CHAPTER VI - Partial Invariance

Abstract

This chapter discusses partial invariance. In the search for solutions of differential equations further possibilities arise if the property of solution invariance is not required. If the group H is given, then the orbit described by the manifold Ψ under the operation of all possible transformations from H is the smallest invariant manifold of the group H containing the given manifold Ψ. The rank ρ of the indicated orbit is called the rank of the manifold Ψ. Also, Ψ is provided with one more numerical characteristic, showing how much Ψ is not invariant relative to the group H. This number, denoted by δ and called the defect of invariance or just defect of the manifold Ψ, is defined as the difference between the dimension of the manifold Ψ's orbit and that of orbit and that of. It is essential that there be an explicit infinitesimal formula for the defect. The manifold orbit and that of is called partially invariant relative to the group H, if its orbit does not coincide with the entire space and its defect δ > 0.