(in Polish) Zasady wariacyjne mechaniki 1000-1M26ZWM
1. Variational problems. Variational derivative. Euler–Lagrange equation.
a) Definition of a functional. The problem of minimizing a functional.
b) Gâteaux derivative, Fréchet derivative.
c) Stationary points of functionals and derivation of the Euler–Lagrange equations.
d) Classical variational problems, e.g. the brachistochrone problem, the isoperimetric problem.
e) The direct method in the calculus of variations.
2. Principles of least action in classical mechanics and their interrelations.
a) Definition of mechanical action. Hamilton’s principle.
b) Maupertuis’ principle and the reciprocal Maupertuis principle.
c) Mechanical–optical analogy. Fermat’s principle.
3. Variational methods in quantum mechanics.
a) The Schrödinger equation as an Euler–Lagrange equation.
b) Bound states, their relation to the calculus of variations, and the problem of stability of atoms and molecules.
c) Nonlinear models in the quantum many-body problem and their relation to variational methods.
Course coordinators
Prerequisites (description)
Learning outcomes
1. Knows the basic concepts of the calculus of variations, in particular functionals, variational derivatives, and Euler–Lagrange equations.
2. Understands the principles of least action in classical mechanics, their relationships, and their physical interpretations.
3. Is able to derive Euler–Lagrange equations for simple variational problems and solve them in selected cases.
4. Is able to apply variational methods to analyze selected problems in classical mechanics.
5. Understands the connection between the calculus of variations and the Schrödinger equation in quantum mechanics.
6. Is able to interpret basic problems concerning the existence of bound states and the stability of atomic systems in a variational framework.
Assessment criteria
Oral examination.
Bibliography
* I. M. Gelfand, S. V. Fomin: Calculus of Variations, Dover Publications.
* E. H. Lieb, M. Loss: Analysis (Chapters 10 and 11), AMS.
* M. Levi: Classical Mechanics with Calculus of Variations, AMS.
* A. Rojo, A. Bloch: The Principle of Least Action: History and Physics, Cambridge University Press.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes: