find the infinitesimal symmetries of a Pfaffian differential system - Maple Programming Help

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ExteriorDifferentialSystems[InfinitesimalSymmetriesOfEDS] - find the infinitesimal symmetries of a Pfaffian differential system

Calling Sequences

InfinitesimalSymmetriesOfEDS(Theta, options)

InfinitesimalSymmetriesOfEDS(ThetaList, options)

Parameters

Theta     - a list of 1-forms on a manifold $M$, the (differential) generators of a Pfaffian differential system $\mathrm{ℐ}$

ThetaList - a list of lists of 1-forms on a manifold $M$, the (differential) generators of a sequence of Pfaffian differential systems ${\mathrm{ℐ}}_{1}$, ${\mathrm{ℐ}}_{2}$, ${\mathrm{ℐ}}_{3}$, ...

options   - (optional keyword arguments) keyword arguments ansatz, unknowns, auxiliaryequations$,$output

Description

 • Let $\mathrm{ℐ}$ be an exterior differential system on a manifold $M$. A local symmetry of $\mathrm{ℐ}$ is a local diffeomorphism such that the pullback map preserves that is, for each differential form  An infinitesimal symmetry of $\mathrm{ℐ}$ is a vector field on $M$ which preserves with respect to Lie differentiation, that is, for each differential form If is a 1-parameter group of local symmetries of, then the infinitesimal transformation determined by is an infinitesimal symmetry and, conversely, if $X$ is an infinitesimal symmetry of, then the flow of $X$ is a 1-parameter group of local symmetries. If $\mathrm{ℐ}$ is a Pfaffian differential system with generators , then $X$ is an infinitesimal symmetry of $\mathrm{ℐ}$ if:

or equivalently,

These conditions readily translate into a system of homogeneous linear first order PDE for the components of the vector field $X.$

 • The first calling sequence InfinitesimalSymmetriesOfEDS(Theta) uses pdsolve to integrate the first order PDE system for the infinitesimal symmetries of the Pfaffian system and returns the general infinitesimal symmetry depending upon a certain number of constants and arbitrary functions.
 • The second calling sequence finds the common symmetries for the given list of differential systems. Here is a simple application. Let be the derived system of the Pfaffian system $\mathrm{ℐ}$. Then every symmetry of is a symmetry of   so that finding the symmetries of $ℐ$ is mathematically equivalent to finding the symmetries of the pair [. But, symbolically, the latter approach generates more defining equations for the symmetry vector, which will often improve the performance of the Maple PDE solver.
 • One can search for special classes of symmetries by specifying the form of the symmetry vector using the keyword argument ansatz= X, where is a vector field depending upon a certain number of arbitrary functions . These functions should be explicitly declared with the keyword argument unknowns. Additional constraints (either algebraic or differential equations or inequalities) can be imposed upon the arbitrary functions  in the symmetry ansatz using the keyword argument auxiliaryequations.
 • If the space of symmetries is finite dimensional, say $m$-dimensional, then the keyword argument output = "list" will change the output of  InfinitesimalSymmetriesOfEDS to a list of vectors. This list of symmetry vectors defines a basis for a Lie algebra, the structure constants of which can be computed using the command LieAlgebraData.

Examples

 > with(DifferentialGeometry): with(ExteriorDifferentialSystems):

Example 1.

We define a rank 3 distribution on a 5-dimensional manifold and calculate its symmetry algebra to be 14-dimensional. Create the 5-dimensional manifold with coordinates

 > DGsetup([x, z, y, y1, y2], M1):

Our Pfaffian system  is

 > Theta := evalDG([dy - y1*dx, dy1 - y2*dx, dz - y2^2*dx]);
 ${\mathrm{Θ}}{:=}\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}{,}{-}{{\mathrm{y2}}}^{{2}}{}{\mathrm{dx}}{+}{\mathrm{dz}}\right]$ (1)
 > Gamma := InfinitesimalSymmetriesOfEDS(Theta, output = "list");
 ${\mathrm{Γ}}{:=}\left[{-}\left({2}{}{\mathrm{y2}}{}{y}{-}\frac{{4}}{{3}}{}{{\mathrm{y1}}}^{{2}}\right){}{\mathrm{D_x}}{-}\left(\frac{{2}}{{3}}{}{y}{}{{\mathrm{y2}}}^{{3}}{-}{{z}}^{{2}}\right){}{\mathrm{D_z}}{-}\left({2}{}{\mathrm{y2}}{}{y}{}{\mathrm{y1}}{-}\frac{{8}}{{9}}{}{{\mathrm{y1}}}^{{3}}{-}{z}{}{y}\right){}{\mathrm{D_y}}{-}\left({y}{}{{\mathrm{y2}}}^{{2}}{-}{z}{}{\mathrm{y1}}\right){}{\mathrm{D_y1}}{-}\left({-}{z}{}{\mathrm{y2}}{+}\frac{{2}}{{3}}{}{\mathrm{y1}}{}{{\mathrm{y2}}}^{{2}}\right){}{\mathrm{D_y2}}{,}{-}\left({2}{}{y}{-}\frac{{8}}{{3}}{}{x}{}{\mathrm{y1}}{+}{{x}}^{{2}}{}{\mathrm{y2}}\right){}{\mathrm{D_x}}{-}\left({-}{2}{}{z}{}{\mathrm{y1}}{+}\frac{{1}}{{3}}{}{{x}}^{{2}}{}{{\mathrm{y2}}}^{{3}}\right){}{\mathrm{D_z}}{-}\left({{x}}^{{2}}{}{\mathrm{y2}}{}{\mathrm{y1}}{-}\frac{{4}}{{3}}{}{x}{}{{\mathrm{y1}}}^{{2}}{-}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{z}\right){}{\mathrm{D_y}}{-}\left(\frac{{1}}{{2}}{}{{x}}^{{2}}{}{{\mathrm{y2}}}^{{2}}{-}\frac{{2}}{{3}}{}{{\mathrm{y1}}}^{{2}}{-}{x}{}{z}\right){}{\mathrm{D_y1}}{-}\left({-}{z}{+}\frac{{2}}{{3}}{}{x}{}{{\mathrm{y2}}}^{{2}}{-}\frac{{2}}{{3}}{}{\mathrm{y1}}{}{\mathrm{y2}}\right){}{\mathrm{D_y2}}{,}{-}\left({2}{}{y}{}{x}{-}\frac{{4}}{{3}}{}{{x}}^{{2}}{}{\mathrm{y1}}{+}\frac{{1}}{{3}}{}{{x}}^{{3}}{}{\mathrm{y2}}\right){}{\mathrm{D_x}}{-}\left({2}{}{z}{}{y}{-}{2}{}{z}{}{\mathrm{y1}}{}{x}{+}\frac{{1}}{{9}}{}{{x}}^{{3}}{}{{\mathrm{y2}}}^{{3}}{+}\frac{{8}}{{9}}{}{{\mathrm{y1}}}^{{3}}\right){}{\mathrm{D_z}}{-}\left({2}{}{{y}}^{{2}}{+}\frac{{1}}{{3}}{}{{x}}^{{3}}{}{\mathrm{y2}}{}{\mathrm{y1}}{-}\frac{{2}}{{3}}{}{{x}}^{{2}}{}{{\mathrm{y1}}}^{{2}}{-}\frac{{1}}{{6}}{}{z}{}{{x}}^{{3}}\right){}{\mathrm{D_y}}{-}\left({2}{}{y}{}{\mathrm{y1}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{}{{\mathrm{y2}}}^{{2}}{-}\frac{{2}}{{3}}{}{x}{}{{\mathrm{y1}}}^{{2}}{-}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{z}\right){}{\mathrm{D_y1}}{-}\left({-}{x}{}{z}{+}\frac{{1}}{{3}}{}{{x}}^{{2}}{}{{\mathrm{y2}}}^{{2}}{-}\frac{{2}}{{3}}{}{\mathrm{y1}}{}{\mathrm{y2}}{}{x}{+}\frac{{4}}{{3}}{}{{\mathrm{y1}}}^{{2}}\right){}{\mathrm{D_y2}}{,}\frac{{1}}{{4}}{}{{x}}^{{2}}{}{\mathrm{D_x}}{+}{{\mathrm{y1}}}^{{2}}{}{\mathrm{D_z}}{+}\frac{{3}}{{4}}{}{y}{}{x}{}{\mathrm{D_y}}{+}\left(\frac{{3}}{{4}}{}{y}{+}\frac{{1}}{{4}}{}{x}{}{\mathrm{y1}}\right){}{\mathrm{D_y1}}{-}\left(\frac{{1}}{{4}}{}{x}{}{\mathrm{y2}}{-}{\mathrm{y1}}\right){}{\mathrm{D_y2}}{,}\left({-}{2}{}{\mathrm{y1}}{+}\frac{{3}}{{2}}{}{x}{}{\mathrm{y2}}\right){}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{{\mathrm{y2}}}^{{3}}{}{x}{}{\mathrm{D_z}}{+}\left(\frac{{3}}{{2}}{}{\mathrm{y1}}{}{\mathrm{y2}}{}{x}{-}{{\mathrm{y1}}}^{{2}}{-}\frac{{3}}{{4}}{}{x}{}{z}\right){}{\mathrm{D_y}}{+}\left(\frac{{3}}{{4}}{}{x}{}{{\mathrm{y2}}}^{{2}}{-}\frac{{3}}{{4}}{}{z}\right){}{\mathrm{D_y1}}{+}\frac{{1}}{{2}}{}{{\mathrm{y2}}}^{{2}}{}{\mathrm{D_y2}}{,}{-}{x}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_z}}{-}{y}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y2}}{,}{2}{}{\mathrm{y1}}{}{\mathrm{D_z}}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{+}{x}{}{\mathrm{D_y1}}{+}{\mathrm{D_y2}}{,}\left({2}{}{x}{}{\mathrm{y1}}{-}{2}{}{y}\right){}{\mathrm{D_z}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{}{\mathrm{D_y}}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y1}}{+}{x}{}{\mathrm{D_y2}}{,}{x}{}{\mathrm{D_y}}{+}{\mathrm{D_y1}}{,}{2}{}{\mathrm{y2}}{}{\mathrm{D_x}}{+}\frac{{2}}{{3}}{}{{\mathrm{y2}}}^{{3}}{}{\mathrm{D_z}}{+}\left({2}{}{\mathrm{y1}}{}{\mathrm{y2}}{-}{z}\right){}{\mathrm{D_y}}{+}{{\mathrm{y2}}}^{{2}}{}{\mathrm{D_y1}}{,}{x}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_z}}{+}{2}{}{y}{}{\mathrm{D_y}}{+}{\mathrm{y1}}{}{\mathrm{D_y1}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}{,}{\mathrm{D_x}}\right]$ (2)

There are 14 infinitesimal symmetries.

 M1 > nops(Gamma);
 ${14}$ (3)

One can calculate the structure equations for this Lie algebra of vector fields with the command LieAlgebraData. The commands CartanSubalgebra, RootSpaceDecomposition, SimpleRoots and CartanMatrix can be used to show that this Lie algebra is the split real form of the exceptional simple Lie algebra ${g}_{2}$. On an historical note, this is one of the first concrete realizations of ${g}_{2}$ and was discovered independently by Cartan and Engel in 1893.

Example 2.

We continue with the Pfaffian system defined by Example 1. We find the derived system $\mathrm{ℐ}\mathit{'}$and re-calculate the symmetries of $ℐ$ by calculating the symmetries of the two systems {$ℐ$ , $ℐ\mathit{'}$}.

 M1 > DF := DerivedFlag(Theta);
 ${\mathrm{DF}}{:=}\left[\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}{,}{-}{{\mathrm{y2}}}^{{2}}{}{\mathrm{dx}}{+}{\mathrm{dz}}\right]{,}\left[{{\mathrm{y2}}}^{{2}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{-}{2}{}{\mathrm{y2}}{}{\mathrm{dy1}}{,}{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}\right]{,}\left[{}\right]\right]$ (4)
 M1 > Theta1 := DF[2];
 ${\mathrm{Θ1}}{:=}\left[{{\mathrm{y2}}}^{{2}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{-}{2}{}{\mathrm{y2}}{}{\mathrm{dy1}}{,}{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}\right]$ (5)
 M1 > Gamma2 := InfinitesimalSymmetriesOfEDS([Theta1,Theta], output = "list"):

The results are the same.

 M1 > seq(Gamma[i] &minus Gamma2[i], i = 1..14);
 ${0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}{,}{0}{}{\mathrm{D_x}}$ (6)



Example 3.

We continue with the Pfaffian system defined by Example 1. In this example we look for the symmetries of $ℐ$ which also preserve the Pfaffian system $\mathrm{𝒞}$ defined by the contact form . Note that $\mathrm{𝒞}$ does appear as a differential system in the derived flag so that in this example we expect to obtain a subalgebra of the full 14-dimensional Lie algebra of infinitesimal symmetries.

 M1 > C := evalDG([dy - y1*dx]);
 ${C}{:=}\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}\right]$ (7)
 M1 > Gamma3 := InfinitesimalSymmetriesOfEDS([C, Theta], output = "list");
 ${\mathrm{Γ3}}{:=}\left[{-}{{x}}^{{2}}{}{\mathrm{D_x}}{-}{4}{}{{\mathrm{y1}}}^{{2}}{}{\mathrm{D_z}}{-}{3}{}{y}{}{x}{}{\mathrm{D_y}}{-}\left({3}{}{y}{+}{x}{}{\mathrm{y1}}\right){}{\mathrm{D_y1}}{+}\left({x}{}{\mathrm{y2}}{-}{4}{}{\mathrm{y1}}\right){}{\mathrm{D_y2}}{,}\left({2}{}{x}{}{\mathrm{y1}}{-}{2}{}{y}\right){}{\mathrm{D_z}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{}{\mathrm{D_y}}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y1}}{+}{x}{}{\mathrm{D_y2}}{,}{-}{x}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_z}}{-}{y}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y2}}{,}{2}{}{\mathrm{y1}}{}{\mathrm{D_z}}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{+}{x}{}{\mathrm{D_y1}}{+}{\mathrm{D_y2}}{,}{x}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_z}}{+}{2}{}{y}{}{\mathrm{D_y}}{+}{\mathrm{y1}}{}{\mathrm{D_y1}}{,}{x}{}{\mathrm{D_y}}{+}{\mathrm{D_y1}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}{,}{\mathrm{D_x}}\right]$ (8)
 M1 > nops(Gamma3);
 ${9}$ (9)

Note that the coefficients of ${\mathrm{D}}_{x}$, ${\mathrm{D}}_{y}$, ${\mathrm{D}}_{z}$ in each of the vector fields in ${\mathrm{Γ}}_{3}$ are independent of the variable ${y}_{2}.$

Example 4.

We continue with the Pfaffian system defined by Example 1. In this example we use the various keyword arguments to look for the symmetries for which the coefficients of ${\mathrm{D}}_{x}$, ${\mathrm{D}}_{y}$, and  ${\mathrm{D}}_{z}$ are functions of variables  First define a general vector field. We shall uses this as our symmetry ansatz.

 M1 > F := [F1, F2, F3, F4, F5](x, y, z, y1, y2);
 ${F}{:=}\left[{\mathrm{F1}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}{\mathrm{F2}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}{\mathrm{F3}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}{\mathrm{F4}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}{\mathrm{F5}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right)\right]$ (10)
 M1 > X := DGzip(F, [D_x, D_y, D_z, D_y1, D_y2]);
 ${X}{:=}{\mathrm{F1}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){}{\mathrm{D_x}}{+}{\mathrm{F3}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){}{\mathrm{D_z}}{+}{\mathrm{F2}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){}{\mathrm{D_y}}{+}{\mathrm{F4}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){}{\mathrm{D_y1}}{+}{\mathrm{F5}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){}{\mathrm{D_y2}}$ (11)

The following equations impose the required functional dependencies on the coefficients of  ${\mathrm{D}}_{x}$, ${\mathrm{D}}_{y}$, and ${\mathrm{D}}_{z}$ .

 M1 > Eq := [seq(seq(diff(G(x, y, z, y1, y2), v), v = [y1, y2]), G = [F1, F2, F3])];
 ${\mathrm{Eq}}{:=}\left[\frac{{\partial }}{{\partial }{\mathrm{y1}}}{}{\mathrm{F1}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}\frac{{\partial }}{{\partial }{\mathrm{y2}}}{}{\mathrm{F1}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}\frac{{\partial }}{{\partial }{\mathrm{y1}}}{}{\mathrm{F2}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}\frac{{\partial }}{{\partial }{\mathrm{y2}}}{}{\mathrm{F2}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}\frac{{\partial }}{{\partial }{\mathrm{y1}}}{}{\mathrm{F3}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right){,}\frac{{\partial }}{{\partial }{\mathrm{y2}}}{}{\mathrm{F3}}{}\left({x}{,}{y}{,}{z}{,}{\mathrm{y1}}{,}{\mathrm{y2}}\right)\right]$ (12)
 M1 > InfinitesimalSymmetriesOfEDS(Theta, ansatz = X, unknowns = F, auxiliaryequations = Eq, output = "list");
 $\left[{-}{x}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_z}}{-}{y}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y2}}{,}{x}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_z}}{+}{2}{}{y}{}{\mathrm{D_y}}{+}{\mathrm{y1}}{}{\mathrm{D_y1}}{,}{x}{}{\mathrm{D_y}}{+}{\mathrm{D_y1}}{,}{\mathrm{D_z}}{,}{\mathrm{D_y}}{,}{\mathrm{D_x}}\right]$ (13)