OrbitDistribution - Maple Help
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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

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

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

 > ${R}_{x}≔\mathrm{VectorField}\left(-z{\mathrm{D}}_{y}+y{\mathrm{D}}_{z},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{x}}{≔}{-}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right){+}{y}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)$ (1)
 > ${R}_{y}≔\mathrm{VectorField}\left(-x{\mathrm{D}}_{z}+z{\mathrm{D}}_{x},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{y}}{≔}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)$ (2)
 > ${R}_{z}≔\mathrm{VectorField}\left(-y{\mathrm{D}}_{x}+x{\mathrm{D}}_{y},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{z}}{≔}{-}{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (3)

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

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

Find the orbit distribution of L...

 > $\mathrm{OD}≔\mathrm{OrbitDistribution}\left(L\right)$
 ${\mathrm{OD}}{≔}\left\{{-}\frac{{z}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{z}}{,}{-}\frac{{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{y}}\right\}$ (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.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.