LieDerivative - Maple Help

LieDerivative

calculate the Lie derivative of an algebraic expression, a vector field or a one-form with respect to a vector field

 Calling Sequence LieDerivative( X, vf )

Parameters

 X - an algebraic expression, a VectorField object, or a OneForm object vf - a VectorField object that is on the same space as X

Description

 • If X is an algebraic expression then LieDerivative(X, vf) is the directional derivative vf(X) of X in the direction of the vector field vf.
 • If X is a VectorField object then LieDerivative(X, vf) is the VectorField object defined by the Lie bracket [vf, X] = LieBracket(vf, X). See LieBracket for more detail.
 • If X is a OneForm object then omega = LieDerivative(X, vf) is the OneForm object defined by omega(Y) = vf(omega(Y)) - omega([Y,vf]), where Y is any vector field on the same space as X.
 • This method is associated with the VectorField and OneForm objects. For more detail, see Overview of the VectorField object, Overview of the OneForm object.

Examples

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

The vector fields X,Y live on the same space (x,y).

 > $X≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)

 > $R≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right]\right)$
 ${R}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)

 > $\mathrm{LieDerivative}\left(a{x}^{2}-{y}^{2},X\right)$
 ${2}{}{a}{}{x}$ (3)

These two commands are equivalent when Y is a vector field. And it returns a vector field.

 > $\mathrm{LieDerivative}\left(R,X\right)$
 $\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4)

 > $\mathrm{LieBracket}\left(X,R\right)$
 $\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (5)

 > $\mathrm{\omega }≔\mathrm{OneForm}\left(xd\left[x\right]-yd\left[y\right]\right)$
 ${\mathrm{\omega }}{≔}{x}{}{\mathrm{dx}}{-}{y}{}{\mathrm{dy}}$ (6)

 > $\mathrm{LieDerivative}\left(\mathrm{\omega },R\right)$
 ${-}{2}{}{y}{}{\mathrm{dx}}{-}{2}{}{x}{}{\mathrm{dy}}$ (7)

Compatibility

 • The LieDerivative command was introduced in Maple 2020.