OrbitDimension - Maple Help

OrbitDimension

calculate the dimension of the orbit distribution of a LAVF object

InvariantCount

calculate the count of invariant of a LAVF object

IsTransitive

check if a LAVF object is transitive.

 Calling Sequence OrbitDimension( obj) InvariantCount( obj, t) IsTransitive( obj)

Parameters

 obj - a LAVF object t - (optional) a string: "all", "essential", or "inessential"

Description

 • The OrbitDimension method calculates the dimension of the orbit distribution of a LAVF object.
 • The InvariantCount method calculates the count of scalar invariants of a LAVF object. By default (t="all"), all invariants are counted.
 • If t="essential" is specified, then only essential invariants are counted. An invariant is essential, roughly speaking, if the group action cannot be expressed without it.
 • Let L be a LAVF object. Then IsTransitive(L) returns true if and only if the action of L is transitive, that is, InvariantCount(L) = 0.
 • Let L be a LAVF object and let OD be the orbit distribution of L. Then OrbitDimension(L) equals to Dimension(OD) and InvariantCount(L) equals to Codimension(OD). See Overview of the Distribution object for more detail.
 • These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

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

Example 1: Build vector fields associated with 3-d spatial rotations...

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

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

 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y,z\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y,z\right)\mathrm{D}\left[y\right]+\mathrm{\zeta }\left(x,y,z\right)\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right){+}{\mathrm{\zeta }}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)$ (4)
 > $L≔\mathrm{EliminationLAVF}\left(V,\left[R\left[x\right],R\left[y\right],R\left[z\right]\right]\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right){+}{\mathrm{\zeta }}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}\frac{{-}{\mathrm{\eta }}{}{y}{-}{\mathrm{\zeta }}{}{z}}{{x}}{,}{{\mathrm{\eta }}}_{{x}}{=}\frac{\left({{\mathrm{\zeta }}}_{{y}}\right){}{z}{+}{\mathrm{\eta }}}{{x}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{z}}{=}{-}{{\mathrm{\zeta }}}_{{y}}{,}{{\mathrm{\zeta }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\zeta }}}_{{x}}{=}\frac{{-}\left({{\mathrm{\zeta }}}_{{y}}\right){}{y}{+}{\mathrm{\zeta }}}{{x}}{,}{{\mathrm{\zeta }}}_{{z}}{=}{0}\right]\right\}$ (5)
 > $\mathrm{OrbitDimension}\left(L\right)$
 ${2}$ (6)
 > $\mathrm{InvariantCount}\left(L\right)$
 ${1}$ (7)

L is not transitive since SO(3) has one invariant.

 > $\mathrm{IsTransitive}\left(L\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{Invariants}\left(L\right)$
 $\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right]$ (9)

Example 2:

 > $Y≔\mathrm{VectorField}\left(ay\mathrm{D}\left[x\right]+bz\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${Y}{≔}\left({a}{}{y}{+}{b}{}{z}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)$ (10)
 > $\mathrm{L2}≔\mathrm{EliminationLAVF}\left(V,\left[Y\right],\mathrm{coefficients}=\left[a,b\right]\right)$
 ${\mathrm{L2}}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right){+}{\mathrm{\zeta }}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{z}{,}{z}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}\frac{{-}\left({{\mathrm{\xi }}}_{{z}}\right){}{z}{+}{\mathrm{\xi }}}{{y}}{,}{\mathrm{\eta }}{=}{0}{,}{\mathrm{\zeta }}{=}{0}\right]\right\}$ (11)
 > $\mathrm{OrbitDimension}\left(\mathrm{L2}\right)$
 ${1}$ (12)
 > $\mathrm{IsTransitive}\left(\mathrm{L2}\right)$
 ${\mathrm{false}}$ (13)
 > $\mathrm{InvariantCount}\left(\mathrm{L2}\right)$
 ${2}$ (14)
 > $\mathrm{InvariantCount}\left(\mathrm{L2},"essential"\right)$
 ${1}$ (15)
 > $\mathrm{InvariantCount}\left(\mathrm{L2},"inessential"\right)$
 ${1}$ (16)

The counts above are found directly from L2. Finding invariants involve integration...

 > $\mathrm{Invariants}\left(\mathrm{L2}\right)$
 $\left[{y}{,}{z}\right]$ (17)
 > $\mathrm{Invariants}\left(\mathrm{L2},"essential"\right)$
 $\left[\frac{{z}}{{y}}\right]$ (18)

Compatibility

 • The OrbitDimension, InvariantCount and IsTransitive commands were introduced in Maple 2020.