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.

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

$U≔\mathrm{VectorField}\left(x\mathrm{D}\left[x\right]+y\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$U≔x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}$

$\mathrm{Tx}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$\mathrm{Tx}≔\frac{\partial }{\partial x}$

$\mathrm{LieBracket}\left(U\,\mathrm{Tx}\right)$

$-\frac{\partial }{\partial x}$

$\mathrm{Commutator}\left(U\,\mathrm{Tx}\right)$

$-\frac{\partial }{\partial x}$

The LieBracket and Commutator commands were introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.