Differential invariant
Encyclopedia
In mathematics
, a differential invariant is an invariant
for the action
of a Lie group
on a space that involves the derivative
s of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry
, and the curvature
is often studied from this point of view. Differential invariants were introduced in special cases by Sophus Lie
in the early 1880s and studied by Georges Henri Halphen
at the same time. was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equation
s, and invariant 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. Élie Cartan
's method of moving frames is a refinement that, while less general than Lie's methods of differential invariants, always yields invariants of the geometrical kind.
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 algebra
s and the Lie derivative
along the G action.
More generally, differential invariants can be considered for mappings from any smooth manifold X into another smooth manifold Y for a Lie group acting on the Cartesian product
X×Y. The graph of a mapping X → Y is a submanifold of X×Y that is everywhere transverse to the fibers over X. The group G acts, locally, on the space of such graphs, and induces an action on the k-th prolongation Y(k) consisting of graphs passing through each point modulo the relation of k-th order contact. A differential invariant is a function on Y(k) that is invariant under the prolongation of the group action.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a differential invariant is an invariant
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
for the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
on a space that involves the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry
Projective differential geometry
In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties that are invariant under the projective group. This is a mixture of attitudes from Riemannian geometry, and the Erlangen program....
, and the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
is often studied from this point of view. Differential invariants were introduced in special cases by Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
in the early 1880s and studied by Georges Henri Halphen
Georges Henri Halphen
George Henri Halphen was a French mathematician. He did his studies at École Polytechnique . He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry...
at the same time. was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, and invariant 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. Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
's method of moving frames is a refinement that, while less general than Lie's methods of differential invariants, always yields invariants of the geometrical kind.
Definition
The simplest case is for differential invariants for one independent variable x and one dependent variable y. Let G be a Lie groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
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 algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s and the Lie derivative
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
along the G action.
More generally, differential invariants can be considered for mappings from any smooth manifold X into another smooth manifold Y for a Lie group acting on the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
X×Y. The graph of a mapping X → Y is a submanifold of X×Y that is everywhere transverse to the fibers over X. The group G acts, locally, on the space of such graphs, and induces an action on the k-th prolongation Y(k) consisting of graphs passing through each point modulo the relation of k-th order contact. A differential invariant is a function on Y(k) that is invariant under the prolongation of the group action.
Applications
- Differential invariants can be applied to the study of systems of partial differential equations: seeking similarity solutionSimilarity solutionIn fluid dynamics, a similarity solution is a form of solution in which at least one co-ordinate lacks a distinguished origin; more physically, it describes a flow which 'looks the same' either at all times, or at all length scales...
s that are invariant under the action of a particular group can reduce the dimension of problem (a "reduced system").
- Noether's theoremNoether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
implies the existence of differential invariants corresponding to every differentiable symmetry of a variational problemCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
.
- Flow characteristicsFluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
using computer visionComputer visionComputer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions... - Geometric integrationGeometric integratorIn the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.-Pendulum example:...