Abstract: When a mechanical or hydrodynamic system, with n degrees of freedom, is modeled in terms of the Cartan-Hilbert-Lagrange differential 1-form of Action, including anholonomic differential constraints, the resulting topological domain is a 2n+2 dimensional non-compact symplectic manifold. Imposition of the constraints of canonical momentum reduce the manifold to the familiar 2n+1 dimensional space of states. On this 2n+1 domain there exists a unique vector field that leaves the Action integral stationary. A unique extremal vector field, similar to that which describes reversible conservative processes on the
space of states, does not exist on the 2n+2 symplectic domain. However, there does exist a unique Torsion vector field which does not leave the Action integral stationary, but only conformally invariant.

Processes represented by the unique Torsion field are irreversible in a thermodynamic sense. In addition to supporting non-canonical momentum, the symplectic domain must support anholonomic differential fluctuations, dv-adt <> 0 in the velocity and/or in position, dx-vdt <> 0. The implication is that such dissipative evolution can not be described kinematically in terms of a single parameter group. Such symplectic systems admit a non-zero temperature gradient and/or a non-zero pressure gradient with a zero point energy. These domains can act as a source of magnetic dynamo action in a plasma, producing an acceleration mechanism in electromagnetic domains where E.B<>0. Anholonomic differential fluctuations lead to the dissipative Navier-Stokes equations. Using Cartan's magic formula, it is possible to deduce a simple non-statistical test to see if a given evolutionary processes is irreversible in a thermodynamic sense. For example, conformal processes in the direction of the Torsion vector produce a heat 1-form, Q, which is non-integrable and does not admit an integrating factor. Therefore such process are irreversible.

The symplectic manifold does admit vector fields that leave the Action integral stationary, but such fields are not unique and can lead to a hierarchy of stationary states. The subset of stationary symplectomorphisms are reversible in the thermodynamic sense described above. As the key feature of the turbulent state admitted to by everyone is irreversibility, turbulence cannot be represented by a symplectomorphism. Arguments are presented that turbulence cannot occur on a domain of less than four variables.

Poster at Marseille chaos workshop

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