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 :
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.
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
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,
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
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,
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
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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:
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(Leibniz rule)
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( 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:
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.
It has the following properties:
literature
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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
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^ 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.
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↑ Anthony M. Bloch: Nonholonomic mechanics and control . Springer, New York 2003, ISBN 0-387-95535-6 , pp. 87 .