Courses

University of Aegean

Department of Mathematics

Graduate Level

Physics I (311-0562)

1st Week: Classification and evolution of the branches of Physics. Relation of Physics with Applied Sciences. The role of experiment in Physics. Mathematical Methods I: Functions of one variable, derivative, indefinite, definite integral and fundamental functions. Solving the second order, linear, non-homogeneous ordinary differential equations with constant coefficients.

2sd Week: Mathematical Methods II: Functions of several variables. Vectors in three-dimensional space. Addition, scalar multiplication, inner product, vector product and mixed product. Scalar and vector fields. The gradient of a scalar field, the divergence and curl of vector fields.

3rd Week: Orthogonal coordinate systems (Cartesian, polar, cylindrical and spherical coordinate systems). Parametrization of curves. Velocity and acceleration. General motion on the plane.

4th Week: Fundamental quantities (mass, space, time, electric charge) and dimensional analysis. Momentum, force, angular momentum, work and energy. Classification of the forces from macrocosmos to microcosmos.

5th Week: Newton's laws. Conservation theorems for linear momentum, angular momentum and energy. Stability and potential energy curves.
6th Week: Relative motion, velocity and acceleration. Galilean transformations. Lorentz's transformations and velocity.

7th Week: Central forces. Conservation of energy. Determination of the orbit from the central force. Conic sections.

8th Week: Oscillations. The simple harmonic oscillator. Energy of the simple harmonic oscillator. Damped and forced vibrations. The simple and physical pendulum.

9th Week: Gravity. Kepler's laws. Gravitational potential energy. Planetary motion.

10th Week: Discrete and continuous systems. Momentum, angular momentum, kinetic energy and potential energy of a system of particles. Center of mass. Motion relative to the center of mass. Conservation laws. Collisions.

11th Week: Rigid body. Translations and rotations. Angular velocity of the rigid body. Moments of inertia and theorems. Kinetic energy and angular momentum about a fixed axis. Motion of a rigid body about a fixed axis.

12th Week: Introduction to the special theory of relativity. Momentum, force and energy in special relativity. The classical limit of special relativity. Energy and momentum transformations.

13th Week: Fluids. Density and pressure. Variation of pressure with respect to the depth. Buoyancy and the Archimedes principle. Characteristic properties of flow. Continuity equation. Bernoulli's equation.

Physics II (311-1004)

1st Week: Electric charge. Coulomb's law. Discrete and continuous charge distributions. The electric field intensity. Flux lines. Quantization of electric charge. Motion of a point electric charge into a uniform electric field.

2sd Week: Electric flux. Gauss's law in integral and differential form. Applications to standard charge configurations. Conductors in electrostatic equilibrium.

3rd Week: Potential energy and electric potential for discrete and continuous charge distributions. The electric potential of a charged conductor.

4th Week: Current density, continuity equation and Ohm's law. A model of electric conductivity. Superconductors.

5th Week: Laplace and Poisson equations. Boundary conditions and the first uniqueness theorem. The methods of images and separation of variables.

6th Week: Conductors, insulators semiconductors. Charge induction, conductors in electrostatic field, the second uniqueness theorem. Applications to charged conductors.

7th Week: Magnetic field, magnetic force, motion of electric charge into a magnetic field. Ampère's law.

8th Week: Biot-Savard law, forces between parallel conductors, magnetic flux. Ferromagnetism, paramagnetism and diamagnetism.

9th Week: Electrodynamics and special theory of relativity. Transformation of electromagnetic field, electromagnetic tensor, invariance of electric charge.

10th Week: Electromagnetism and the principle of relativity, the electromagnetic field of a moving charge, interaction among moving charges.

11th Week: Faraday's law, Lenz's rule, displacement current, charge conservation and the Ampère-Maxwell law.

12th Week: Maxwell's equations in differential and integral form, scalar and vector potentials, magnetic monopoles.

13th Week: The wave nature of light, electromagnetic theory of light, the electromagnetic spectrum, speed of light, Doppler's phenomenon.

CLASSICAL MECHANICS (311-0266 )

1st Week: Newton equations of motion. Conservative forces. Conservation laws. Galilean transformations. Introduction to the calculus of variations (the notion of functional, variation of a functional, necessary conditions for the existence of extremals).

2sd Week: Examples in the calculus of variations.

3rd Week: An introduction to Lagrangian mechanics. The notion of generalized coordinates. Lagrangians and Hamilton’s principle. Euler-Lagrange equations.

4th Week: An introduction to symmetries. Noether’s theorem. Energy and momentum conservation. Problem and examples.

5th Week: The notion of constraints and Lagrange multipliers. Systems with moving constraints and non-conservation of energy. Euler-Lagrange equations for moving constraints. Examples.

6th Week: Hamiltonian Mechanics. Hamiltonian and canonical equations (Hamilton’s equations).

7th Week: Introduction to the notion of phase space. Phase plain, phase orbits, the notion of flow. The phase plain of conservative systems.

8th Week: The structure of the phase space for Hamilton’s canonical equations. Examples for Hamiltonian systems in the plane.

9th Week: An introduction to the notion of canonical transformations. Liouville’s theorem. The generating functions approach.

10th Week: An introduction to Hamilton-Jacobi (HJ) equations. The time-independent HJ equations. Comments on the notion of integrable systems.

11th Week: The notion of Poisson brackets and symplectic matrices. Examples.

12th Week: Applications.  

Postgraduate Level

Mathematical Physics (B1.1)

States of a system, dynamical laws and observables in Classical and Quantum Mechanics. Experimental evidence and interpretation according to Quantum Mechanics. The correspondence principle and implications.

Inner product and normed vector spaces. Orthogonal complements and direct sums. Orthonormal sets and sequences. Total orthonormal sets and sequences. Applications to Legendre, Hermite and Laguerre polynomials. Hilbert space. Representation of functionals on Hilbert spaces. Hilbert-adjoint, unitary and normal operators.

Unbounded linear operators in Hilbert space. Symmetric and self-adjoint operators. Closed linear operators. The multiplication and differentiation operators.

Von Neumann's axioms. Mean value and variance of observables. Ehrenfest's theorem. Schrödinger and Heisenberg representations.

Heisenberg's uncertainty principle and consequences. A refined uncertainty principle. Application to the hydrogen atom.

The classical and quantum harmonic oscillators. The phase space solution. Creation, annihilation operators and the energy spectrum. The virial theorem.

Schrödinger's equation in three spatial dimensions. Continuity equation. Applications to potential wells and scattering theory in more than two spatial dimensions.