Lie derivative

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 : ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle f}$

${\displaystyle {\mathcal {L}}_{X}f=Xf}$.

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 : ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {C}}^{\infty }}$ ${\displaystyle f\colon M\to \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {L}}_{X}f(p)}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle p\in M}$ ${\displaystyle f}$ ${\displaystyle X(p)}$

${\displaystyle {\mathcal {L}}_{X}f(p):=X_{p}(f)=d_{p}f(X(p))}$

The vector field can be represented in local coordinates as ${\displaystyle (x_{1},\ldots ,x_{n})\colon U\subseteq M\to \mathbb {R} ^{n}}$

${\displaystyle X=\sum _{j=1}^{n}X_{j}{\frac {\partial }{\partial x_{j}}}}$, with . ${\displaystyle X_{j}\colon U\to \mathbb {R} }$

The Lie derivative then results

${\displaystyle {\mathcal {L}}_{X}f(p)=\sum _{j=1}^{n}X_{j}(p){\frac {\partial f}{\partial x_{j}}}(p)}$.

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 ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle F_{t}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {L}}_{X}Y}$ ${\displaystyle Y}$ ${\displaystyle X}$

${\displaystyle {\mathcal {L}}_{X}Y=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}(F_{t}^{*}Y)}$,

which means the return of the river . ${\displaystyle F_{t}^{*}}$ ${\displaystyle F_{t}}$

properties

Lie bracket

If and are again two vector fields, then the identity holds for the Lie derivative ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle ({\mathcal {L}}_{X}Y)f=X(Y(f))-Y(X(f))}$,

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. ${\displaystyle f}$ ${\displaystyle (X,Y)\mapsto {\mathcal {L}}_{X}Y}$ ${\displaystyle [X,Y]:={\mathcal {L}}_{X}Y}$ ${\displaystyle [\cdot ,\cdot ]}$

Sometimes the Lie derivative or Lie bracket is defined directly by the term . Sometimes the reverse sign convention is also used. ${\displaystyle X(Y(f))-Y(X(f))}$ ${\displaystyle [X,Y]:=Y\circ X-X\circ Y}$

Local coordinates

The vector fields or have representations in local coordinates ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle X=\sum _{j=1}^{n}X_{j}{\frac {\partial }{\partial x_{j}}}}$

respectively

${\displaystyle Y=\sum _{j=1}^{n}Y_{j}{\frac {\partial }{\partial x_{j}}}}$.

The following then applies to the Lie derivative or Lie bracket

${\displaystyle [X,Y]=\sum _{j=1}^{n}\left(\sum _{k=1}^{n}X_{k}{\frac {\partial Y_{j}}{\partial x_{k}}}-\sum _{k=1}^{n}Y_{k}{\frac {\partial X_{j}}{\partial x_{k}}}\right){\frac {\partial }{\partial x_{j}}}\,.}$

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 ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle \Phi _{t}}$ ${\displaystyle T}$ ${\displaystyle X}$

${\displaystyle {\mathcal {L}}_{X}T={\frac {\mathrm {d} }{\mathrm {d} t}}\Phi _{t}^{*}T|_{t=0}\,.}$

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. ${\displaystyle {\mathcal {L}}_{X}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle X}$

In contrast to a relationship , non- linear is in . ${\displaystyle {\mathcal {L}}_{X}}$ ${\displaystyle {\mathcal {C}}^{\infty }}$ ${\displaystyle X}$

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 ${\displaystyle {\mathcal {C}}^{\infty }(M,\mathbb {R} )}$ ${\displaystyle M\to \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {L}}_{X}:{\mathcal {C}}^{\infty }(M,\mathbb {R} )\to {\mathcal {C}}^{\infty }(M,\mathbb {R} )}$

• ${\displaystyle {\mathcal {L}}_{X}}$is -linear ${\displaystyle \mathbb {R} }$
• ${\displaystyle {\mathcal {L}}_{X}(fg)=({\mathcal {L}}_{X}f)g+f{\mathcal {L}}_{X}g}$ (Leibniz rule)

Denote the set of all smooth vector fields on , then the Lie derivative is also a -linear derivation on , and we have: ${\displaystyle {\mathcal {X}}(M)}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {C}}^{\infty }(M,\mathbb {R} )\times {\mathcal {X}}(M)}$

• ${\displaystyle {\mathcal {L}}_{X}(fY)=({\mathcal {L}}_{X}f)Y+f{\mathcal {L}}_{X}Y}$ (Leibniz rule)
• ${\displaystyle [X,[Y,Z]]={\mathcal {L}}_{X}[Y,Z]=[{\mathcal {L}}_{X}Y,Z]+[Y,{\mathcal {L}}_{X}Z]=[[X,Y],Z]+[Y,[X,Z]]}$( Jacobi identity )

This becomes a Lie algebra . ${\displaystyle {\mathcal {X}}(M)}$

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 : ${\displaystyle M}$ ${\displaystyle {\mathcal {C}}^{\infty }}$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle \alpha \in \Lambda ^{k+1}(M)}$ ${\displaystyle (k+1)}$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle \alpha }$

${\displaystyle (i_{X}\alpha )(X_{1},\ldots ,X_{k})=(k+1)\alpha (X,X_{1},\ldots ,X_{k})\,}$

and receives the image:

${\displaystyle i_{X}:\Lambda ^{k+1}(M)\to \Lambda ^{k}(M),\;\alpha \mapsto i_{X}\alpha }$

This figure has the following properties:

• ${\displaystyle i_{X}}$is -linear, ${\displaystyle \mathbb {R} }$
• for any valid , ${\displaystyle f\in \Lambda ^{0}(M)}$ ${\displaystyle i_{fX}\alpha =fi_{X}\alpha }$
• for any differential form over and holds ${\displaystyle \beta }$ ${\displaystyle M}$ ${\displaystyle \alpha \in \Lambda ^{k}(M)}$

${\displaystyle i_{X}(\alpha \wedge \beta )=(i_{X}\alpha )\wedge \beta +(-1)^{k}\alpha \wedge (i_{X}\beta )}$.

The Lie derivative was defined above with respect to a vector field for functions via : ${\displaystyle X}$ ${\displaystyle M}$

${\displaystyle {\mathcal {L}}_{X}f=i_{X}df}$

For real differential forms, the Lie derivative with respect to a vector field is defined as follows: ${\displaystyle X}$

${\displaystyle {\mathcal {L}}_{X}\alpha =\left(i_{X}\circ d\ +d\circ i_{X}\right)\alpha }$.

It has the following properties:

• ${\displaystyle {\mathcal {L}}_{fX}\alpha =f{\mathcal {L}}_{X}\alpha +df\wedge i_{X}\alpha }$
• ${\displaystyle {\mathcal {L}}_{X}(\alpha \wedge \beta )=({\mathcal {L}}_{X}\alpha )\wedge \beta +\alpha \wedge ({\mathcal {L}}_{X}\beta )}$
• ${\displaystyle [{\mathcal {L}}_{X},{\mathcal {L}}_{Y}]\alpha :={\mathcal {L}}_{X}{\mathcal {L}}_{Y}\alpha -{\mathcal {L}}_{Y}{\mathcal {L}}_{X}\alpha ={\mathcal {L}}_{[X,Y]}\alpha }$
• ${\displaystyle [{\mathcal {L}}_{X},i_{Y}]\alpha =[i_{X},{\mathcal {L}}_{Y}]\alpha =i_{[X,Y]}\alpha }$

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 .