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Mathematical Aspects of Theoretical Physics

1. Analytical Dynamics

Lagrange’s form of dynamics. Newtonian laws of motion. D’Alembert’s principle. Virtual displacements. Lagrange’s equations of the first and the second type. Hamilton’s form of dynamics. Hamilton canonical equations.
Variational principles of Mechanics. Calculus of variations. Euler - Lagrange equations. Hamilton’s variational principle. Gauss variational principle.
Canonical transforms in Mechanics. Infinitesimal transforms. Poisson’s brackets. Lagrange brackets. Multiple Poisson brackets. Jacobi’s identity. Hamilton - Jacobi’s partial differential equation. Complete Integral. Integral invariants in Mechanics. Liouville’s equation.

2. Quantum Mechanics

Heisenberg’s uncertainty principle. Shroedinger’s equation. Eigenvalues and eigenfunctions. Wave function. Operators of physical quantities. Harmonic oscillator. Heisenberg’s matrix mechanics.
Fock’s representation in Quantum Mechanics. Creation and annihilation operators. Second quantization. Dirac’s form of Quantum Mechanics.
Stationary and Nonstationary Perturbation theories. Hartree- Fock approximation. Transition probabilities in Quantum Mechanics. Many - body systems.
Magnetic field. Spin operators. Pauli’s equation.

3. Classical Electrodynamics

Maxwell’s equations in integral and differential forms. Vector analysis. Gradient, Divergence and Curl vector operators. Electric charge conservation law. Continuity equation. Displacement current.
Scalar and Vector potentials of Elecromagnetic field. Equations for Electromagnetic potentials. Hertz’s vector. Gradient invariance of the equations for Electromagnetic field.
Wave equation. Electromagnetic waves and their properties.


4. Statistical Physics and Thermodynamics

Phenomenological Thermodynamics. Thermodynamic Potentials. Fundamental Gibbs’ equation. Thermodynamical identities.
Ensembles in Equilibrium Statistical Physics. Microcanonical, Canonical and Grand Canonical Ensembles. Distribution functions. Boltzman’s and Gibbs’ distributions. Statistical integrals in different Ensembles. Laplace’s transforms.
Fermi and Bose statistics of quantum gases. Fermi - Dirac integrals in Quantum Statistics.
Nonequilibrium systems. Density matrix. Liouville’s and von Neimann’s Equations for classical and quantum systems. Complete descriptions of a physical systems. Reduced description. Pauli’s Master Equation.
Introduction to Plasma Physics. Self-consistent Electromagnetic field. Vlasov’s Equation. Electromagnetic oscillations and waves in plasmas.
Bogolubov-Born-Green-Kirkwood-ivon hierarchy of equations.


Introduction to Mathematical Modeling

1. Rigid and soft mathematical models. Simple (intuitive) models of real world phenomena. Lanchester’s model of a battle. Logistic model of population.
Prey - Predator models of Lottka - Volterra. Models of competing species.
Multi-steps steering models.
2. Mathematical methods for mathematical modeling. Ordinary and Partial Differential Equations. Stochastic Differential Equations. Integral, Integrodifferential and Functional Equations.
3. Physical foundations of mathematical modeling. Statistical description of dynamical systems. Complete and reduced descriptions. Hierarchy of time scales. Rigorous verification of the possibility of the reduced description of complex dynamical systems. Memory function.
4. Measurements in physical systems. Brownian motion. Physical stage of mathematical modeling.
5. Phenomenelogical model of Quantum electron Fermi - liquid in normal metals. Properties of neutral Fermi - liquid. Zero sound. Coulumb systems. Dynamical screening of Coulumb interaction. Electron quasiparticles.
6. Mathematical models in biology and medicine. Mathematical models in immunology: immune response to infection. Mathematical model of laser-induced thrombosis phenomenon.


ORDINARY DIFFERENTIAL EQUATIONS

Ordinary differential equations. First-order equations. General solution. Initial value problem. Particular solution. Classification of ordinary differential equations. Highorder differential equations. Linear and nonlinear equations. “A general solution” and “the general solution”. The precise meaning of the word “solution”. Singular solution. General form of a differential equation. Normal form of a differential equation. Explicit solution, implicit solution.
First-order differential equations. Separable equations. First-order linear equations. Integrating factor. Method of undetermined coefficients. Substitution methods. Special equations. Logistic equation. Bernoulli equation. Lagrange equation. Clairaut equation. Homogeneous differential equations. Exact differential equations. Riccati equation. Existence and uniqueness of solutions. Direction fields and solution curves.
Linear equations of higher order. Second-order linear equations. General solution of linear equations. Existence and uniqueness theorem. Wronskians of solutions. Abel’s formulae. Reduction of order of a differential equation. Nonhomogeneous equations. Complementary function. Homogeneous differential equations with constant coefficients. Characteristic equation. Mechanical and Electromagnetic natural oscillations. Nonhomogeneous differential equations with constant coefficients. General solution. Particular solution. Method of undetermined coefficients. Variations of parameters. Forced oscillations and Resonance. Beats. Electrical circuits. The mechanical-electrical analogy. Method of complex function.
Power series methods. Power series. Analytic functions. Termwise differentiation of power series. Radius of convergence. Series solutions near ordinary points. Regular singular points. Irregular singular points. Translated series solutions. Method of Frobenius Series solutions. Bessel’s equations. Bessel function of zero-order of the first kind. Frobenius solutions. The exceptional cases. Reduction of order. The Logarithmic cases. Bessel function of zero order of the second type. The Gamma function. Bessel functions of the first kind. Bessel function identities. Applications of Bessel functions. Gauss’s hypergeometric function. The point of Infinity. Euler’s equation.
Fourier Series. Fourier coefficients. The mean convergence of Fourier Series. Bessel’s inequality. Parseval’s equation. The completeness of the sets orthonormal functions. Pointwise convergence. Periodic functions. Trigonometric Fourier Series. Gibb’s phenomenon. Even and odd functions. Even and odd extensions of the functions. Termwise differentiation and integration of Fourier Series. Dirichlet’s theorem. The convergence theorem. Fourier Series in a complex form. Formal Fourier Series solutions of differential equations. Boundary value problems. Eigenvalues and eigenfunctions. One-dimensional wave equation. Solution by Fourier method.
Fourier integral. Transition from Fourier Integral to Fourier Transform. Direct and Inverse Fourier Transforms. Dirac’s delta-function. Fourier Cosine and Sine Transforms.
Laplace Transform. Properties of Laplace Transform. Initial value problems. Passage from Fourier Transform to Laplace Transform. Mellin’s Transform. Duhamel’s Principle. Laplace Transform for integral and Integro-differential equations.
Systems of ordinary differential equations. Autonomous equations. Phase plane and phase trajectories. Phase portrait of a system. Stability theory. Elementary singularities. Critical points, their types, geometry and stability. Asymptotic stability. Nodes, saddle points, centers, spiral points. Linear and Nonlinear systems of differential equations. Bendixson’s negative criterion. Hartman — Grobman theorem. Nonelementary singularities.