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Last update: T_KFES (29.04.2016)
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Last update: prof. RNDr. Jiří Podolský, CSc., DSc. (04.10.2011)
Classical mechanics of point particles in the Lagrangian and Hamiltonian formalisms. Kinematics and dynamics of rigid bodies (the inertia tensor, Euler's angles and equations). Oscillations of a string and solutions of the wave equation. Introduction to relativistic mechanics. Main topics: 1. Introduction and motivation 2. Lagrangian formalism and Lagrange's equations 3. Motion of planets and further applications 4. Hamilton's canonical equations and the Poisson brackets 5. Mechanics of rigid bodies 6. Wave equation and its solutions 7. Foundations of relativistic mechanics |
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Last update: doc. Mgr. David Heyrovský, Ph.D. (11.10.2017)
Zápočet je nutnou podmínkou účasti u zkoušky. K jeho dosažení studenti během semestru získávají body za řešení čtyř domácích úkolů a jedné zápočtové písemky. Za každý úkol lze získat maximálně 10 bodů a za zápočtovou písemku 60 bodů, maximální celkový počet je tedy 100 bodů. Podmínkou zápočtu je dosažení minimálně 60 bodů. Při dosažení 80 a více bodů studenti nemusí studenti řešit písemnou část zkoušky. Průběžné získávání zápočtových bodů vylučuje opakování zápočtu. |
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Last update: prof. RNDr. Jiří Podolský, CSc., DSc. (04.10.2011)
[1] J. Horský, J. Novotný, M. Štefaník: Mechanika ve fyzice, Academia, Praha, 2001
[2] J. Kvasnica a kol.: Mechanika, Academia, Praha, 1988.
[3] J. W. Leech: Klasická mechanika, SNTL, Praha, 1970.
[4] K. R. Symon: Mechanics, Addison-Wesley, Reading, 1971.
[5] L. D. Landau, E. M. Lifšic: Mechanika, Fizmatgiz, Moskva, 1958. |
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Last update: doc. Mgr. David Heyrovský, Ph.D. (06.10.2017)
Zkouška sestává z písemné a ústní části. Při dosažení minimálně 80 zápočtových bodů stačí složit ústní část zkoušky.
Písemná část bude sestávat ze tří příkladů z témat probraných na přednášce a procvičených na cvičení.
Ústní část zkoušky bude pokrývat sylabus předmětu v rozsahu probraném na přednášce. |
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Last update: prof. RNDr. Jiří Podolský, CSc., DSc. (04.10.2011)
Advantages of alternative formulations of some problems in physics. Illustrated by theories of gravity: Newton's gravitational force -> Poisson's equation (potential field) -> Einstein's equation (metric field, general relativity). Theoretical mechanics as formulation of Newton's laws of motion in various formalisms: for point masses, rigid bodies, and continuum. Motivation and outline of the course. Recalling the main ideas and principles of Newtonian mechanics. Limits of classical mechanics (relativistic and quantum mechanics). Lagrangian formalism and Lagrange's equations Generalized coordinates: don't use only (x,y,z). Occam's razor: don't use more coordinates than necessary. Configuration space: Zeno's paradox and independence of generalized velocities on generalized coordinates. Derivation of Lagrange's equations of the second kind. Lagrange's function L: cases without potential, with potential, with generalized potential (motion of a particle in a given electromagnetic field). Illustration: motion of a particle in the field of a central force. First integrals (cyclic coordinate -> conservation of the corresponding generalized momentum, explicit independence of L on time -> conservation of a generalized energy). Illustration: Binet's equation for motion in a central field. Motion of planets and further applications Kepler's problem: revolution of planets around the Sun. Derivation of Kepler's laws of planetary motion. Effective potential method. Comparison of classical and relativistic mechanics: motion around the Sun versus motion around a black hole, perihelion shift. Simplification of the problem of two bodies to motion of a single particle with reduced mass. The 3-body problem and celestial mechanics: a few words about deterministic chaos. Scattering of particles, the Rutherford formula for cross-section. Hamilton's canonical equations and the Poisson brackets Generalized momentum as a canonically conjugate variable. The concept of phase space with some illustrations (oscillator, damping, chaos). Hamiltonian function. Derivation of Hamilton's canonical equations. Illustrations of canonical equations (harmonic oscillator, particle in electromagnetic field). Importance of Hamiltonian formalism for quantum theory (the Schroedinger equation, the Feynman diagrams as an expansion of interaction Hamiltonian) and statistical physics (partition function). Definition, basic properties, and the algebra of Poisson's brackets. Their analogy with commutators in quantum mechanics. Mechanics of rigid bodies Recalling vectors and tensors in Euclidean space. Group of finite rotations and algebra of infinitesimal rotations. Definition of the vector of angular velocity. The rotation of a rigid body around fixed axis, the inertia tensor. Eigenvalues and eigenvectors, including an interpretation of the inertia ellipsoid. Kinetic energy of rotational motion. Euler's angles and Euler's kinematic equations. Euler's dynamical equations. Explicit examples: motion of a symmetrical gyroscope and symmetrical top. Wave equation and its solutions Transition from a finite system of point masses to a continuous system. Illustration: longitudinal and transverse oscillations of a string. Wave equation and basic methods of its solution: a) the d'Alembert method, b) separation of variables (normal modes, boundary and initial conditions, the Fourier analysis). Foundations of relativistic mechanics |