In his book on Lecons sur les Invariant Integraux , Cartan introduces a
combination of the exterior derivative and the interior product which (so I am
told) was coined as the "Lie derivative" by Sledbodzinsky. Marsden calls this
Cartan's Magic formula. The formula is also attributed to Cartan's son, Henri
The Lie derivative (with respect to a contravariant vector field, V) acts on
exterior differential forms in the following way:
L(V)A = i(V)dA + d(i(V)A).
On 1-forms, the second term is a perfect differential, dU, and the first term is
a usually non-integrable 1-form, W = i(V)dA , defined herein as the 1-form of
Cartan's magic formula acting on a 1-form of Action, A, is thereby an abstract
equivalent to the first law of thermodynamic processes:
L(V)A = W + d(U) = Q.
The first law of thermodynamics is obviously a statement about cohomology. The
motivation for much of the work in this article is to interpret Cartan's magic
formula literally as an equivalent of the thermodynamic expression. The
interpretation permits a direct contact to be made between the laws of dynamical
systems and thermodynamic concepts and irreversibility. No statistical
assumptions are required.
A cyclically irreversible process will be one for which the 1-form of Heat, Q
does not admit an integrating factor. Namely, Q^dQ<>0.