Lie derivative

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In analysis , the Lie derivative (according to Sophus Lie ) denotes the derivative of a vector field or, more generally, a tensor field along a vector field. In the space of the vector fields, the Lie derivative defines a Lie bracket , which is called a Jacobi-Lie bracket . This operation turns the space of the vector fields into a Lie algebra .

In general relativity and in the geometric formulation of Hamiltonian mechanics , the Lie derivative is used to reveal symmetries , to exploit them to solve problems and, for example, to find constants of motion.

Lie derivative for functions

If a vector field , then the Lie derivative of a differentiable function is the application of to :

.

More precisely: Let there be a -dimensional -manifold, a smooth function and a smooth vector field . The Lie derivative of the function to in point is defined as the directional derivative of to :

The vector field can be represented in local coordinates as

, with .

The Lie derivative then results

.

Lie derivative of vector fields

definition

Let and be two vector fields at the -dimensional smooth manifold and the flow of the vector field . Then the Lie derivative of in direction is defined by

,

which means the return of the river .

properties

Lie bracket

If and are again two vector fields, then the identity holds for the Lie derivative

,

where is a smooth function on an open subset of the underlying manifold. From this equation it can be shown that satisfies the properties of a Lie bracket . Therefore one also writes . In particular, the set of vector fields with the Lie derivative forms a Lie algebra and its Lie bracket is called a Jacobi-Lie bracket.

Sometimes the Lie derivative or Lie bracket is defined directly by the term . Sometimes the reverse sign convention is also used.

Local coordinates

The vector fields or have representations in local coordinates

respectively

.

The following then applies to the Lie derivative or Lie bracket

Lie derivative of tensor fields

definition

For a tensor field and a vector field with local flow the Lie derivative is with respect to defined as

properties

The Lie derivative is -linear in and for solid a derivation of the tensor algebra , which is compatible with the contraction. The Lie derivative is already clearly characterized by this and by its values ​​on functions and vector fields.

In contrast to a relationship , non- linear is in .

Properties and Lie Algebra

The vector space of all smooth functions is an algebra with respect to pointwise multiplication . The Lie derivative with respect to a vector field is then a -linear derivation , i.e. i.e., it has the properties

  • is -linear
  • (Leibniz rule)

Denote the set of all smooth vector fields on , then the Lie derivative is also a -linear derivation on , and we have:

  • (Leibniz rule)
  • ( Jacobi identity )

This becomes a Lie algebra .

Definition of the Lie derivative on differential forms

Let be a -manifold, a vector field on and a - differential form on . Through evaluation one can define a kind of inner product between and :

and receives the image:

This figure has the following properties:

  • is -linear,
  • for any valid ,
  • for any differential form over and holds
.

The Lie derivative was defined above with respect to a vector field for functions via :

For real differential forms, the Lie derivative with respect to a vector field is defined as follows:

.

It has the following properties:

literature

  • Uwe Storch , Hartmut Wiebe: Textbook of Mathematics. Volume 4: Analysis on Manifolds - Function Theory - Functional Analysis. Spectrum, Heidelberg 2001, ISBN 3-8274-0137-2 .
  • Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine: Riemannian Geometry. 2nd Edition. Springer-Verlag, Berlin / Heidelberg 1990, ISBN 3-540-52401-0

Individual evidence

  1. ^ R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 , pp. 277-279.
  2. Anthony M. Bloch: Nonholonomic mechanics and control . Springer, New York 2003, ISBN 0-387-95535-6 , pp. 87 .