Selected Chapters in the Calculus of Variations Lecture Notes by Oliver Knill Birkhauser Verlag (original) (raw)

Lecture Notes on Calculus of Variations

The lecture notes consist of 53 lectures and are intended for senior undergraduate students. Topics convered include the first and second variations, the Euler–Lagrange equation, the Hamilton’s canonical system, the Hamilton-Jacobi equation, brachistochrone curve, geodesics, and necessary / sufficient conditions.

The Calculus of Variations

The variational principles of mechanics are firmly rooted in the soil of that great century of Liberalism which starts with Descartes and ends with the French Revolution and which has witnessed the lives of Leibniz, Spinoza, Goethe, and Johann Sebastian Bach. It is the only period of cosmic thinking in the entire history of Europe since the time of the Greeks. 1

En Route for the Calculus of Variations

2019

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. An optimal control is an extension of the calculus of variations. It is a mathematical optimization method for deriving control policies. The calculus of variations is concerned with the extrema of functionals. The different approaches tried out in its solution may be considered, in a more or less direct way, as the starting point for new theories. While the true “mathematical” demonstration involves what we now call the calculus of variations, a theory for which Euler and then Lagrange established the foundations, the solution which Johann Bernoulli originally produced, obtained with the help analogy with the law of refraction on optics, was empirical. A similar analogy between optics and mechanics reappears when Hamilton applied the principle of least action in mechanics which Maupertuis justified in the first instance, on the basis of the laws o...

Calculus of Variations in Mechanics and Related Fields

Journal of Optimization Theory and Applications

With Calculus of Variations-we know-we indicate a body of mathematical techniques aiming at determining existence and properties of functions on which certain classes of functionals attain optimal values, according to some prescribed criteria, and pertinent results. Our common belief that nature minimizes energy at constant entropy and maximizes entropy at constant energy-a belief not falsified so far by common experiments-is at roots of the continuous interest in calculus of variations, besides aesthetic evaluation of the theory per se. Companion motivation for the attention on such a topic is also the technological interest for designing objects with some optimal property-e.g., shape, strength, conductivity-under some constraints, or to get optimal control of processes, e.g., in mechanics or economy. Problems requiring recourse to calculus of variations techniques to be tackled emerge in several sectors, even in social sciences, but above all in mechanics (be it classical, quantum, or relativistic), condensed matter physics, chemistry, and else. The motion of a three-dimensional rigid body can be viewed as a geodetic curve (the one with minimal length) over the special orthogonal group, while perfect fluids move along geodetic paths over the special group of diffeomorphisms. We can also aim at controlling optimally the motion of certain systems, be them multi-rigid-bodies with flexible mutual constraints or continua suffering distributed strain, as, e.g., rods are. We may ask to find the minimal energy of an atom, a molecule, a thin film, or we may tackle optimality questions connected with chemical reactions, or we aim at printing and connecting microstructures, in order to obtain an artifact with some optimized properties, what we call a metamaterial. Also, we may be interested in optimal transportation of mass, charges, and their like. Energy minimization characterizes equilibrium configurations. When coupled with appropriate monotonicity conditions mimicking irreversible behavior, such a minimization procedure may allow us to describe classes of (rate-independent) dissipative processes, such as plastic flows, damage, some phase transitions, or nucleation of fractures, by adapting Ennio De Giorgi's idea of minimizing movements. The idea is to partition the time interval into finitely many sub-intervals, presuming to go from

Field theories in the modern calculus of variations

Transactions of the American Mathematical Society, 1988

Two methods of construction of fields of extremals ("geodesic coverings") in the generalized problem of Bolza are given and, as a consequence, sufficient conditions for optimality in a form similar to Weierstrass' are formulated. The first field theory is an extension of Young's field theory-"concourse of flights" for our problem; the other describes a nonclassical treatment of field theory which allows one to reject the "self-multiplier restriction".