Inmathematics, adifferential invariant is aninvariant for theaction of aLie group on a space that involves thederivatives of graphs of functions in the space. Differential invariants are fundamental inprojective differential geometry, and thecurvature is often studied from this point of view.^[1] Differential invariants were introduced in special cases bySophus Lie in the early 1880s and studied byGeorges Henri Halphen at the same time.Lie (1884) was the first general work on differential invariants, and established the relationship between differential invariants, invariantdifferential equations, andinvariant differential operators.

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. Themoving frames method, which is a refinement ofÉlie Cartan'smethod of moving frames, gives several new powerful tools for finding and classifying the equivalence and symmetry properties of submanifolds, differential invariants, and their syzygies. Although, the moving frames method is less general than Lie's methods of differential invariants, it always yields invariants of the geometrical kind.

The simplest case is for differential invariants for one independent variablex and one dependent variabley. LetG be aLie group acting onR². ThenG also acts, locally, on the space of all graphs of the formy = ƒ(x). Roughly speaking, ak-th order differential invariant is a function

${\displaystyle I\left(x,y,\frac{dy}{dx},\dots ,\frac{d^{k}y}{d{x}^k}\right)}$

depending ony and its firstk derivatives with respect tox, 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 theprolongation of the group action. The action ofG on the first derivative, for instance, is such that thechain rule continues to hold: if

${\displaystyle (\overline{x},\overline{y})=g\cdot (x,y),}$

then

${\displaystyle g\cdot \left(x,y,\frac{dy}{dx}\right)\stackrel{\text{def}}{=}\left(\overline{x},\overline{y},\frac{d\overline{y}}{d\overline{x}}\right).}$

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 ofLie algebras and theLie derivative along theG action.

More generally, differential invariants can be considered for mappings from anysmooth manifoldX into another smooth manifoldY for a Lie group acting on theCartesian productX×Y. The graph of a mappingX → Y is a submanifold ofX×Y that is everywhere transverse to the fibers overX. The groupG acts, locally, on the space of such graphs, and induces an action on thek-th prolongationY^(k) consisting of graphs passing through each point modulo the relation ofk-th order contact. A differential invariant is a function onY^(k) that is invariant under the prolongation of the group action.

• Solving equivalence problems
• Differential invariants can be applied to the study of systems ofpartial differential equations: seekingsimilarity 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 avariational problem.
• Flow characteristics usingcomputer vision^[3]
• Geometric integration

• Cartan's equivalence method

• Guggenheimer, Heinrich (1977),Differential Geometry, New York:Dover Publications,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.
• Olver, Peter J. (1993),Applications of Lie groups to differential equations (2nd ed.), Berlin, New York:Springer-Verlag,ISBN 978-0-387-94007-6.
• Olver, Peter J. (1995),Equivalence, Invariants, and Symmetry,Cambridge University Press,ISBN 978-0-521-47811-3.
• Mansfield, Elizabeth Louise (2010),A Practical Guide to the Invariant Calculus, Cambridge University Press,ISBN 978-0-521-85701-7

• Invariant Variation Problems

• Differential geometry
• Invariant theory
• Projective geometry