SymmetricProductsOfKillingTensors - Maple Help

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Tensor[SymmetricProductsOfKillingTensors] - form all possible symmetric tensors of a given rank (linearly independent over the real numbers) from a list of symmetric tensors

Calling Sequences

SymmetricProducstOfKillingTensors(K, p, ptlist )

Parameters

K         - a list of lists of Killing tensors where is a list of Killing tensors of rank r on a manifold $M$

p         - a positive integer.

ptlist    - (optional) a list of points on the manifold $M$ Description

 • This program first computes all partitions $\left[{p}_{1},{p}_{2},...\right]$of the integer $p$, that is, all non-decreasing sequences ${p}_{1},{p}_{2},...$with ${p}_{1}+{p}_{2}+\cdot \cdot \cdot =p$. Then, for each partition, all possible rank $p$ symmetric tensors of the form ${T}_{1}\odot {T}_{2}\odot {T}_{3}\odot ...$ ,where the tensor ${T}_{i}$ is taken from the list $K$, are generated. From the totality of tensors so obtained a maximal set of linearly independent tensors (over the real numbers) is selected. Each symmetric tensor in the returned list is a Killing tensor if each of the ${K}_{i}$are.
 • The independent tensors are generated by a call to the DifferentialGeometry command DGbasis. For tensors with coefficients which are not rational functions, the DGbasis program may work faster using a Wronskian approach which requires the specification of a list of points on the manifold $M$.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SymmetricProductsOfKillingTensors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-SymmetricProductsOfKillingTensors. Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $g≔\mathrm{evalDG}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${g}{≔}\frac{{\mathrm{dx}}}{{2}}{}{\mathrm{dy}}{+}\frac{{\mathrm{dy}}}{{2}}{}{\mathrm{dx}}$ (2.2)

Calculate the rank 1 Killing tensors.

 M > $\mathrm{K1}≔\mathrm{KillingTensors}\left(g,1\right)$
 ${\mathrm{K1}}{≔}\left[{-}{y}{}{\mathrm{dx}}{+}{x}{}{\mathrm{dy}}{,}{\mathrm{dy}}{,}{\mathrm{dx}}\right]$ (2.3)

Calculate all the rank 2 and rank 3 Killing tensors generated by $\mathrm{K1}$.

 M > $\mathrm{S2}≔\mathrm{SymmetricProductsOfKillingTensors}\left(\left[\mathrm{K1}\right],2\right)$
 ${\mathrm{S2}}{≔}\left[{{y}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{y}{}{x}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{y}{}{x}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{{x}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{,}{-}\frac{{y}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{y}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{x}{}{\mathrm{dy}}{}{\mathrm{dy}}{,}{-}{y}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{x}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{x}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{\mathrm{dx}}}{{2}}{}{\mathrm{dy}}{+}\frac{{\mathrm{dy}}}{{2}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{dx}}\right]$ (2.4)
 M > $\mathrm{nops}\left(\mathrm{S2}\right)$
 ${6}$ (2.5)
 M > $\mathrm{S3}≔\mathrm{SymmetricProductsOfKillingTensors}\left(\left[\mathrm{K1}\right],3\right)$
 ${\mathrm{S3}}{≔}\left[{-}{{y}}^{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{y}}^{{2}}{}{x}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{{y}}^{{2}}{}{x}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}{y}{}{{x}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{{y}}^{{2}}{}{x}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{y}{}{{x}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{y}{}{{x}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{{x}}^{{3}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{{y}}^{{2}}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{{y}}^{{2}}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{2}{}{y}{}{x}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{{{y}}^{{2}}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{2}{}{y}{}{x}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{2}{}{y}{}{x}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{{x}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{,}{{y}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{2}{}{y}{}{x}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{2}{}{y}{}{x}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}\frac{{{x}}^{{2}}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}\frac{{2}{}{y}{}{x}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{{x}}^{{2}}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{{x}}^{{2}}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}{-}\frac{{y}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}\frac{{y}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{y}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{x}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{,}{-}\frac{{y}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{y}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}\frac{{x}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}\frac{{y}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{x}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{x}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}{-}{y}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{x}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{x}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}\frac{{x}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{\mathrm{dx}}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{{\mathrm{dy}}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{\mathrm{dy}}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}\frac{{\mathrm{dx}}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{\mathrm{dx}}}{{3}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}\frac{{\mathrm{dy}}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}\right]$ (2.6)
 M > $\mathrm{nops}\left(\mathrm{S3}\right)$
 ${10}$ (2.7)

Check that all the tensors in $\mathrm{S2}$ are indeed Killing tensors using the CheckKillingTensor command.

 M > $\mathrm{map2}\left(\mathrm{CheckKillingTensor},g,\mathrm{S2}\right)$
 $\left[{0}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{,}{0}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{,}{0}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{,}{0}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{,}{0}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{,}{0}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}\right]$ (2.8)

Calculate all the rank 3 Killing tensors generated by $\mathrm{S3}$ and the metric $g$. We get the list $\mathrm{S3}$ again because the metric is already a constant linear combination of the rank 1 Killing tensors.

 M > $\mathrm{S3a}≔\mathrm{SymmetricProductsOfKillingTensors}\left(\left[\mathrm{K1},\left[g\right]\right],3\right):$
 M > $\mathrm{nops}\left(\mathrm{S3}\right)$
 ${10}$ (2.9)
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