Invariants - Maple Help

Invariants

attempt to find invariants of a LAVF object.

 Calling Sequence Invariants( obj)

Parameters

 obj - a LAVF object.

Description

 • The Invariants method attempts to find the invariants of a LAVF object via integration. If successful, it returns the invariants as a list of expressions.
 • Let L be a LAVF object and OD be the orbit distribution of L. Then Invariants(L) is equivalent to Integrals(OD). For more detail of the Distribution's methods, see Overview of the Distribution object.
 • 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...

 > $R\left[x\right]≔\mathrm{VectorField}\left(-z\mathrm{D}\left[y\right]+y\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{x}}{≔}{-}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right){+}{y}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)$ (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 vector fields system (as 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)
 > $\mathrm{Invariants}\left(L\right)$
 $\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right]$ (6)

Compatibility

 • The Invariants command was introduced in Maple 2020.