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 :
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
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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 :
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ mathcal {C}} ^ {\ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ed72bb2fb83c421d84887d252bbee98aa6eae3)
![f \ colon M \ to \ R](https://wikimedia.org/api/rest_v1/media/math/render/svg/b28c87bb68bdeda0931c6fa7210a19a7da47528a)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ mathcal {L}} _ {X} f (p)](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5adf061127a2d06e21c45b8d2d0ac79409edeb)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![p \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad2c18a15749505c928763cd4fdb56f4982816)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![X (p)](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7775144bb022929b79a824dcfebbbaaef9eed4)
![{\ mathcal {L}} _ {X} f (p): = X_ {p} (f) = d_ {p} f (X (p))](https://wikimedia.org/api/rest_v1/media/math/render/svg/30dc41506135770412678bd162a223124a9a33ad)
The vector field can be represented in local coordinates as
![{\ displaystyle (x_ {1}, \ ldots, x_ {n}) \ colon U \ subseteq M \ to \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e878909facc5cf993d45dda23c28ff9a6eb575f3)
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, with .![X_ {j} \ colon U \ to \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ded0f7d8c99316a562f20822c510be63757ab78)
The Lie derivative then results
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.
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
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ mathcal {L}} _ {X} Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1e6c305b263c2c55fdac7e57f69fbc6a505c4d)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
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,
which means the return of the river .
![F_ {t} ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c37e7ff30c060195f639c8711505316b90550501)
![F_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42cf9007bfa40c26216b648363be8f67e73d38cb)
properties
Lie bracket
If and are again two vector fields, then the identity holds for the Lie derivative
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
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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.
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![(X, Y) \ mapsto \ mathcal {L} _X Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/8554d8fa2006188911f9b3e57cccd7c4622dc301)
![[X, Y]: = \ mathcal {L} _X Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7a1865530b22d00954b64471970ed0cffee7b3)
![[\ cdot, \ cdot]](https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9)
Sometimes the Lie derivative or Lie bracket is defined directly by the term . Sometimes the reverse sign convention is also used.
![X (Y (f)) - Y (X (f))](https://wikimedia.org/api/rest_v1/media/math/render/svg/66cfc02b35bf84c597b307a967d36d7e36262213)
![[X, Y]: = Y \ circ XX \ circ Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff8e51063fa4f7ca3bda5fcc07695c70b381fbf)
Local coordinates
The vector fields or have representations
in local coordinates![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![X = \ sum _ {{j = 1}} ^ {n} X_ {j} {\ frac {\ partial} {\ partial x_ {j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d098f71cd82207188c83821659eb30b9a066efe5)
respectively
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.
The following then applies to the Lie derivative or Lie bracket
![[X, Y] = \ sum _ {{j = 1}} ^ {n} \ left (\ sum _ {{k = 1}} ^ {n} X_ {k} {\ frac {\ partial Y_ {j }} {\ partial x_ {k}}} - \ sum _ {{k = 1}} ^ {n} Y_ {k} {\ frac {\ partial X_ {j}} {\ partial x_ {k}}} \ right) {\ frac {\ partial} {\ partial x_ {j}}} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9ca85af5e09f8a0659c7d37e173e6a175860b6)
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
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![\ Phi_t](https://wikimedia.org/api/rest_v1/media/math/render/svg/6573886a4aa9f86bb43953426774b27682cf3e12)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ mathcal L} _ {X} T = {\ frac {{\ mathrm {d}}} {{\ mathrm {d}} t}} \ Phi _ {{t}} ^ {*} T | _ { {t = 0}} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/112eb185db5ffcead91e20581c92fb94df26b7f7)
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.
![\ mathcal L_X](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4db2d17b365a7321dbfdb8f8bc512dd911ea54)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
In contrast to a relationship , non- linear is in .
![\ mathcal L_X](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4db2d17b365a7321dbfdb8f8bc512dd911ea54)
![{\ mathcal C} ^ {\ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ed72bb2fb83c421d84887d252bbee98aa6eae3)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
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
![{\ mathcal {C}} ^ {\ infty} (M, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1a5413f922b163ca21f60044efe79586c89de8)
![M \ to \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d72976e9f605fb47c68a0b7824079f183aff7f53)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ mathcal {L}} _ {X}: {\ mathcal {C}} ^ {\ infty} (M, \ mathbb {R}) \ to {\ mathcal {C}} ^ {\ infty} (M, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/a846d8f155b1179b0cb85638e9f20128f30104d1)
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is -linear![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
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(Leibniz rule)
Denote the set of all smooth vector fields on , then the Lie derivative is also a -linear derivation on , and we have:
![{\ mathcal {X}} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7825afa749a3316e9ae61baee20f11da2ea4edd5)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![{\ mathcal {C}} ^ {\ infty} (M, \ mathbb {R}) \ times {\ mathcal {X}} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f48e248dd4c9e7626773bc1ae88452855136ac)
-
(Leibniz rule)
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( Jacobi identity )
This becomes a Lie algebra .
![{\ mathcal {X}} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7825afa749a3316e9ae61baee20f11da2ea4edd5)
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 :
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ mathcal {C}} ^ {\ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ed72bb2fb83c421d84887d252bbee98aa6eae3)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![\ alpha \ in \ Lambda ^ {{k + 1}} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8af1f1c608cbc8094b2932e726be969414bc23c0)
![(k + 1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9f13644a6be482d7ddb19a6e0c706564773085)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![(i_ {X} \ alpha) (X_ {1}, \ ldots, X_ {k}) = (k + 1) \ alpha (X, X_ {1}, \ ldots, X_ {k}) \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/108cf7a3ff913d05f01ce19aa1938bba0dbbbe9f)
and receives the image:
![i_ {X}: \ Lambda ^ {{k + 1}} (M) \ to \ Lambda ^ {k} (M), \; \ alpha \ mapsto i_ {X} \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1c24504ac524aa09759f6ceb71db8f4eea17c8)
This figure has the following properties:
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is -linear,![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
- for any valid ,
![f \ in \ Lambda ^ {0} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddaa86bb47a89962f320c194c81df024f236e572)
![i _ {{fX}} \ alpha = fi_ {X} \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9d0c4959e52de27d45f410acaac63fc0d329db)
- for any differential form over and holds
![\beta](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![\ alpha \ in \ Lambda ^ {k} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5416839db771bffc18606a8245743d96bf4428)
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.
The Lie derivative was defined above with respect to a vector field for functions via :
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ mathcal {L}} _ {X} f = i_ {X} df](https://wikimedia.org/api/rest_v1/media/math/render/svg/b950989979e69f56a55bc0a80782f2b320dc632e)
For real differential forms, the Lie derivative with respect to a vector field is defined as follows:
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
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.
It has the following properties:
![{\ mathcal {L}} _ {{fX}} \ alpha = f {\ mathcal {L}} _ {X} \ alpha + df \ wedge i_ {X} \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb5426f747424b7c7c8305c26c600b6c4adea8a)
![{\ mathcal {L}} _ {X} (\ alpha \ wedge \ beta) = ({\ mathcal {L}} _ {X} \ alpha) \ wedge \ beta + \ alpha \ wedge ({\ mathcal {L }} _ {X} \ beta)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d53bf16e98b4884717bd224683244775cf75de10)
![[{\ mathcal {L}} _ {X}, {\ mathcal {L}} _ {Y}] \ alpha: = {\ mathcal {L}} _ {X} {\ mathcal {L}} _ {Y } \ alpha - {\ mathcal {L}} _ {Y} {\ mathcal {L}} _ {X} \ alpha = {\ mathcal {L}} _ {{[X, Y]}} \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/65dc5117a4e5efcb8122cc7ccf43221086ae68db)
![[{\ mathcal {L}} _ {X}, i_ {Y}] \ alpha = [i_ {X}, {\ mathcal {L}} _ {Y}] \ alpha = i _ {{[X, Y] }}\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b537fbea4b8b073cead77755ab63f08658de91c1)
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
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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 .