Calculus of variations (I)

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

Heat Kernels for Elliptic and Sub-elliptic Operators, 2010

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...

Rendiconti del Seminario Matematico e Fisico di Milano, 1994

2013

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at

Nonlinear Dynamics and Systems Theory

Employing the contemporary theory of functional differential equations, we propose an effective test on the existence of a minimum for a wide class of functionals in various Banach spaces.

A foray into variational calculus and functionals. The mantelpiece of the subject, the Euler-Lagrange equation, is derived and applied to several canonical examples, namely Hamilton's principle. Hamilton's principle, expressed as the principle of least Action, is also derived, whose importance and power is demonstrated on examples in classical mechanics, and discussed in the context of general relativity and quantum field theory.

Reports on Mathematical Physics, 1992

We show that the homological algebra provides the most natural language for extension of the calculus of variations to multivalued functions.