domain with an axisymmetric inclusion. European journal of mechanics. A, Solids, ISSN 0997-7538. [Print ed.], 2011, vol. 30, iss. 4, str. 477-490. [COBISS-SI-ID 16025177]