DerivedFlag - Maple Help

ExteriorDifferentialSystems[DerivedFlag] - calculate the derived flag for a distribution of vector fields or a Pfaffian differential system

Calling Sequences

DerivedFlag(A,  options)

DerivedFlag(Theta,  options)

Parameters

A        - a list of vector fields on a manifold M, defining a distribution

Theta    - a lists of 1-forms on a manifold M, defining the (differential) generators for a Pfaffian differential system

options  - (optional keyword arguments) flagtype = "WeakDerivedFlag"

Description

 • In differential geometry a distribution is a set of vector fields $\mathrm{𝒜}$defined on a manifold $M.$ Associated to any distribution are two increasing nested sequences of distributions called the derived flag ${\mathrm{𝒟}}^{\left(i\right)}$and the weak derived flag ${\mathrm{𝒲}}^{\left(i\right)}$.  They are defined inductively by:

=  =  [ +   = [+

and:

${\mathrm{𝒲}}^{\left(0\right)}$ =  = [ +   = [+ .

 • These distributions satisfy and [${\mathrm{𝒟}}^{\left(j\right)}]\subset {\mathrm{𝒟}}^{\left(i+j\right)}$ and, likewise,  and [${\mathrm{𝒲}}^{\left(j\right)}]\subset {\mathrm{𝒲}}^{\left(i+j\right)}$.  Note that these are called the derived distribution of $\mathrm{𝒜}$and are often denoted by$\mathrm{𝒜}$.
 • Let be a Pfaffian differential system generated by a space of 1-forms $\mathrm{Θ}$.  Then the derived differential system of is the Pfaffian systemgenerated by the 1 forms mod $\mathrm{Θ}}$.  If  is the distribution annihilated by $\mathrm{Θ}$, that is, if ann($\mathrm{Θ})$, then ann($\mathrm{𝒜}')$. The derived flag of $\mathrm{ℐ}$ is defined inductively by  and .
 • The command DerivedFlag returns a list of lists of vector fields or a list of lists of 1-forms. The first list $\mathrm{DF}\left[1\right]$ is always the original list of vector fields or 1-forms specified in the calling sequence and or . The computation of the derived flag terminates at or  if    or . If the last differential system in the derived flag for a Pfaffian system is then an empty list is given.

Examples

 > with(DifferentialGeometry): with(ExteriorDifferentialSystems):

Example 1.

In this example, we calculate the derived flags for the standard contact system on the jet space . First, we introduce the coordinates that are needed.

 > DGsetup([x, y, y1, y2, y3, y4], M);
 ${\mathrm{frame name: M}}$ (1)

Here are the 1-form generators for the contact system.

 M > Theta := evalDG([dy - y1*dx, dy1 - y2*dx, dy2 - y3*dx, dy3 - y4*dx]);
 ${\mathrm{\Theta }}{≔}\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}{,}{-}{\mathrm{y3}}{}{\mathrm{dx}}{+}{\mathrm{dy2}}{,}{-}{\mathrm{y4}}{}{\mathrm{dx}}{+}{\mathrm{dy3}}\right]$ (2)

Here is the derived flag.

 M > DF1 := DerivedFlag(Theta);
 ${\mathrm{DF1}}{≔}\left[\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}{,}{-}{\mathrm{y3}}{}{\mathrm{dx}}{+}{\mathrm{dy2}}{,}{-}{\mathrm{y4}}{}{\mathrm{dx}}{+}{\mathrm{dy3}}\right]{,}\left[{-}{\mathrm{y3}}{}{\mathrm{dx}}{+}{\mathrm{dy2}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}{,}{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}\right]{,}\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}\right]{,}\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}\right]{,}\left[\right]\right]$ (3)

We see that the rank of the differential systems in the derived flag decreases by one at each iteration.

 M > map(nops, DF1);
 $\left[{4}{,}{3}{,}{2}{,}{1}{,}{0}\right]$ (4)

Here is the distribution defined by as the annihilator of the contact system.

 M > A := Annihilator(Theta);
 ${A}{≔}\left[{\mathrm{D_y4}}{,}\frac{{1}}{{\mathrm{y4}}}{}{\mathrm{D_x}}{+}\frac{{\mathrm{y1}}}{{\mathrm{y4}}}{}{\mathrm{D_y}}{+}\frac{{\mathrm{y2}}}{{\mathrm{y4}}}{}{\mathrm{D_y1}}{+}\frac{{\mathrm{y3}}}{{\mathrm{y4}}}{}{\mathrm{D_y2}}{+}{\mathrm{D_y3}}\right]$ (5)

The command CanonicalBasis can be used to simplify this output.

 M > A := Tools:-CanonicalBasis(A);
 ${A}{≔}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}\right]$ (6)

Here is the derived flag for the distribution $A.$

 M > DF2 := DerivedFlag(A);
 ${\mathrm{DF2}}{≔}\left[\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}{,}{\mathrm{D_y2}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}{,}{\mathrm{D_y2}}{,}{-}{\mathrm{D_y1}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}{,}{\mathrm{D_y2}}{,}{-}{\mathrm{D_y1}}{,}{\mathrm{D_y}}\right]\right]$ (7)

We see that the rank of the distributions in the derived flag increases by one at each iteration.

 M > map(nops, DF2);
 $\left[{2}{,}{3}{,}{4}{,}{5}{,}{6}\right]$ (8)

Here is the weak derived flag. In this simple example, the derived flag and weak derived flag coincide.

 M > DerivedFlag(A, flagtype = "WeakDerivedFlag");
 $\left[\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}{,}{\mathrm{D_y2}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}{,}{\mathrm{D_y2}}{,}{-}{\mathrm{D_y1}}\right]{,}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{\mathrm{y3}}{}{\mathrm{D_y2}}{+}{\mathrm{y4}}{}{\mathrm{D_y3}}{,}{\mathrm{D_y4}}{,}{-}{\mathrm{D_y3}}{,}{\mathrm{D_y2}}{,}{-}{\mathrm{D_y1}}{,}{\mathrm{D_y}}\right]\right]$ (9)

Example 2

In this example, we calculate the derived flag for the Pfaffian system defined by the underdetermined ODE  .

 M > DGsetup([x, z, z1, y, y1, y2], M2);
 ${\mathrm{frame name: M2}}$ (10)

The generators for the Pfaffian system are:

 M2 > Theta := evalDG([dy - y1*dx, dy1 - y2*dx, dz - z1*dx, dz1 - y2^2*dx]);
 ${\mathrm{\Theta }}{≔}\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}{,}{-}{\mathrm{z1}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{,}{-}{{\mathrm{y2}}}^{{2}}{}{\mathrm{dx}}{+}{\mathrm{dz1}}\right]$ (11)
 M2 > DF := DerivedFlag(Theta);
 ${\mathrm{DF}}{≔}\left[\left[{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{-}{\mathrm{y2}}{}{\mathrm{dx}}{+}{\mathrm{dy1}}{,}{-}{\mathrm{z1}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{,}{-}{{\mathrm{y2}}}^{{2}}{}{\mathrm{dx}}{+}{\mathrm{dz1}}\right]{,}\left[{-}{\mathrm{z1}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{,}{-}{\mathrm{y1}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{,}{{\mathrm{y2}}}^{{2}}{}{\mathrm{dx}}{+}{\mathrm{dz1}}{-}{2}{}{\mathrm{y2}}{}{\mathrm{dy1}}\right]{,}\left[{-}\frac{{2}{}{\mathrm{y1}}{}{\mathrm{y2}}{-}{\mathrm{z1}}}{{2}{}{\mathrm{y2}}}{}{\mathrm{dx}}{-}\frac{{1}}{{2}{}{\mathrm{y2}}}{}{\mathrm{dz}}{+}{\mathrm{dy}}\right]{,}\left[\right]\right]$ (12)

Notice that the rank of second derived system is two less than that of the first derived system.

 M2 > map(nops,DF);
 $\left[{4}{,}{3}{,}{1}{,}{0}\right]$ (13)