The integrable structure of the three-wave equations is discussed in the setting of geometric mechanics. Lie-Poisson structures with quadratic Hamiltonian are associated with the three-wave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigid-body or Euler equations is discusse...

Computers have emerged as essential tools in the modern scientific analysis and simulationbased design of complex physical systems. The deeply-seated abstraction of continuity immanent to many physical systems inherently clashes with a digital computer’s ability of storing and manipulating only finite sets of numbers. While there has been a number of computational techniques that proposed discr...

The molecular electrostatic potential inside the potassium channel protein from Streptomyces liVidans has been investigated using linear scaling semiempirical quantum chemical method, for a variety of geometries, with and without solvating water molecules. The results are compared with those given by a number of popular molecular mechanics force-fields. The difference between the quantum and mo...

In this paper we present a specialised algebraic manipulation package devoted to Celestial Mechanics. The system, called Piranha, is built on top of a generic and extensible framework, which allows to treat e ciently and in a uni ed way the algebraic structures most commonly encountered in Celestial Mechanics (such as multivariate polynomials and Poisson series). In this contribution we explain...

A structural similarity between Classical Mechanics (CM) and Quantum Mechanics (QM) was revealed by P.A.M. Dirac in terms of Lie Algebras: while in CM the dynamics is determined by the Lie algebra of Poisson brackets on the manifold of scalar fields for classical position/momentum observables Q/P , QM evolves (in Heisenberg’s picture) according to the formally similar Lie algebra of commutator ...

The basic concepts and equations of classical fluid mechanics are presented in the form necessary for the formulation of Newtonian cosmology and for derivation and analysis of a system of the averaged Navier-Stokes-Poisson equations. A special attention is paid to the analytic formulation of the definitions and equations of moving fluids and to their physical content.

Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics. MSC (2000): 37J05, 53A40, 53D99, 55R10, 70H99.

We present discrete analogues of Dirac structures and the Tulczyjew’s triple by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete analogues of implicit Lagrangian and Hamiltonian systems. In particular, this yields implicit nonholonomic Lagrangian and Hamiltonian integrators. We also in...

The orbit method of Kirillov is used to derive the p-mechanical brackets [25]. They generate the quantum (Moyal) and classical (Poisson) brackets on respective orbits corresponding to representations of the Heisenberg group. The extension of p-mechanics to field theory is made through the De Donder–Weyl Hamiltonian formulation. The principal step is the substitution of the Heisenberg group with...