Dynamic Geometry and Gravitational Energy

The Hamiltonian of dynamic geometry as it is applied to gravitational energy is examined in this modern but classical approach to the subject. It is a comprehensive development of the variational principle for general dynamic fields, which is applied in particular to dynamic geometry, and results in an improved description of gravitational energy. Indeed, using differential forms and a first-order Lagrangian, a covariant Hamiltonian formalism is presented here in complete form for the first time. Unique, outstanding features include: A systematic, user-friendly, self-contained exposition with numerous examples * Essential tools and principles are given for differential forms, differential geometry, variational principles, Noether theorems, the gauge theories of fundamental interactions, as well as theories of gravity * Focus on the symplectic structure provides new possibilities, insights and ways to solve some outstanding problems * Key topics: differential geometry, the gauge theory of interactions, variational principles for differential forms, a covariant Hamiltonian formulation, the boundary symplectic structure, and gravitational energy: its positivity and localization * Prerequisites: complex variables, linear algebra and some basic ideas of physics: classical mechanics (including the least action principle and Hamiltonian mechanics), electrodynamics, and special relativity. An understanding of classical field theory is desirable but not essential A broad audience of advanced undergraduate and graduate students, as well as researchers, in mathematics and physics who are interested in the theoretical principles and mathematical formulation of fundamental physical interactions, including gravity as a manifestation of dynamic spacetime geometry, variational principles and symplectic structure, the canonical Hamiltonian analysis and gravitational energy will find this a stimulating text, full of challenging ideas at the frontier of present-day research.