The linear algebra of (Dirac-) Jacobi geometry

This book studies the linear algebra ensuing from several types of geometric structures, such as:

• symplectic and contact structures, which become central to classical mechanics through the work of Vladimir Arnold in the 60s;

• Poisson, Jacobi and Dirac structures, which have been studied since the 70s by Andre Lichnerowicz, Alan Weinstein, Alexandre Kirillov, and many others;

• Dirac-Jacobi structures, which encompass all the above classical structures, introduced by Aissa Wade and, in a slightly more general setting, by Luca Vitagliano, one of the authors.

 

All these geometries originate in the Hamiltonian formulation of classical mechanics, as they appear naturally on the phase space of many mechanical systems, and are used to understand the equations of motion of such systems.

 

lt studies the linear algebra of such geometries, in the sense that it describes what happens at a single point on a manifold endowed with such a structure. This step is necessary for the deeper understanding of these geometries, and therefore the book will be very useful for any researcher who begin studying these geometric structures. Such beginning researchers will be also pleased: the book contains all definitions of the used notions, and all the proofs of the stated results. For more advanced researchers in the field, the book can serve as a reference to all these linear algebra results related to Jacobi geometry.