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 |
Variational Approach to Mechanics. Calculus of Variations. Hamilton's Principle: General Mathematical Formulation.
Lecture 3: 2.3-2.5, 1.3-1.4Lagrange'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, 2.7General Structure of Solutions. First Integrals and Conservation Theorems. Elimination of Cyclic Variables.
Lecture 5-6: 6.1-6.4Theory of Small Oscillations: Lagrangian and Equations of Motion. Eigenvalue Problem. Normal Coordinates. Example: Free Vibrations of Triatomic Molecule.
Lecture 7: 6.5Forced Oscillations: Sinusoidal Driving Forces. Dissipative Forces. Combined Effect of Sinusoidal and Dissipative Forces.
Lecture 8: 3.1-3.4Central Force Problem: Lagrangian. Lagrange's Equations and First Integrals. Classification of Orbits. Virial Theorem.
Lecture 9: 3.5-3.8Motion in Central Force: Differential Equations for Orbits. Bertrand's Theorem. Kepler Problem. Kepler's Laws.
Lecture 10: 8.1-8.2, 8.5Hamiltonian 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, 9.6Canonical Transformations: Generating Functions. Examples: Identity, Point and Exchange Transformations, Harmonic Oscillator. Poisson Bracket Formulation of Mechanics.
Lecture 12: 10.1-10.3, 10.8Hamilton-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, 10.6-10.8Hamilton-Jacobi Theory (Continuation): Separable Variables. Periodic Motion. Action-Angle Variables. Kepler Problem in Action-Angle Variables.
Lecture 14: 13.1-13.4Introduction 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.