Introduction to the Fractional Calculus of Variations

This invaluable book provides a broad introduction to the fascinating and beautiful subject of Fractional Calculus of Variations (FCV). In 1996, FVC evolved in order to better describe non-conservative systems in mechanics. The inclusion of non-conservatism is extremely important from the point of view of applications. Forces that do not store energy are always present in real systems. They remove energy from the systems and, as a consequence, Noether's conservation laws cease to be valid. However, it is still possible to obtain the validity of Noether's principle using FCV. The new theory provides a more realistic approach to physics, allowing us to consider non-conservative systems in a natural way. The authors prove the necessary Euler–Lagrange conditions and corresponding Noether theorems for several types of fractional variational problems, with and without constraints, using Lagrangian and Hamiltonian formalisms. Sufficient optimality conditions are also obtained under convexity, and Leitmann's direct method is discussed within the framework of FCV.

The book is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students and ambitious undergraduates in mathematics and mechanics. It provides an opportunity for an introduction to FCV for experienced researchers. The explanations in the book are detailed, in order to capture the interest of the curious reader, and the book provides the necessary background material required to go further into the subject and explore the rich research literature.

Contents:

• The Classical Calculus of Variations
• Fractional Calculus of Variations via Riemann–Liouville Operators
• Fractional Calculus of Variations via Caputo Operators
• Other Approaches to the Fractional Calculus of Variations
• Towards a Combined Fractional Mechanics and Quantization

Readership: Advanced undergraduate, graduate students and researchers in mathematics, physics, operations research and applied sciences.
Key Features:

• This is the first and the only book on Fractional Calculus of Variations
• The first comprehensive and unified mathematical approach to Fractional Calculus of Variations, making this recent and interesting subject accessible to a wider community
• This book contains the authors' research results and perspectives on Fractional Calculus of Variations