Topology and Topological Evolution

In order to understand irreversible evolution it is necessary to develop a theory of topological evolution. The key to such a theory is the recognition that Cartan's magic formula is the "evolution operator". The rule of covariant transplantation is inadequate. Evolutionary mechanisms that are based upon groups can not describe a process that does not have an inverse! The reason that quantum mechanics does not describe the details of the transition from state a to state b is that it is founded on a Unitary evolution process. True, the group methods are useful, but they cannot be used to describe irreversibility.

When a physical system can be described by a system of differential forms ( in effect the usual starting point of the calculus of variations with constraints), a search can be made for those processes (singly parametrized vector fields) which leave all of the differential forms, or the integrals of differential forms, invariant. Such processes preserve topology. In certain processes, some of the differential forms are invariant while others are not. The invariant forms develop the concept of conservation laws.

Quite often in physical systems, the even dimensional differential forms are invariant while the odd dimensional differential forms can change under certain classes of evolutionary processes. For example, the 2-forms of charge (charge conservation), or vorticity (angular momentum conservation) lead to stronger conservation laws than the conservation of flux, energy, and torsion, which depend upon 1 and 3 dimensional differential forms. There are wider classes of processes that preserve the former, and a smaller class of processes that preserve both. In classical mechanics, for example, the Poincare invariants belong to the even dimensional class of differential forms.

It is remarkable that the statistical theory of fluctuations has its dual in the application of differential topology. The key idea is the concept of differential topological fluctuations. Why is it necessary to insist upon the topological constraint of a single parameter evolutionary process? Why should neighborhoods of a variety be topologically constrained such that

dx - Vdt = 0.

For example, in string theory, the displacement is constrained as a function of 2 parameters ( not 1).

Why not study the differential topological fluctuations given by the statement that

w = dx - Vdt <> 0.

Then define the "equilibrium" state by the subspace where w = 0. When such a mechanism is introduced into Lagrangian mechanics using the Lagrange multipliers, p,

Action A = L(x,V,t)dt + p.w

It might be thought that the maximum dimension of the domain is 3n+1. However, it can be shown that the maximum topological dimension (Pfaff dimension or class of the Action 1-form) is of dimension 2n+2 !!!

The topological dimension is reduced to 2n+1 when the Lagrange multipliers, p, are constrained to be equal to a jet of "canonical momenta" defined as the partial dervatives of L with respect to V.

With out such a restriction, in the 2n+2 dimensional space, there exists a unique "torsion" vector field proportional to the 2n+1 form A^dA^dA^dA... This vector (current) - to within a factor - can be used to describe evolutionary processes that are thermodynamically irreversible !!!

The irreversible processes decay (conformally) until the process approaches the domain where dA^dA^dA^dA... = 0. From then on the topological Pfaff dimension is 2n+1, and there exists a unique Hamiltonian vector field that leaves that Action integral a relative integral invariant. The process becomes conservative in a Bernoulli sense.

 Hopf Topological Torsion ... A^dA# 0 Magnetic Helicity and Jets from Rotating Stars!

 Falaco Solitons Cosmological Strings in a Swimming Pool!

 Continuous Topological Evolution