OrbitDistribution - Maple Programming Help

Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : LAVF : LieAlgebrasOfVectorFields/LAVF/OrbitDistribution

OrbitDistribution

calculate the orbit distribution of a LAVF object.

 Calling Sequence OrbitDistribution( obj)

Parameters

 obj - a LAVF object.

Description

 • Let L be a LAVF object and is Lie algebra (see IsLieAlgebra). Then OrbitDistribution method returns the orbit distribution of L.
 • The returned orbit distribution of obj is a Distribution object. A Distribution object has access to various methods, see Overview of the Distribution object for more detail.
 • The orbit distribution is the infinitesimal version of the group orbit.
 • This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y,z\right),\mathrm{\eta }\left(x,y,z\right),\mathrm{zeta}\left(x,y,z\right)\right]\right):$

Build vector fields associated with 3-d spatial rotations...

 > $R\left[x\right]≔\mathrm{VectorField}\left(-z\mathrm{D}\left[y\right]+y\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{x}}{≔}{-}{z}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $R\left[y\right]≔\mathrm{VectorField}\left(-x\mathrm{D}\left[z\right]+z\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{y}}{≔}{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{-}{x}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $R\left[z\right]≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{z}}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)

We now construct a LAVF object for SO(3) that are generated by these rotation vector fields.

 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y,z\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y,z\right)\mathrm{D}\left[y\right]+\mathrm{zeta}\left(x,y,z\right)\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{ζ}}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4)
 > $L≔\mathrm{EliminationLAVF}\left(V,\left[R\left[x\right],R\left[y\right],R\left[z\right]\right]\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{ζ}}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}\frac{{-}{\mathrm{\eta }}{}{y}{-}{\mathrm{ζ}}{}{z}}{{x}}{,}{{\mathrm{\eta }}}_{{x}}{=}\frac{\left({{\mathrm{ζ}}}_{{y}}\right){}{z}{+}{\mathrm{\eta }}}{{x}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{z}}{=}{-}{{\mathrm{ζ}}}_{{y}}{,}{{\mathrm{ζ}}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{ζ}}}_{{x}}{=}\frac{{-}\left({{\mathrm{ζ}}}_{{y}}\right){}{y}{+}{\mathrm{ζ}}}{{x}}{,}{{\mathrm{ζ}}}_{{z}}{=}{0}\right]\right\}$ (5)

Find the orbit distribution of L...

 > $\mathrm{OD}≔\mathrm{OrbitDistribution}\left(L\right)$
 ${\mathrm{OD}}{≔}\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}\frac{{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (6)

OD is a Distribution object which has access to various methods, for example,

 > $\mathrm{Dimension}\left(\mathrm{OD}\right)$
 ${2}$ (7)
 > $\mathrm{Integrals}\left(\mathrm{OD}\right)$
 $\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right]$ (8)
 > $\mathrm{GetAnnihilator}\left(\mathrm{OD}\right)$
 $\left[{\mathrm{dx}}{+}\frac{{y}{}{\mathrm{dy}}}{{x}}{+}\frac{{z}{}{\mathrm{dz}}}{{x}}\right]$ (9)

Compatibility

 • The OrbitDistribution command was introduced in Maple 2020.