Differential invariant
In
Differential invariants are contrasted with geometric invariants. Whereas differential invariants can involve a distinguished choice of independent variables (or a parameterization), geometric invariants do not.
Definition
The simplest case is for differential invariants for one independent variable x and one dependent variable y. Let G be a Lie group acting on R2. Then G also acts, locally, on the space of all graphs of the form y = ƒ(x). Roughly speaking, a k-th order differential invariant is a function
depending on y and its first k derivatives with respect to x, that is invariant under the action of the group.
The group can act on the higher-order derivatives in a nontrivial manner that requires computing the prolongation of the group action. The action of G on the first derivative, for instance, is such that the chain rule continues to hold: if
then
Similar considerations apply for the computation of higher prolongations. This method of computing the prolongation is impractical, however, and it is much simpler to work infinitesimally at the level of Lie algebras and the Lie derivative along the G action.
More generally, differential invariants can be considered for mappings from any
Applications
- Solving equivalence problems
- Differential invariants can be applied to the study of systems of similarity solutions that are invariant under the action of a particular group can reduce the dimension of the problem (i.e. yield a "reduced system").[2]
- Noether's theorem implies the existence of differential invariants corresponding to every differentiable symmetry of a variational problem.
- Flow characteristics using computer vision[3]
- Geometric integration
See also
Notes
- ^ Guggenheimer 1977
- ^ Olver 1995, Chapter 3
- ISBN 90-481-4461-2
References
- ISBN 978-0-486-63433-3.
- Lie, Sophus (1884), "Über Differentialinvarianten", Gesammelte Adhandlungen, vol. 6, Leipzig: B.G. Teubner, pp. 95–138; English translation: Ackerman, M; Hermann, R (1975), Sophus Lie's 1884 Differential Invariant Paper, Brookline, Mass.: Math Sci Press.
- ISBN 978-0-387-94007-6.
- ISBN 978-0-521-47811-3.
- ISBN 978-0-521-85701-7