 LieDerivative - Maple Help

DifferentialGeometry

 LieDerivative
 calculate the Lie derivative of a vector field, differential form, tensor, or connection with respect to a vector field Calling Sequence LieDerivative(X, T) Parameters

 X - a vector field on a manifold M or a vector in a Lie algebra A T - a vector field, a Maple expression, a differential form or a tensor field on the manifold M or the Lie algebra A Description

 • If T is a Maple expression, then LieDerivative(X, T) is the directional derivative X(T) of T in the direction of the vector field X.
 • If T is a vector field, then LieDerivative(X, T) coincides with the Lie bracket [X, T] = LieBracket(X, T).
 • If T is a differential 1-form, then alpha = LieDerivative(X, T) is the 1-form defined by alpha(Y) = X(alpha(Y)) - alpha([X,Y]), where Y is any vector field on M.
 • The Lie derivative operator acts as a derivation with respect to both the wedge and tensor products.  If alpha and beta are differential forms and T and S are tensors, then LieDerivative(X, alpha &w beta) = LieDerivative(X, alpha) &w beta + alpha &w LieDerivative(X, beta), and LieDerivative(X, S &t T) = LieDerivative(X, S) &t T + S &w LieDerivative(X, T).
 • The Lie derivative of a differential form can also be calculated from the Cartan formula, LieDerivative(X, alpha) = ExteriorDerivative(Hook(X, alpha)) + Hook(X, ExteriorDerivative(alpha))
 • The Lie derivative of a connection nabla_Y(Z) is the type (1, 2) tensor field S = LieDerivative(X, nabla), defined (when viewed as mapping from pairs of vector fields to vector fields) by S(Y, Z) = LieDerivative(X, nabla_Y(Z)) - nabla_{LieDerivative(X, Y)}(Z) - nabla_X(LieDerivative(Y, Z)).
 • For the definition of the Lie derivative of these geometric objects in terms of the flow of the vector field X see, for example, Spivak page 207-208.
 • The Lie derivative of a tensor defined on a Lie algebra can also be computed.
 • The first argument also be a list of vectors. The second argument can be a list of a vectors, Maple  expressions, a differential forms or tensors.
 • This command is part of the DifferentialGeometry package, and so can be used in the form LieDerivative(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-LieDerivative. Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

First initialize a manifold M with local coordinates [x, y, z].

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$

Example 1.

First we calculate the Lie derivative of a function f and note that it agrees with the directional derivative f.

 > $X≔\mathrm{evalDG}\left(a\mathrm{D_x}+b\mathrm{D_y}+c\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{c}\right]\right]\right]\right)$ (1)
 > $\mathrm{LieDerivative}\left(X,f\left(x,y,z\right)\right)$
 ${a}{}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{b}{}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{c}{}\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right)$ (2)

Example 2.

First we calculate the Lie derivative of a vector field and check that it coincides with the Lie bracket.

 > $X≔\mathrm{evalDG}\left(x\mathrm{D_y}-{y}^{2}z\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{x}\right]{,}\left[\left[{3}\right]{,}{-}{z}{}{{y}}^{{2}}\right]\right]\right]\right)$ (3)
 > $Y≔\mathrm{evalDG}\left(y\mathrm{D_x}+{z}^{2}\mathrm{D_y}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{y}\right]{,}\left[\left[{2}\right]{,}{{z}}^{{2}}\right]\right]\right]\right)$ (4)
 > $\mathrm{LieDerivative}\left(X,Y\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]{,}\left[\left[{2}\right]{,}{-}{2}{}{{y}}^{{2}}{}{{z}}^{{2}}{-}{y}\right]{,}\left[\left[{3}\right]{,}{2}{}{{z}}^{{3}}{}{y}\right]\right]\right]\right)$ (5)
 > $\mathrm{LieBracket}\left(X,Y\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]{,}\left[\left[{2}\right]{,}{-}{2}{}{{y}}^{{2}}{}{{z}}^{{2}}{-}{y}\right]{,}\left[\left[{3}\right]{,}{2}{}{{z}}^{{3}}{}{y}\right]\right]\right]\right)$ (6)

Example 3.

First we calculate the Lie derivative of a differential form and check the result against Cartan's formula.

 > $X≔\mathrm{evalDG}\left({z}^{2}\mathrm{D_x}-y\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{z}}^{{2}}\right]{,}\left[\left[{3}\right]{,}{-}{y}\right]\right]\right]\right)$ (7)
 > $\mathrm{ω}≔\mathrm{evalDG}\left(y\mathrm{dx}&w\mathrm{dz}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{y}\right]\right]\right]\right)$ (8)
 > $\mathrm{LieDerivative}\left(X,\mathrm{ω}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{y}\right]\right]\right]\right)$ (9)
 > $\left(\mathrm{Hook}\left(X,\mathrm{ExteriorDerivative}\left(\mathrm{ω}\right)\right)\right)&plus\left(\mathrm{ExteriorDerivative}\left(\mathrm{Hook}\left(X,\mathrm{ω}\right)\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{y}\right]\right]\right]\right)$ (10)

Example 4.

We calculate the Lie derivative of a tensor field.

 > $X≔\mathrm{evalDG}\left({z}^{2}\mathrm{D_x}-y\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{z}}^{{2}}\right]{,}\left[\left[{3}\right]{,}{-}{y}\right]\right]\right]\right)$ (11)
 > $T≔\mathrm{evalDG}\left(z\left(\mathrm{D_x}&t\mathrm{dy}\right)&t\mathrm{dz}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{z}\right]\right]\right]\right)$ (12)
 > $\mathrm{LieDerivative}\left(X,T\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{-}{z}\right]{,}\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{y}\right]\right]\right]\right)$ (13)

Example 5.

We calculate the Lie derivative of the zero connection.

 > $X≔\mathrm{evalDG}\left({z}^{2}\mathrm{D_x}-{y}^{2}\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{z}}^{{2}}\right]{,}\left[\left[{3}\right]{,}{-}{{y}}^{{2}}\right]\right]\right]\right)$ (14)
 > $T≔\mathrm{Tensor}:-\mathrm{Connection}\left(0&mult\left(\left(\mathrm{D_x}&tensor\mathrm{dx}\right)&tensor\mathrm{dx}\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (15)
 > $\mathrm{LieDerivative}\left(X,T\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{2}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{1}{,}{3}\right]{,}{2}\right]\right]\right]\right)$ (16)

Example 6.

The Lie derivative with respect to a list of vectors can be calculated simultaneously.

 > $\mathrm{LieDerivative}\left(\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right],xyz\mathrm{Dz}\right)$
 $\left[{y}{}{z}{}{\mathrm{Dz}}{,}{x}{}{z}{}{\mathrm{Dz}}{,}{x}{}{y}{}{\mathrm{Dz}}\right]$ (17)

The Lie derivative of a list of tensors can be calculated simultaneously.

 > $\mathrm{LieDerivative}\left(\mathrm{D_x},\left[\mathrm{D_x}&t\mathrm{D_x},x\mathrm{D_x}&t\mathrm{Dy},{x}^{2}\mathrm{D_x}&t\mathrm{Dz}\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}{\mathrm{Dy}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}{2}{}{x}{}{\mathrm{Dz}}\right]\right]\right]\right)\right]$ (18)

Both arguments to LieDerivative can be lists.

 > $\mathrm{LieDerivative}\left(\left[\mathrm{D_x},\mathrm{D_y}\right],\left[\mathrm{D_x},x\mathrm{D_x},y\mathrm{D_x},xy\mathrm{Dz}\right]\right)$
 $\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{y}{}{\mathrm{Dz}}\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{x}{}{\mathrm{Dz}}\right]\right]$ (19)

The Lie derivative of a Matrix of differential 2-forms can be calculated simultaneously.

 > $T≔\mathrm{map}\left(\mathrm{evalDG},\mathrm{Matrix}\left(\left[\left[{y}^{2}\mathrm{dx}&w\mathrm{dy},{x}^{2}\mathrm{dy}&w\mathrm{dz}\right],\left[xy\mathrm{dx}&w\mathrm{dz},{z}^{2}\mathrm{dx}&w\mathrm{dy}\right]\right]\right)\right)$
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],{y}^{2}\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[2,3\right],{x}^{2}\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,3\right],x{}y\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],{z}^{2}\right]\right]\right]\right)\end{array}\right]$ (20)
 > $\mathrm{LieDerivative}\left(x\mathrm{D_x}+y\mathrm{D_y}+z\mathrm{D_z},T\right)$
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],4{}{y}^{2}\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[2,3\right],4{}{x}^{2}\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,3\right],4{}x{}y\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],4{}{z}^{2}\right]\right]\right]\right)\end{array}\right]$ (21)

Example 7.

The Lie derivative can be calculated in anholonomic frames. Use FrameData to find the structure equations for an anholonomic frame and initialize with DGsetup.

 > $\mathrm{FD}≔\mathrm{FrameData}\left(\left[y\mathrm{dx},\mathrm{dx}+z\mathrm{dy},\mathrm{dz}\right],P\right)$
 ${\mathrm{FD}}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{-}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{z}}{,}{d}{}{\mathrm{Θ2}}{=}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{y}{}{z}}{-}\frac{{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{z}}{,}{d}{}{\mathrm{Θ3}}{=}{0}\right]$ (22)
 > $\mathrm{DGsetup}\left(\mathrm{FD},\left[U\right],\left[\mathrm{σ}\right]\right)$
 ${\mathrm{frame name: P}}$ (23)
 > $\mathrm{LieDerivative}\left(\mathrm{U1},\mathrm{σ1}&w\mathrm{σ3}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{y}{}{z}}\right]\right]\right]\right)$ (24)
 > $\mathrm{LieDerivative}\left(x\mathrm{U1},\mathrm{U1}&t\mathrm{σ3}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{P}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-}\frac{{1}}{{y}}\right]\right]\right]\right)$ (25)

Example 8.

The Lie derivative can be calculated for abstract forms.

 > $\mathrm{DGsetup}\left(\left[\left[\mathrm{ω1},\mathrm{ω2},\mathrm{ω3}\right],\mathrm{β1}=\mathrm{dgform}\left(2\right)\right],\left[d\left(\mathrm{ω1}\right)=\mathrm{ω2}&w\mathrm{ω3},d\left(\mathrm{ω2}\right)=\mathrm{β1}\right],N\right)$
 ${\mathrm{frame name: N}}$ (26)
 > $\mathrm{LieDerivative}\left(\mathrm{D_omega2},\mathrm{ω1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)$ (27)
 > $\mathrm{LieDerivative}\left(\mathrm{D_omega2},\mathrm{β1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]{,}\left[\left[{8}\right]{,}{1}\right]\right]\right]\right)$ (28)

Example 9.

The Lie derivative can be calculated for tensors on a Lie algebra. Use LieAlgebraData and DGsetup to initialize a Lie algebra.

 > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x1},\mathrm{x2}\right]=\mathrm{x3},\left[\mathrm{x3},\mathrm{x1}\right]=2\mathrm{x1},\left[\mathrm{x3},\mathrm{x2}\right]=-2\mathrm{x2}\right],\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right],\mathrm{alg}\right)$
 ${\mathrm{LD}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e2}}\right]$ (29)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: alg}}$ (30)
 > $\mathrm{MultiplicationTable}\left("LieTable"\right)$
 $\left[\begin{array}{ccccc}{}& |& \mathrm{e1}& \mathrm{e2}& \mathrm{e3}\\ {}& \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}\\ \mathrm{e1}& |& 0& \mathrm{_DG}{}\left(\left[\left[{"vector"},\mathrm{alg},\left[{}\right]\right],\left[\left[\left[3\right],1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"vector"},\mathrm{alg},\left[{}\right]\right],\left[\left[\left[1\right],-2\right]\right]\right]\right)\\ \mathrm{e2}& |& \mathrm{_DG}{}\left(\left[\left[{"vector"},\mathrm{alg},\left[{}\right]\right],\left[\left[\left[3\right],-1\right]\right]\right]\right)& 0& \mathrm{_DG}{}\left(\left[\left[{"vector"},\mathrm{alg},\left[{}\right]\right],\left[\left[\left[2\right],2\right]\right]\right]\right)\\ \mathrm{e3}& |& \mathrm{_DG}{}\left(\left[\left[{"vector"},\mathrm{alg},\left[{}\right]\right],\left[\left[\left[1\right],2\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"vector"},\mathrm{alg},\left[{}\right]\right],\left[\left[\left[2\right],-2\right]\right]\right]\right)& 0\end{array}\right]$ (31)

Calculate the Killing form for the Lie algebra and show that its Lie derivative is zero for all vectors in the Lie algebra.

 > $K≔\mathrm{KillingForm}\left(\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{alg}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{4}\right]{,}\left[\left[{2}{,}{1}\right]{,}{4}\right]{,}\left[\left[{3}{,}{3}\right]{,}{8}\right]\right]\right]\right)$ (32)
 > $\mathrm{LieDerivative}\left(\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right],K\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{alg}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{alg}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{alg}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)\right]$ (33)