COURSE GOALS: The principle objectives of the course Classical Mechanics 2 are the introduction of fundamental laws and methods of classical mechanics, further development of acquired mathematical skills and their applications to selected physical problems, and the preparation of students for more advanced courses in theoretical physics.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.1 formulate, discuss and explain the basic laws of physics including mechanics, electromagnetism and thermodynamics
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
4. COMMUNICATION SKILLS
4.2 present one's own research or literature search results to professional as well as to lay audiences
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required)
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon completing the course Classical mechanics 2, students will be able:
- to derive the equation of motion for a particle in a noninertial system, analyze the effects of noninertial forces, in particular the influence of the Coriolis force on the motion of a particle close to the surface of the Earth;
- to determine the points of equilibrium for a system with an arbitrary number of degrees of freedom, examine their stability and linearize the equations of motion for stable equilibrium points;
- to determine the normal modes for a system of coupled harmonic oscillators, sketch the Lissajous curves and describe the vibrations of molecules, in particular those with two and three atoms;
- to describe the motion of forced harmonic oscillator with or without damping and analyze the occurrence of resonance;
- to write the Hill's equation and derive the Floquet's theorem, derive exact and perturbative solution of the Mathieu's equation and sketch the Ince diagram, analyze the occurrence of the parametric resonance;
- to discuss Hamilton's formulation of classical mechanics, explain the concept of the phase space and sketch the phase portrait of an arbitrary conservative system with one degree of freedom;
- to discuss the Hamilton-Jacobi formulation of classical mechanics, construct global constants of motion by separating Hamilton-Jacobi equation and derive the Schrödinger's equation by using the analogy between geometrical and physical optics;
- to apply the action-angle variables in order to analyze the behaviour of conservative systems with several degrees of freedom;
- to use the Poisson brackets, discuss the implications of the Noether's theorem and discuss the generalization to quantum mechanics through Poisson brackets;
- to formulate and the canonical perturbation theory and apply it to the adiabatic invariants.
- Motion in a non-inertial frame of reference. The effects of the Coriolis force. Example: a particle falling freely close to the surface of the Earth.
- Oscillations of systems with more than one degree of freedom. Normal modes of oscillations. Lissajous curves.
- Vibrations of molecules.
- Forced oscillations, Greens function, resonance. Forces oscillations under friction.
- Parametric resonance.
- Hamilton's formulation of classical mechanics. Phase space. Canonical transformations.
- Hamilton-Jacobi formulation of classical mechanics. Hamilton-Jacobi equation.
- Geometrical interpretation of the Hamilton-Jacobi function, relation to the geometrical optics. Connections with quantum mechanics.
- Action-angle variables. Periodical and quasiperiodical motion, degeneracy and global constants of motion. Examples: harmonic oscillator and Kepler problem.
- Liouville's theorem. Poincare recurrence theorem. Poincare's integral invariants.
- The algebra of Poisson brackets. Poisson brackets and canonical transformations. Infinitesimal canonical transformations and Noether's theorem. Connections with quantum mechanics.
- Canonical perturbation theory. Adiabatic invariants.
- Integrable and nonintegrable systems. Hamiltonian chaos.
REQUIREMENTS FOR STUDENTS:
Students are required to regularly attend classes, participate actively in solving problems and solve homework. Furthermore, students are required to pass two written examinations during the semester.
GRADING AND ASSESSING THE WORK OF STUDENTS:
At the end of the course a written and oral examination is held for students who have successfully completed the requirements of the course.
- 1) H. Goldstein, C.P. Poole, J.L. Safko : Classical Mechanics 3rd Edition, Addison-Wesley Publishing Company, 2001
2) L.D. Landau, E.M. Lifschitz: Mechanics, Buttenworth-Heinemann, 2001
- 1) V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1991
2) Spiegel M.R.: Theoretical Mechanics, Schaum's Outline Series, McGraw-Hill, 1967
3) G.L. Kotkin, V.G. Serbo: Collection of Problems in Classical Mechanics