 solve a list of tensor equations for an unknown list of tensors - Maple Programming Help

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DifferentialGeometry[DGsolve] - solve a list of tensor equations for an unknown list of tensors

 Calling Sequence DGsolve(Eq, T, options)

Parameters

 Eq - a vector, differential form or tensor constructed from the objects in the 2nd argument; or list of such. The vanishing of these tensors defines the equations to be solved. T - a vector, differential form, or tensor, depending upon a number of arbitrary parameters or functions; or a list of such auxiliaryequations - (optional) a keyword argument to specify a set of auxiliary equations, to be solved in conjunction with the equations specified by the first argument unknowns - (optional) list of parameters and functions, explicitly specifying the unknowns to be solved for. method - (optional) a Maple procedure which will be used to solve the equations other - (optional) additional arguments to be passed to the procedure used the solve the equations

Description

 • Let  be a vector, a differential form, or a tensor which depends upon a number of parameters . These parameters may be constants or functions. Now let $\mathrm{ℰ}$ be a differential-geometric construction depending upon which can be implemented in Maple by a sequence of commands in the DifferentialGeometry package. For example, $T$ could be a metric tensor and the Einstein tensor constructed from $g$. The command DGsolve will solve the equations obtained by setting to zero all the components of for the unknowns . The output is a set containing those $T$ solving $ℰ$=0 (obtainable by Maple).
 • Additional constraints (for example, initial conditions or inequalities) can be imposed upon the unknowns with the keyword argument auxiliaryequations.
 • The command DGsolve uses the general purpose solver PDEtools:-Solve to solve the system =0 for the unknowns . The keyword argument method can be used to specify a particular Maple solver (for example, solve, pdsolve, dsolve) or a customized solver created by the user.
 • If the equations defined by =0 are homogenous linear algebraic equations, then the command DGNullSpace can also be used.

Examples

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

Example 1.

Let  be a 4-dimensional space. We define a metric tensor depending upon an arbitrary function. We find the metrics which have vanishing Einstein tensor, and vanishing Bach tensor.

 > $\mathrm{DGsetup}\left(\left[x,y,u,v\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.1)
 > $g≔\mathrm{evalDG}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+f\left(x,u\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}\right)$
 ${g}{:=}{\mathrm{dx}}{\mathrm{dx}}{+}{\mathrm{dy}}{\mathrm{dy}}{+}{f}\left({x}{,}{u}\right){\mathrm{du}}{\mathrm{du}}{+}\frac{{1}}{{2}}{\mathrm{du}}{\mathrm{dv}}{+}\frac{{1}}{{2}}{\mathrm{dv}}{\mathrm{du}}$ (4.2)

Here are the metrics of the form (4.2) with vanishing Einstein tensor.

 M > $\mathrm{DGsolve}\left(\mathrm{EinsteinTensor}\left(g\right),g\right)$
 $\left\{{\mathrm{dx}}{\mathrm{dx}}{+}{\mathrm{dy}}{\mathrm{dy}}{+}\left({\mathrm{_F1}}\left({u}\right){x}{+}{\mathrm{_F2}}\left({u}\right)\right){\mathrm{du}}{\mathrm{du}}{+}\frac{{1}}{{2}}{\mathrm{du}}{\mathrm{dv}}{+}\frac{{1}}{{2}}{\mathrm{dv}}{\mathrm{du}}\right\}$ (4.3)

Here are the metrics of the form (4.2) with vanishing Bach tensor.

 M > $\mathrm{DGsolve}\left(\mathrm{BachTensor}\left(g\right),g\right)$
 $\left\{{\mathrm{dx}}{\mathrm{dx}}{+}{\mathrm{dy}}{\mathrm{dy}}{+}\left(\frac{{1}}{{6}}{\mathrm{_F1}}\left({u}\right){{x}}^{{3}}{+}\frac{{1}}{{2}}{\mathrm{_F2}}\left({u}\right){{x}}^{{2}}{+}{\mathrm{_F3}}\left({u}\right){x}{+}{\mathrm{_F4}}\left({u}\right)\right){\mathrm{du}}{\mathrm{du}}{+}\frac{{1}}{{2}}{\mathrm{du}}{\mathrm{dv}}{+}\frac{{1}}{{2}}{\mathrm{dv}}{\mathrm{du}}\right\}$ (4.4)

Example 2.

In this example we define a 2-form which depends upon parameters . We find those values of the parameters for which

 M > $\mathrm{DGsetup}\left(\left[x,y,u,v\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.5)
 M > $\mathrm{\alpha }≔\mathrm{evalDG}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+r\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}+s\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}\right):$
 M > $\mathrm{DGsolve}\left(\mathrm{\alpha }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\alpha },\mathrm{\alpha },\left\{r,s\right\}\right)$
 $\left\{{\mathrm{dx}}{\bigwedge }{\mathrm{dy}}{+}{r}{\mathrm{dx}}{\bigwedge }{\mathrm{du}}{,}{\mathrm{dx}}{\bigwedge }{\mathrm{dy}}{+}{s}{\mathrm{dy}}{\bigwedge }{\mathrm{dv}}\right\}$ (4.6)

Example 3.

We define a connection $\mathrm{Γ}$ and calculate the parallel transport of a vector $X\left(t\right)$ along a curve $C\left(t\right)$.

 M > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.7)
 M > $\mathrm{Gamma}≔\mathrm{Connection}\left(-\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\left(\mathrm{D_y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${\mathrm{Γ}}{:=}{-}{\mathrm{D_x}}{\mathrm{dx}}{\mathrm{dy}}{+}{\mathrm{D_y}}{\mathrm{dy}}{\mathrm{dx}}$ (4.8)
 M > $C≔\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right]$
 ${C}{:=}\left[{\mathrm{cos}}\left({t}\right){,}{\mathrm{sin}}\left({t}\right)\right]$ (4.9)
 M > $X≔\mathrm{evalDG}\left(A\left(t\right)\mathrm{D_x}+B\left(t\right)\mathrm{D_y}\right)$
 ${X}{:=}{A}\left({t}\right){\mathrm{D_x}}{+}{B}\left({t}\right){\mathrm{D_y}}$ (4.10)
 M > $\mathrm{DGsolve}\left(\mathrm{ParallelTransportEquations}\left(C,X,\mathrm{Gamma},t\right),X\right)$
 $\left\{{\mathrm{_C2}}{{ⅇ}}^{{\mathrm{sin}}\left({t}\right)}{\mathrm{D_x}}{+}{\mathrm{_C1}}{{ⅇ}}^{{-}{\mathrm{cos}}\left({t}\right)}{\mathrm{D_y}}\right\}$ (4.11)

We can use the keyword argument auxiliaryequations to specify an initial position for the vector $X.$

 M > $\mathrm{DGsolve}\left(\mathrm{ParallelTransportEquations}\left(C,X,\mathrm{Gamma},t\right),X,\mathrm{auxiliaryequations}=\left\{A\left(0\right)=1,B\left(0\right)=0\right\}\right)$
 $\left\{{{ⅇ}}^{{\mathrm{sin}}\left({t}\right)}{\mathrm{D_x}}\right\}$ (4.12)

Example 4.

The source-free Maxwell equations may be expressed in terms of a 2-form $F$ by the equations  and , where $d$ is the exterior derivative and $*$ is the Hodge star operator. In this example we define a 2-form depending on 2 functions of 4 variables and solve the Maxwell equations for $F.$

 M > $\mathrm{DGsetup}\left(\left[x,y,z,t\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.13)
 M > $g≔\mathrm{evalDG}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}-\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}\right)$
 ${g}{:=}{\mathrm{dx}}{\mathrm{dx}}{+}{\mathrm{dy}}{\mathrm{dy}}{+}{\mathrm{dz}}{\mathrm{dz}}{-}{\mathrm{dt}}{\mathrm{dt}}$ (4.14)
 M > $F≔\mathrm{evalDG}\left(A\left(x,y,z,t\right)\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+B\left(x,y,z,t\right)\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}\right)$
 ${F}{:=}{A}\left({x}{,}{y}{,}{z}{,}{t}\right){\mathrm{dx}}{\bigwedge }{\mathrm{dy}}{+}{B}\left({x}{,}{y}{,}{z}{,}{t}\right){\mathrm{dx}}{\bigwedge }{\mathrm{dt}}$ (4.15)
 M > $\mathrm{DGsolve}\left(\left[\mathrm{ExteriorDerivative}\left(F\right),\mathrm{ExteriorDerivative}\left(\mathrm{HodgeStar}\left(g,F,\mathrm{detmetric}=-1\right)\right)\right],F\right)$
 $\left\{\left({\mathrm{_F1}}\left({t}{+}{y}\right){+}{\mathrm{_F2}}\left({t}{-}{y}\right)\right){\mathrm{dx}}{\bigwedge }{\mathrm{dy}}{+}\left({\mathrm{_F1}}\left({t}{+}{y}\right){-}{\mathrm{_F2}}\left({t}{-}{y}\right){+}{\mathrm{_C1}}\right){\mathrm{dx}}{\bigwedge }{\mathrm{dt}}\right\}$ (4.16)

 See Also