Cartan's Magic formula: The Lie derivative

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 Cartan.

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 virtual work.

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.

image As I have time I will replace this under construction
symbol with a link to a more detailed pdf or tex file download.

Copyright © CSDC Inc. All rights reserved.
Last update 07/19/2001
to HomePage