By Magri's Theorem, Self-Dual Gravity is Completely Integrable

By Magri's theorem the bi-Hamiltonian structure of Plebanski's second heavenly equation proves that (anti)-self-dual gravity is a completely integrable system in four dimensions.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Covariant Symplectic Structure and Conserved Charges of Topologically Massive Gravity

We present the covariant symplectic structure of the Topologically Massive Gravity and find a compact expression for the conserved charges of generic spacetimes with Killing symmetries.Comment: 10 pages, Dedicated to the memory of Yavuz Nutku (1943-2010), References added, Conserved charges of non-Einstein solutions of TMG are added, To appear in Phys. Rev.

AN INTEGRABLE FAMILY OF MONGE-AMPERE EQUATIONS AND THEIR MULTI-HAMILTONIAN STRUCTURE

We have identified a completely integrable family of Monge-Ampère equations through an examination of their Hamiltonian structure. Starting with a variational formulation of the Monge-Ampère equations we have constructed the first Hamiltonian operator through an application of Dirac's theory of constraints. The completely integrable class of Monge-Ampère equations are then obtained by solving the Jacobi identities for a sufficiently general form of the second Hamiltonian operator that is compatible with the first

Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems

Geometry and integrability

Articles from leading researchers to introduce the reader to cutting-edge topics in integrable systems theory