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Last update: T_KG (16.05.2002)
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Last update: T_KG (11.04.2008)
Students will be acquainted with the mathematical methods that are commonly used in the classical theories of the gravitational, electrostatic and magnetostatic fields. |
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Last update: RNDr. Pavel Zakouřil, Ph.D. (05.08.2002)
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Last update: T_KG (11.04.2008)
Lecture + exercises |
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Last update: T_KG (19.01.2003)
Forces acting on a point mass. Conservative forces and their properties. Intensity and potential. Lines of force and equipotential surfaces. Special cases of conservative forces; central forces. 2. Newtonian potential Newton's law of universal gravitation; Coulomb's law. Superposition principle. Singularities of the field. Limits of the applicability of Newton's gravitational law and of Coulomb's law. 3. Potential of a system of point masses Two fixed point masses. Dipole. Two revolving point masses; Lagrange's libration points; tidal potential. Laplace's equation. Gauss's law. Self-energy of a system of point masses. 4. On approaches to solving the potential theory problems Basic formulae. Various expressions for potential. 5. Potential in an integral form Amplified statement of Newton's gravitational law. General formulae for intensity: volume, surface, and line distributions of masses or charges; volume and surface distributions of dipoles; double layer. General formulae for potential: Newtonian potentials; potentials of a magnetized body and of a double layer. Self-energy of a system. Capacity. Gauss's law and Poisson's equation. Two dimensional problems; logarithmic potential. 6. Legendre polynomials Generating function. Some properties of Legendre polynomials following from the generating function. Convergence of the corresponding series. Recurrence relations. Legendre's differential equation. Orthogonality and norm. Alternate definitions: Rodrigues' formula; Schlaefi integral. Legendre functions of the second kind. Relation to the solution of Laplace's equation. 7. Associated Legendre functions Expansion of the reciprocal distance of two arbitrary points into a series of Legendre polynomials. Associated Legendre functions. The addition theorem for Legendre polynomials. Expansion of the reciprocal distance into a series of spherical harmonics. 8. Newtonian potential in a form of a series Expansion of a gravitational potential into a series of spherical harmonics; physical interpretation of the first terms of the series. Expansion of a magnetostatic potential; Gauss's theory of the Earth's magnetic field; comparison of the expansions for the gravitational and geomagnetic fields. Expansion of an electrostatic potential; multipole expansion. Dipole. 9. Properties of the associated Legendre functions Differential equation. Integral properties. Integral properties of spherical functions. Computing the coefficients in the potential series from surface measurements. 10. Green's theorems Gauss's theorem. Green's preliminary formula. Green's first theorem. Gauss's integral. Green's second theorem. Harmonic functions and their properties. Applications to the study of the external gravitational potential. |