Information about the course TFYA Analytical Mechanics, 6 ECTS-points

Advancement level: A
Aim: The course represents the basic principles of classical mechanics within the framework of the variational approach. The knowledge of these principles is necessary for a deeper penetration into the structure of the classical mechanics itself, and for a proper understanding of subsequent courses such as thermodynamics and statistical mechanics, quantum mechanics etc. Furthermore, the methods of the analytical mechanics can serve as a set of the efficient tools for the solution of the majority of hard problems of mechanics and related topics.
Course organization: The course contains 28 lecture hours and 28 seminar hours with solution of problems in connection with the lectures.
Course content: The variational approach to mechanics, Hamilton's principle, Lagrange's equations, first integrals and conservation theorems, elimination of cyclic variables, small oscillations, central force problem, Hamiltonian formulation of mechanics, canonical transformations, Hamilton-Jacobi theory, introduction to analytical mechanics of continuous systems.
Course literature: I. Yakimenko. Lecture notes in Analytical mechanics. I. Yakimenko. Problems in Analytical mechanics. Goldstein H., Poole Ch., Safko J.: Classical Mechanics Addison & Wesley, 3d edition, 2001 (selected parts). You can buy it in Linnus & Linnea AB.
Exam: A written examination with questions in theory and solution of problems.
Course language: English

Plan of Lectures

Lecture 1-2: 2.1-2.2 (according to Goldstein H., Poole Ch., Safko J.: Classical Mechanics, third edition)

Variational Approach to Mechanics. Calculus of Variations. Hamilton's Principle: General Mathematical Formulation.

Lecture 3: 2.3-2.5, 1.3-1.4

Lagrange's Equations for Conservative Systems. Forces of Constraints: Lagrange's Multipliers. Constraints: Examples. Lagrange's Equations: Derivation from D'Alambert's Principle.

Lecture 4: 2.6, 2.7

General Structure of Solutions. First Integrals and Conservation Theorems. Elimination of Cyclic Variables.

Lecture 5-6: 6.1-6.4

Theory of Small Oscillations: Lagrangian and Equations of Motion. Eigenvalue Problem. Normal Coordinates. Example: Free Vibrations of Triatomic Molecule.

Lecture 7: 6.5

Forced Oscillations: Sinusoidal Driving Forces. Dissipative Forces. Combined Effect of Sinusoidal and Dissipative Forces.

Lecture 8: 3.1-3.4

Central Force Problem: Lagrangian. Lagrange's Equations and First Integrals. Classification of Orbits. Virial Theorem.

Lecture 9: 3.5-3.8

Motion in Central Force: Differential Equations for Orbits. Bertrand's Theorem. Kepler Problem. Kepler's Laws.

Lecture 10: 8.1-8.2, 8.5

Hamiltonian Formulation of Mechanics: Hamilton's Equations. Canonical Integral. Cyclic Coordinates and Conservation Laws. Evaluation of Hamiltonian: Hamiltonians for Particle in Central Force and Electromagnetic Field.

Lecture 11: 9.1-9.2, 9.6

Canonical Transformations: Generating Functions. Examples: Identity, Point and Exchange Transformations, Harmonic Oscillator. Poisson Bracket Formulation of Mechanics.

Lecture 12: 10.1-10.3, 10.8

Hamilton-Jacobi Theory: Hamilton-Jacobi Equation. Hamilton's Principal Function. Harmonic Oscillator in Hamilton-Jacobi Theory. Hamilton's Characteristic Function. Eikonal Equation.

Lecture 13: 10.4, 10.6-10.8

Hamilton-Jacobi Theory (Continuation): Separable Variables. Periodic Motion. Action-Angle Variables. Kepler Problem in Action-Angle Variables.

Lecture 14: 13.1-13.4

Introduction to Analytical Mechanics of Continuous Systems: Lagrangian Density. Three- and Four-Dimensional Lagrangian Formulation. Stress-Energy Tensor and Conservation Theorems. Hamiltonian and Poisson Bracket Formulations.

Plan of seminars

(following I.Yakymenko. Problems in Analytical mechanics. You can find it in Linus & Linnea AB.)

Seminar 1. 1, 2, 3, 4, H5, H6
Seminar 2. 7, 8, H9
Seminar 3. 10, 11, 12, H13
Seminar 4. 14, 15, 16, 17, H18
Seminar 5. 19, 20, H21, H22
Seminar 6. 23, H24, H25
Seminar 7. 26, 27, H28, H29, H30
Seminar 8. 31, 32, 33, H34, H35
Seminar 9. 36, 37, 38, H39
Seminar 10. 40, 41, H42
Seminar 11. 43, 44, H45, H46
Seminar 12. 47, 48, 49, H50
Seminar 13. 51, 52, 53, H54
Seminar 14. Repetition of problems solution.

H cooresponds to home problems