# Mathematical methods in Physics. I (5 hrs/w)

(one-semester course)

Prof. Lipovka A.A.

1. Tensors. (2 weeks)
• Definitions
• Properties.
• Fundamental tensor. Examples for curved spaces.
• Co- and contra-variant tensors. Geometric interpretation.
• Dual tensors, invariant values.
• Geodesic lines.
• Christoffel symbols.
• Parallel shift and covariant derivative.
• Contravariant derivative. Equations for geodesic line.
• Generalization of differential operators to the case of N-space.
• Curvature tensor (the tensor of Riemann – Christoffel).
• Identity of Bianchi.
• Ricci tensor, Ricci scalar, Einstein tensor.
• Problem solving, midterm exam.
2. The calculus of variations based on the tensors formalism. (2 weeks)
• Introduction
• The brachistochrone problem.
• Euler’s method.
• Classical mechanics, the laws of conservation. Lagrange function.
• Action for a free particle. Lagrange function.
• Action for a charged particle in the electromagnetic field. Lagrange function.
• Derivative of Lagrange – Euler.
• Field equations. Lagrangian function, the energy – momentum tensor.
• Examples (method applied to scalar, vector, tensor fields).
• Problem solving, midterm exam.
3. Typical problems in physics, based on 1-st and 2-nd parts. (1 week)
• Differential operators in curved coordinates. Lamé coefficients as components of the metric tensor, operators Δ and ▼ in curved spaces
• Equation of vibrations for a string, membrane, bar. (deduction)
• Equation of diffusion and heat flow. Border conditions. (deduction)
• Equation of the long line (telegraph equation). (deduction)
• Electrostatic equation. (deduction)
• Ideal liquid dynamics equations. (deduction)
• Equations of the elasticity theory. (deduction)
• Classification of differential equations.
• Dirichlet problem for Poisson equation, conditions for unique solution.
• Neumann’s problem for Poisson’s equation, conditions for a unique solution.
• Problem with type III border conditions.
• Laplace’s external problem.
• Boundary conditions of IV-th type. Typical problems.
4. Methods of solution. (2 weeks)
• The general solution structure for the partial differential equation.
• Problem of an infinite string (Method of d’Alembert).
• Separation of variables (Fourier method).
• Infinite plate cooling problem. Evaluation of convergence of the solution.
• Problem of a finite string.
• Dirichlet’s problem for a circle. Solution with the Poisson integral.
• External Dirichlet problem for a circle.
• Frobenius method.
• Example 1 – Legendre equation.
• Example 2 – Bessel equation.
5. Sturm-Liouville problem (2 weeks)
• Hilbert space, its properties, operators, set of orthogonal functions.
• Gram-Schmidt orthogonalization process. Example: Legendre polynomials.
• Definition of SLP, operator properties.
• Sturm-Liouville problem regular inside the boundaries.
• Properties of eigenvalues and eigenfunctions.
• Lower boundedness of the spectrum of the Sturm-Liouville problem (SLP).
• Determination of the eigenvalues and eigenfunctions of the SLP.
• SLP. Relationship between the norm of eigenfunctions and the characteristic equation.
• Regular SLP with periodic conditions at the border. The foundations of the appearance of quantization in quantum mechanics.
• Singular Sturm-Liouville problem. Properties of eigenvalues and eigenfunctions.
• General scheme for the Fourier method in the case on N variables.
• Example 1 – Thermal flow.
• Example 2 – problem for spherical shell.
• Problem solving, midterm exam.
6. Non-homogeneous problems. Based on parts 3,4,5. (2 weeks)
• Reduction to the homogeneous problem. General scheme.
• Example 1 – infinite plate.
• Example 2 – infinite cylinder.
• Example 3 – stimulated vibrations of a string.
• Thermal problem for a ball with variable external temperature.
• Method of Finite Integral Transformations (FIT). (Scheme by G.A. Grinberg)
• Neumann’s problem for a rectangle.
• Bar vibration problem.
• A ball heating problem.
• Electrostatic problem for a sector of the circle. U (φ = 0) = Vr / a, 0 <φ <π / 2.
• Dirichlet’s problem for a sector of the circle. U (φ = 0, α.) = F (φ), 0 <φ <α.
• Dirichlet’s problem for a sector of the circle. General case.
• Problem solving, midterm exam.
7. Special functions. Based on parts 3,4,5,6 (3 weeks)
• Bessel functions. Definitions, the radius of convergence for the series.
• Recurring relationships.
• Linear dependence (independence) of the Bessel functions with negative and positive index.
• Weber (Neumann) functions Yν(z)
• Asymptotic of Bessel functions 1st and 2nd kind for small and infinite argument.
• Integral representation.
• Bessel functions of semi-integer order.
• Bessel equation with parameter.
• Some useful integrals with Bessel function.
• SLP based on the Bessel equation.
• Fourier-Bessel and Dini series
• Dirichlet problem for a finite cylinder.
• Thermal flow problem for an infinite cylinder.
• Modified Bessel functions. Properties, recurring relationships, particular cases, asymptotic.
• Mc-Donald functions.
• Dirichlet problem for a short cylinder.
• Inhomogeneous Neumann problem for a finite cylinder.
• Legendre polynomials. Definition, properties.
• Legendre equation, show that Pn is a solution.
• General solution of Legendre equation. Second solution.
• SLP based on Legendre equation.
• Internal Dirichlet problem for a sphere.
• External Dirichlet problem for a sphere.
• A sphere within an ideal liquid flow.
• A point charge near the surface of a conducting sphere.
• Associated Legendre polynomials.
• SLP based on associated Legendre functions.
• Problem solving, midterm exam.
8. Green’s function and its relationship with the FIT method. Based on part 5. (2 weeks)
• Generalized functions, Delta-Dirac function.
• Inhomogeneous differential equations. Green’s function method.
• Green’s function in quantum theory and perturbation theory.
• Solution in general case.
• Modified Green function.
• Type II Green function.
• Fundamental solution for the Laplace operator. Space dimension = 1,2,3.
• Relationship between Grinberg’s methods (FIT) and Green’s method.
• Problem solving, midterm exam.

Bibliography:

1. G.B. Arfken, H.J. Weber, “Mathematical methods for physicists” Elsevier Academic Press 2005
2. E. Butkov “Mathematical Physics” Adisson-Wesley, Readings, Massachusetts 1968
3. P.Dennery, A. Krzywicki “Mathematics for Physicists” Harper & Row, New York 1967
4. J.Mathews, R.L.Walker, “Mathematical Methods of Physics” Carnegie-Melon U 1984
5. P.M.Morse, H.Feshbach, ,“Methods of Theoretical Physics” V.1,2, McGraw Hill, New York 1953
6. A.J.Diaz, A.U.Araujo “Funciones Especiales” Universidad de Sonora, 2007
7. Robert D. Richtmyer, “Principles of advanced mathematical physics”, Spring-Verlag.