If X is a vector field on a manifold M and f is a real-valued function on M, then X may be applied to f to give a new real valued function.  In coordinates, X(f) is the directional derivative of f with respect to X.  The Lie bracket of two vector fields X, Y , defined on a manifold M, is the vector field Z defined by the commutator rule Z(f) = X(Y(f)) - Y(X(f)).  The standard notation for the Lie bracket is Z = [X, Y].

The LieBracket command is also used to calculate brackets in an abstract Lie algebra.

This command is part of the DifferentialGeometry package, and so can be used in the form LieBracket(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-LieBracket.

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

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

$\mathrm{X1}≔\mathrm{evalDG}\left(y^2\mathrm{D\_x}+x^2\mathrm{D\_y}\right)$

$\mathrm{X1}≔y^2\mathrm{D\_x}+x^2\mathrm{D\_y}$

$\mathrm{Y1}≔\mathrm{evalDG}\left(x\mathrm{D\_x}\right)$

$\mathrm{Y1}≔x\mathrm{D\_x}$

$Z≔\mathrm{LieBracket}\left(\mathrm{X1}\,\mathrm{Y1}\right)$

$Z≔y^2\mathrm{D\_x}-2x^2\mathrm{D\_y}$

$\mathrm{LHSofDefinition}≔\mathrm{LieDerivative}\left(Z\,F\left(x\,y\right)\right)$

$\mathrm{LHSofDefinition}≔y^2\left(\frac{\partial }{\partial x}F\left(x\,y\right)\right)-2x^2\left(\frac{\partial }{\partial y}F\left(x\,y\right)\right)$

$\mathrm{RHSofDefinition}≔\mathrm{LieDerivative}\left(\mathrm{X1}\,\mathrm{LieDerivative}\left(\mathrm{Y1}\,F\left(x\,y\right)\right)\right)-\mathrm{LieDerivative}\left(\mathrm{Y1}\,\mathrm{LieDerivative}\left(\mathrm{X1}\,F\left(x\,y\right)\right)\right)$

$\mathrm{RHSofDefinition}≔y^2\left(\frac{\partial }{\partial x}F\left(x\,y\right)+x\left(\frac{{\partial }^2}{\partial x^2}F\left(x\,y\right)\right)\right)+x^3\left(\frac{{\partial }^2}{\partial x\partial y}F\left(x\,y\right)\right)-x\left(y^2\left(\frac{{\partial }^2}{\partial x^2}F\left(x\,y\right)\right)+2x\left(\frac{\partial }{\partial y}F\left(x\,y\right)\right)+x^2\left(\frac{{\partial }^2}{\partial x\partial y}F\left(x\,y\right)\right)\right)$

$\mathrm{simplify}\left(\mathrm{RHSofDefinition}\right)$

$y^2\left(\frac{\partial }{\partial x}F\left(x\,y\right)\right)-2x^2\left(\frac{\partial }{\partial y}F\left(x\,y\right)\right)$

$\mathrm{X2}≔\mathrm{evalDG}\left(a\left(x\,y\right)\mathrm{D\_x}+b\left(x\,y\right)\mathrm{D\_y}\right)$

$\mathrm{X2}≔a\left(x\,y\right)\mathrm{D\_x}+b\left(x\,y\right)\mathrm{D\_y}$

$\mathrm{Y2}≔\mathrm{evalDG}\left(c\left(x\,y\right)\mathrm{D\_x}+d\left(x\,y\right)\mathrm{D\_y}\right)$

$\mathrm{Y2}≔c\left(x\,y\right)\mathrm{D\_x}+d\left(x\,y\right)\mathrm{D\_y}$

$\mathrm{LieBracket}\left(\mathrm{X2}\,\mathrm{Y2}\right)$

$\left(-c\left(x\,y\right)\left(\frac{\partial }{\partial x}a\left(x\,y\right)\right)-d\left(x\,y\right)\left(\frac{\partial }{\partial y}a\left(x\,y\right)\right)+a\left(x\,y\right)\left(\frac{\partial }{\partial x}c\left(x\,y\right)\right)+b\left(x\,y\right)\left(\frac{\partial }{\partial y}c\left(x\,y\right)\right)\right)\mathrm{D\_x}+\left(-c\left(x\,y\right)\left(\frac{\partial }{\partial x}b\left(x\,y\right)\right)-d\left(x\,y\right)\left(\frac{\partial }{\partial y}b\left(x\,y\right)\right)+a\left(x\,y\right)\left(\frac{\partial }{\partial x}d\left(x\,y\right)\right)+b\left(x\,y\right)\left(\frac{\partial }{\partial y}d\left(x\,y\right)\right)\right)\mathrm{D\_y}$

$\mathrm{X3}≔\mathrm{evalDG}\left(x\mathrm{D\_x}+y\mathrm{D\_y}\right)$

$\mathrm{X3}≔x\mathrm{D\_x}+y\mathrm{D\_y}$

$\mathrm{Y3}≔\mathrm{evalDG}\left(\left(-\frac{y{x}^2}{x^2+y^2}\right)\mathrm{D\_x}+\frac{x^3}{x^2+y^2}\mathrm{D\_y}\right)$

$\mathrm{Y3}≔-\frac{y{x}^2\mathrm{D\_x}}{x^2+y^2}+\frac{x^3\mathrm{D\_y}}{x^2+y^2}$

$\mathrm{LieBracket}\left(\mathrm{X3}\,\mathrm{Y3}\right)$

$0\mathrm{D\_x}$

$X≔\mathrm{evalDG}\left(\sin \left(x\right)\mathrm{D\_x}+\ln \left(x\right)y\mathrm{D\_y}\right)$

$X≔\sin \left(x\right)\mathrm{D\_x}+\ln \left(x\right)y\mathrm{D\_y}$

$Y≔\mathrm{evalDG}\left(\cos \left(y\right)\mathrm{D\_x}+\exp \left(x\right)\mathrm{D\_y}\right)$

$Y≔\cos \left(y\right)\mathrm{D\_x}+{\ⅇ}^x\mathrm{D\_y}$

$Z≔\mathrm{evalDG}\left(x{y}^3\mathrm{D\_x}-y{x}^3\mathrm{D\_y}\right)$

$Z≔y^{3}x\mathrm{D\_x}-y{x}^3\mathrm{D\_y}$

$\left(\mathrm{LieBracket}\left(\mathrm{LieBracket}\left(X\,Y\right)\,Z\right)\&plus\mathrm{LieBracket}\left(\mathrm{LieBracket}\left(Z\,X\right)\,Y\right)\right)\&plus\mathrm{LieBracket}\left(\mathrm{LieBracket}\left(Y\,Z\right)\,X\right)$

$0\mathrm{D\_x}$

$\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x1}\,\mathrm{x2}\right]=\mathrm{x3}\,\left[\mathrm{x3}\,\mathrm{x1}\right]=2\mathrm{x1}\,\left[\mathrm{x3}\,\mathrm{x2}\right]=-2\mathrm{x2}\right]\,\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\right]\,\mathrm{alg}\right)$

$\mathrm{LD}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]=\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]=-2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=2\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; alg}$

$\mathrm{MultiplicationTable}\left(''LieTable''\right)$

$\left[\begin{array}{ccccc}& |& \mathrm{e1}& \mathrm{e2}& \mathrm{e3}\\ & \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}\\ \mathrm{e1}& |& 0& \mathrm{e3}& -2\mathrm{e1}\\ \mathrm{e2}& |& -\mathrm{e3}& 0& 2\mathrm{e2}\\ \mathrm{e3}& |& 2\mathrm{e1}& -2\mathrm{e2}& 0\end{array}\right]$

$\mathrm{LieBracket}\left(\mathrm{e1}+\mathrm{e2}\,\mathrm{e2}+\mathrm{e3}\right)$

$-2\mathrm{e1}+2\mathrm{e2}+\mathrm{e3}$