Modern Problems in Calculus of Variations
Office: JWB 225
Every problem of the calculus of variations has a solution,
provided that the word `solution' is suitably understood.
The course introduces classical methods of Calculus of Variations, Legendre
transform, conservation laws and symmetries. The attention is paid to variational
problems with unstable (highly oscillatory) solutions, especially in multidimensional
problems. These problems arrive in large number of applications, including
structural optimization, phase transitions, composites, inverse problems,
etc., where an optimal layout are characterized by short scale inhomogenuities:
patterns of unknown shapes. We discuss methods of effective description
of such solutions. They are called relaxation methods and are based on
theory of quasiconvexity.
Topics to be covered:
Clear and elegant methods of modern Calculus of Variations allow to solve
large number of problems in Science and Engineering. Originated by Bernoulli,
Newton, Euler, and systematically developed beginning from XVIII century,
these days Calculus of Variations attracts attention of mathematicians
and provides new tools to find the best possible solutions, and to understand
the essence of optimality.
Basic techniques of Calculus of Variations. Euler equations, Legendre and
Weierstrass tests, direct methods.
Noether theory: symmetries and invariants.
Duality and Legendre transform. Two sides bounds.
Traditional and novel variational principles.
Ill-posed variational problems for multiple integrals:
Quasiconvexity of Lagrangian and existence of a solution; relaxation of
Translation method (sufficient conditions),
Weierstrass-type conditions (necessary conditions),
Addressed to graduate and to senior undergraduate students in math and
Instructors notes (will be distributed).
Robert Weinstock. Calculus of Variations with Applications to Physics and
Engineering. Dover, 1974.
Charles Fox. An Introduction to the calculus of variations. Dover, 1987.
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The main question of the classical theory of Calculus of Variation is:
What curve or surface is in a sense `the best one'? For example,
In these problems, the extremal property is attributed to an entire curve
(function). A group of methods aimed to find `optimal' functions is called
Calculus of Variations.
What curve with the fixed length covers maximal area? (of course, the circle!)
What curve corresponds to the fastest glide along it? (the brachistochrone
What surface with a fixed boundary has the smallest area? (the minimal
What variable thickness of a plate maximizes its stiffness? (an optimal
From ancient times, geometers noticed extremal properties of symmetric
figures and bodies. The circle has maximal area among all figures with
fixed perimeter; the right triangular and the square have maximal area
among all triangles and quadrangles with fixed perimeter, respectively,
etc. However, regular proofs of these extremal properties usually are not
easy. Even more difficult task was to develop a regular theory that was
able to search for optimal curves and surfaces. This theory of extremal
problem has been actively developed for the last three centuries.
Calculus of Variations has been originated by Bernoulli, Newton, Euler;
systematically developed beginning from XVIII century; it still attracts
attention of mathematicians and it helps scientists and engineers to find
the best possible solutions. The study of Calculus of Variations was fruitful
for mathematics: it leaded to development of Analysis, Harmonic analysis,
Operator theory and distributions, and other important branches of math.
Modern branches of Calculus of Variations include Control theory, Optimal
design theory or Structural Optimization,
games theory, Operation research, Programming, etc. The goal of the theory
is to analyse extremal trajectories. In the last decades, interest to these
approaches grows thanks to advances in numerical methods: it becomes possible
to solve equations for extremals and to use them in many scientific and
Extremal problems are attractive due to human's natural desire to find
perfect solutions, they also root in natural laws of physics. The last
ones can often be formulated as extremal problems: the true trajectory
delivers minimum of a functional called `the energy' among all admissible
trajectories. In mechanics, a real motion minimizes the mechanical energy;
in thermodynamics, the dissipative processes maximizes the dissipation
rate, etc. This phenomenon looks mysterious, and it has caused a lot of
The course covers classical techniques of Calculus of Variations, discusses
natural variational principles in classical and continuum mechanics, and
introduces modern applications.
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