# 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