COURSE GOALS: The principle objectives of the course Classical Mechanics 1 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 1, students will be able:
- to formulate basic principles of classical mechanics including Newton's determinism, Galilean invariance and the laws of conservation of momentum, angular momentum and energy;
- to analyze the motion of a general mechanical system with one degree of freedom and solve Newton's equation analytically for a number of standard problems;
- to sketch possible trajectories of a particle moving in a central force field and solve Newton's equation analytically for several standard types of central forces, including the Kepler problem;
- to describe the Rutherford's experiment, derive analytical expressions for differential and total cross section and provide physical interpretation of these results;
- to formulate the variational principle, derive Euler-Lagrange equations and apply them to various physical systems, including those with forces of constraints;
- to formulate the D'Alambert's principle and apply it to problems of static equilibrium;
- to calculate the angular velocity and kinetic energy of an arbitrarily shaped rigid body;
- to construct the inertia tensor for several selected examples of rigid bodies (sphere, cube, cylinder, cone, triangle, etc.) and determine the principle axes of an arbitrarily shaped rigid body;
- to derive the Euler-Lagrange equations for an arbitrarily shaped rigid body and apply them in order to describe the motion of free and heavy symmetric top;
- to derive the Euler's equations for an arbitrarily shaped rigid body and apply them in order to describe the motion of free symmetric and asymmetric top.
COURSE DESCRIPTION:
- Introduction and historical development of classical mechanics. Space and time in classical mechanics. Galilean transformations. Newton's formulation of classical mechanics.
- The laws of conservation of momentum, angular momentum and energy. Conservative systems.
- Two body problem. Particle moving in a central force field. Kepler's problem. Scattering and Rutheford's experiment.
- Mechanical similarity and virial theorem.
- Variational principle. Lagrange's formulation of classical mechanics. Systems with constraints.
- D'Alambert's principle and conditions for static equilibrium.
- Rigid body kinematics. Fixed and moving system and the concept of angular velocity. Euler's angles. The kinetic energy and tensor of inertial for an arbitrary rigid body. The principle axis of an arbitrary rigid body.
- Rigid body dynamics. Euler-Lagrange equations for an arbitrary rigid body. Examples: cylinder rolling on a horizontal surface, free and heavy symmetric top.
- Time rate of change of a vector in the moving system. Euler's equations for an arbitrary rigid body. Examples: free symmetric and asymmetric top.
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.
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- 1) V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1991
2) L.D. Landau, E.M. Lifschitz: Mechanics, Buttenworth-Heinemann, 2001
3) H. Goldstein, C.P. Poole, J.L. Safko : Classical Mechanics 3rd Edition, Addison-Wesley Publishing Company, 2001
- 1) Spiegel M.R.: Theoretical Mechanics, Schaum's Outline Series, McGraw-Hill, 1967
2) G.L. Kotkin, V.G. Serbo: Collection of Problems in Classical Mechanics
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