Hamiltonian Extremal and Symplectic Physics

Cartan developed the idea, given a 1-form of Action describing a phsycial system, that if the Pfaff dimension was 2n+1 (state space) then there existed a unique vector field that had a null eigen value relative to the matrix of two forms, F=dA. Such processes defined by this unique vector field are said to be extremals (the variation of the action integral is zero along such directions.) The extremal vector fields have a Hamiltonian representation, for which the closed integral of Action (flux) is an evolutionary invariant and the open integrals of F are evolutionary invariants. The latter concept corresponds to Helmholtz conservation of vorticity theorem.

It the Pfaff dimension of the action 1-form is even, the extremal field does not exist (assuming maximal rank). However, there still can exist a Hamiltonian representation (only now the Hamiltonian generating function is called the Casimir-Bernoulli invariant) for the flow if the Virtual Work 1-form, W=i(V)dA = dT, is a perfect differential. An even more general situation that still preserves the open integrals of F=dA as an evolutionary invariant is given by those situations where the Work 1-form is closed, dW=0. Such cases define a symplectic evolutionary process. They are thermodynamically reversible, for Q^dQ = L(V)A^L(V)dA= 0.

 Extremal and Symplectic evolution

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