R. M. Kiehn Physics Department, University of Houston

Abstract : Recent activity in topological classifications of closed symplectic integrable Hamiltonian systems focuses attention on those properties of a Lagrangian formulation for which the fundamental 2-form is exact. The Lagrangian formulation, based on a Cartan-Hilbert Action which has n degrees of freedom, leads to an unconstrained symplectic system which is dissipative and of dimension 2n+2. Canonical momentum constraints lead to a contact submanifold of dimension 2n+1 with a unique extremal field. If the 2n+2 symplectic system is to exist, it is necessary that the momenta are not defined canonically and that there must exist anholonomic differential fluctuations delta v = dv -­ Adt <> 0 in the velocity and/or in position, delata x = dx - V dt <> 0. The implication is that (non-extremal) evolution on the 2n+2 symplectic domain can be dissipative but the process is not described kinematically in terms of a single parameter group. The fluctuations in velocity lead to non-zero temperature gradients and the fluctuations in position lead to non-zero pressure gradients. Both types of fluctuations lead to distinct contributions to a zero point energy. These 2n+2 domains can act as a source of magnetic dynamo action in a plasma, where velocity ĝuctuations associated with temperature produce a charge acceleration mechanism in regions where E . B <> 0: Anholonomic direrential fluctuations in position lead to the dissipative terms in the Navier-Stokes equations. Using the fact that Cartan's Lie derivative of the Action with respect to a vector field V is a cohomological equivalent to the First Law of Thermodynamics, it is possible to decide if a given process V is irreversible or not. On the 2n+2 symplectic domain, defined as Thermodynamic Space, two distinct evolutionary processes may be defined in terms of the Adiabatic Vector and the Torsion Current. The first process is a symplectomorphism, and therefore is reversible; the second process is not a symplectomorphism, and is irreversible in a thermodynamic sense.

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