OrderOfInvolution - Maple Help

OrderOfInvolution

find the (possible) minimal differential order that a LHPDEs system is in involution

 Calling Sequence OrderOfInvolution( obj)

Parameters

 obj - a LHPDE object that is in rif-reduced form with respect to a total degree ranking (see IsTotalDegreeRanking, IsRifReduced)

Description

 • The OrderOfInvolution method returns the order at which a LHPDE object is involutive, or a bound for this order. Note that the LHPDE object must be rif-reduced, with respect to a total degree ranking (see IsTotalDegreeRanking).
 • So far, the only implementation in this method is the Mansfield bound (ref: E. Mansfield. A Simple Criterion for Involutivity. Journal of the London Mathematics Society 54: 323-345,1996).  This gives an upper bound for the order of involutivity. It is often -- but not always -- exact.
 • This method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object.

Examples

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

Create an LHPDE object (these are the determining equations for the Euclidean group E(2))...

 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (1)
 > $\mathrm{IsFiniteType}\left(\mathrm{E2}\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{ParametricDerivatives}\left(\mathrm{E2}\right)$
 $\left[{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}\right]$ (3)

Create another LHPDE object that is rif-reduced with respect to a total degree ranking....

 > $\mathrm{E2red}≔\mathrm{RifReduce}\left(\mathrm{E2},\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2red}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (4)

Now this can be checked for the order at which it becomes involutive....

 > $\mathrm{OrderOfInvolution}\left(\mathrm{E2red}\right)$
 ${2}$ (5)

Compatibility

 • The OrderOfInvolution command was introduced in Maple 2020.