 LieBracket - Maple Help

LieBracket

calculate the Lie bracket of two VectorField objects

Commutator

a synonym for LieBracket Calling Sequence LieBracket (X, Y) Commutator (X, Y) Parameters

 X - a VectorField object Y - a VectorField object on the same space as X Description

 • The Lie bracket of two vector fields $X$, $Y$, defined on the same space, is the vector field $Z$ such that $Z\left(f\right)=X\left(Y\left(f\right)\right)-Y\left(X\left(f\right)\right)$ where $f$ is a real-valued function on the same space as $X$ and $Y$. The standard notation is $Z=\left[X,Y\right]$.
 • The LieBracket(X,Y) method returns a VectorField object on the same space as $X$ and $Y$.
 • The Commutator method is provided as an alias.
 • This method is associated with the VectorField object. For more detail, see Overview of the VectorField object. Examples

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

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

 > $\mathrm{LieBracket}\left(U,\mathrm{Tx}\right)$
 ${-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)

which is the same as

 > $\mathrm{Commutator}\left(U,\mathrm{Tx}\right)$
 ${-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4) Compatibility

 • The LieBracket and Commutator commands were introduced in Maple 2020.