Infinitesimal transformation

Encyclopedia

In mathematics

, an

form of

, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix

in space; but for small real values of a parameter ε we have

a small rotation, up to quantities of order ε

A comprehensive theory of infinitesimal transformations was first given by Sophus Lie

. Indeed this was at the heart of his work, on what are now called Lie group

s and their accompanying Lie algebra

s; and the identification of their role in geometry

and especially the theory of differential equation

s. The properties of an abstract Lie algebra

are exactly those definitive of infinitesimal transformations, just as the axioms of group theory

embody symmetry

. The term "Lie algebra" was introduced in 1934 by Hermann Weyl

, for what had until then been known as the

For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the cross product

, once a skew-symmetric matrix has been identified with a 3-vector. This amounts to choosing an axis vector for the rotations; the defining Jacobi identity

is a well-known property of cross products.

The earliest example of an infinitesimal transformation that may have been recognised as such was in Euler's theorem on homogeneous functions. Here it is stated that a function

with

a differential operator

. That is, from the property

we can in effect differentiate with respect to λ and then set λ equal to 1. This then becomes a necessary condition on a smooth function

considerations here). This setting is typical, in that we have a one-parameter group

of scalings operating; and the information is in fact coded in an infinitesimal transformation that is a first-order differential operator.

The operator equation

where

is an operator version of Taylor's theorem

— and is therefore only valid under

. Concentrating on the operator part, it shows in effect that

. In Lie's theory, this is generalised a long way. Any connected

Lie group can be built up by means of its infinitesimal generators (a basis for the Lie algebra of the group); with explicit if not always useful information given in the Baker–Campbell–Hausdorff formula.

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...

, an

**infinitesimal transformation**is a limitingLimit (mathematics)

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

form of

*small*transformation. For example one may talk about an**infinitesimal rotation**of a rigid bodyRigid body

In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix

Skew-symmetric matrix

In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...

*A*. It is not the matrix of an actual rotationRotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

in space; but for small real values of a parameter ε we have

a small rotation, up to quantities of order ε

^{2}.A comprehensive theory of infinitesimal transformations was first given 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 :...

. Indeed this was at the heart of his work, on what are now called 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...

s and their accompanying 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 identification of their role in geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

and especially the theory of 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. The properties of an abstract 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...

are exactly those definitive of infinitesimal transformations, just as the axioms of group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

embody symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

. The term "Lie algebra" was introduced in 1934 by Hermann Weyl

Hermann Weyl

Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

, for what had until then been known as the

*algebra of infinitesimal transformations*of a Lie group.For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the cross product

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

, once a skew-symmetric matrix has been identified with a 3-vector. This amounts to choosing an axis vector for the rotations; the defining Jacobi identity

Jacobi identity

In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...

is a well-known property of cross products.

The earliest example of an infinitesimal transformation that may have been recognised as such was in Euler's theorem on homogeneous functions. Here it is stated that a function

*F*of*n*variables*x*_{1}, ...,*x*_{n}that is homogeneous of degree*r*, satisfieswith

a differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

. That is, from the property

we can in effect differentiate with respect to λ and then set λ equal to 1. This then becomes a necessary condition on a smooth function

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

*F*to have the homogeneity property; it is also sufficient (by using Schwartz distributions one can reduce the mathematical analysisMathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

considerations here). This setting is typical, in that we have a one-parameter group

One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G...

of scalings operating; and the information is in fact coded in an infinitesimal transformation that is a first-order differential operator.

The operator equation

where

is an operator version of Taylor's theorem

Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...

— and is therefore only valid under

*caveats*about*f*being an analytic functionAnalytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

. Concentrating on the operator part, it shows in effect that

*D*is an infinitesimal transformation, generating translations of the real line via the exponentialExponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

. In Lie's theory, this is generalised a long way. Any connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

Lie group can be built up by means of its infinitesimal generators (a basis for the Lie algebra of the group); with explicit if not always useful information given in the Baker–Campbell–Hausdorff formula.