Abstract: When a mechanical or hydrodynamic system, with n degrees of
freedom, is modeled in terms of the CartanHilbertLagrange differential 1form
of Action, including anholonomic differential constraints, the resulting
topological domain is a 2n+2 dimensional noncompact 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
Processes represented by the unique Torsion field are irreversible in a
thermodynamic sense. In addition to supporting noncanonical momentum, the
symplectic domain must support anholonomic differential fluctuations, dvadt <>
0 in the velocity and/or in position, dxvdt <> 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 nonzero temperature gradient
and/or a nonzero 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 NavierStokes equations. Using Cartan's
magic formula, it is possible to deduce a simple nonstatistical 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 1form, Q, which is nonintegrable 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
