Envelopes, Non-Uniqueness, and Topological Torsion

A family of surfaces may or may not have an envelope. The shock cone is an example of an envelope of a set of translating and expanding Huygen spheres. At the point of contact, the envelope and the spherical wavefront represent a multiple (non-unique) solution to the problem of supersonic flow. The key idea is that of non-uniqueness. Non-uniqueness implies either a point of intersection or a point of tangency. An envelope is built from the points of tangency.

Cartan's methods can be applied to such problems, for consider a family of implicit surfaces given by the zero set of some function F(x,y,z; t), with parameter t. Then construct the 1-form which is equal to the perfect differential of the given function, and subtract from it those terms which multiply the differential dt. The resulting 1-form,

A = dF-Ftdt

need not be exact, nor closed, nor integrable. If A^dA = 0, the Frobenious integrability conditions for uniqueness are satisfied for the equation A = 0, and no envelope exists.

However, when A^dA # 0, the Torsion Vector is not zero, and the conditions for uniqueness are not satisfied. The equation A = 0 then does not have a unique solution, and an envelope can exist.

If Ftt = 0, and A^dA # 0, intersections of the family can occur, but to obtain an envelope it is necessary that both Ftt # 0, and A^dA # 0.

The concept of an envelope also can be associated with the Jacobian neighborhood of any vector field representing an evolutionary process. The family of surfaces is given by the Hamilton characteristic polynomial, which is globally zero for every Jacobian matrix. The parameter of the family is the eigenvalue parameter. Of particular significance are those polynomials which are homogeneous of degree 1, for then the surfaces can be put into correspondence with the primitive equilibrium thermodynamic surfaces of Gibbs. Such a method gives an explanation for the universal behavior of the Van der Waals gas, and the chemical engineering law of corresponding states.


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Last update 01/23/2009
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