This course is devoted to selected problems of classical (theoretical) and fluid mechanics which are usually remain outside the standard course of mechanics. Despite the fact that the course is aimed at students with an understanding of the methods and approaches of classical and fluid mechanics, several lectures of the course are devoted to revision of material from the course of classical (theoretical) mechanics.
The course is aimed at an audience interested in theoretical physics methods for solving problems of classical and fluid mechanics.
The course is designed for an audience that has previously attended general course of classical and fluid mechanics and courses of higher mathematics: mathematical analysis and differential equations theory.
The first week starts with the brief overview of the whole course, and continues with introduction to Lagrangian mechanics. We introduce the notion of constraints and discuss Lagrange equation of the first kind.
The second week is devoted to Lagrange equation of the second kind. We formulate a concept of generalized coordinates, introduce Lagrange function and action function, analyze the role of the least action principle in mechanics and derive Lagrange equation of the second kind from it.
Lagrangian Formalism: Specific Problems
The third week continuous the study of Lagrangian formalism. We investigate the general features of an open system dynamics by solving Kelly’s problem. We also analyze the connection between the symmetry of Lagrangian function and the conservation lows for the basic physical quantities, and formulate the main issue of dynamics in central force field.
The fourth week is devoted to scattering. We analyze general features of the scattering in the central field, and study the scattering on a solid sphere and Rutherford’s scattering. We also compare the main issues of the scattering for short-range and long-range potentials.
The theory of Oscillations
In the fifth week we investigate two interesting problems from the aria of oscillation theory: Kapitza pendulum and parametric resonance phenomenon.
Elements of Solid Mechanics
In the sixth week we deal with the dynamics of an asymmetric rigid body and investigate an unusual Dzhanibekov effect (intermediate axis theorem).
Hamiltonian Formalism and Canonical Transformations
In the seventh week we proceed to the Hamiltonian formalism in classical mechanics. We introduce Hamilton function, form the system of Hamiltonian equations and establish the connection between Lagrangian and Hamiltonian mechanics. We also investigate the issues of canonical transformations.
In eighth week we introduce Hamilton-Jacobi equation and analyze the role of the action function in the framework of Hamiltonian mechanics. In this week we also summarize our study of the general methods and approaches in classical mechanics.
In the ninth week we investigate the perturbation theory in classical mechanics, which is based on the canonical transformation theory.
Elements of Fluid Mechanics
The last tenth week is devoted to selected problems of fluid mechanics, such as shock waves and nozzles.