A vector field $X$is called a homothety vector for the metric $g$ if$\mathit{ }{\mathrm{\ℒ}}_{X}g \= \mathrm{\λ} g$, where${\mathrm{\ℒ}}_X$ denotes Lie differentiation with respect to $X$and $\mathrm{\λ}$ is a constant (proportional to the divergence of $X$). If $X_j \= g_{\mathrm{jk}} X^k$ and $▿$denotes covariant differentiation with respect to the given metric, then this equation can be written as $2{▿}_{\(i} X_{j\)} \= \mathrm{\λ} g_{\mathrm{ij}} \.$The Killing vectors of $g$are the solutions to this equation with $\mathrm{\λ} \=0.$The set of all homothety vector fields forms a Lie algebra of vector fields of dimension at most 1 greater than the dimension of the Lie algebra of Killing vector fields. The Killing vector fields are an ideal in the Lie algebra of homotheties.

The command HomothetyVectors generates the defining system of 1st order PDE for a homothety vector field and uses pdsolve to find the solutions to these PDE.

The command HomothetyVectors returns a sequence of two lists. The first list contains the homothety vector and the second the Killing vectors. If there are no genuine homothety vector fields then the first list is empty.

The keyword argument coefficientvariables $\= \left[x^1 \, x^2\, ..\. \, x^k\right]$ allows the user to specify the coefficient functions in the homothety $X$as functions of the variables $\left[x^1\, x^2\, ..\. \, x^k\right]$ .

The exact form of the homothety vector $X$can be specified with the keyword argument ansatz $\= X$ . For example, if the coordinates on the underlying manifold are $\left[x\, y \, z\right]$and $X_1\, X_2$ are defined vectors, then one may solve for homothety vectors of the form$X \= f\left(y\,z\right)X_1 \+ g\left(y\,z\right)X_2$. In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns $\= \left[f\left(y\,z\right)\, g\left(y\,z\right)\right]\.$

When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations $\= \mathrm{EqList}\.$Here $\mathrm{EqList}$is a list of the auxiliary equations to be added to the homothety equations.

If the metric $g$depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameters$\= \mathrm{ParList}\,$where $\mathrm{ParList}$is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the homothety vectors changes, are calculated.

With keyword argument output = $''pde''\,$the defining partial differential equations for the homothety vectors are returned. The option output = $''general''$returns the general homothety vector in terms of a number of arbitrary constants ${\mathrm{\_C}}_1$, ${\mathrm{\_C}}_{2 }\,$...  . The option output = $''list''$returns a list of vectors which form a basis for the solution space. The default value of this keyword argument is output = $''list''\.$

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form HomothetyVectors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-HomothetyVectors(...).

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

$\mathrm{DGsetup}\left(\left[u\,v\,x\,y\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(\mathrm{du}\&t\mathrm{du}-\exp \left(u\right)\mathrm{du}\&s\mathrm{dv}+\exp \left(u\right)\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)$

$g:=\mathrm{du}\mathrm{du}-\frac{{\ⅇ}^u}{2}\mathrm{du}\mathrm{dv}-\frac{{\ⅇ}^u}{2}\mathrm{dv}\mathrm{du}+{\ⅇ}^u\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}$

$H,K≔\mathrm{HomothetyVectors}\left(g\right)$

$H\,K:=\left[2\mathrm{D\_u}+2{\ⅇ}^{-u}\mathrm{D\_v}+y\mathrm{D\_y}\right]\,\left[\mathrm{D\_y}\,2y\mathrm{D\_v}+{\ⅇ}^u\mathrm{D\_y}\,-2\mathrm{D\_u}+2v\mathrm{D\_v}+x\mathrm{D\_x}\,\mathrm{D\_x}\,2x\mathrm{D\_v}+u\mathrm{D\_x}\,-2\mathrm{D\_v}\right]$

$\mathrm{evalDG}\left(\mathrm{LieDerivative}\left(H\left[1\right]\,g\right)-2g\right)$

$0\mathrm{du}\mathrm{du}$

$\mathrm{LieDerivative}\left(K\,g\right)$

$\left[0\mathrm{du}\mathrm{du}\,0\mathrm{du}\mathrm{du}\,0\mathrm{du}\mathrm{du}\,0\mathrm{du}\mathrm{du}\,0\mathrm{du}\mathrm{du}\,0\mathrm{du}\mathrm{du}\right]$

$\mathrm{Gamma}≔\left[\mathrm{op}\left(K\right)\,\mathrm{op}\left(H\right)\right]$

$\mathrm{\Γ}:=\left[\mathrm{D\_y}\,y\mathrm{D\_v}\+{\ⅇ}^u\mathrm{D\_y}\,\mathrm{D\_x}\,x\mathrm{D\_v}\+u\mathrm{D\_x}\,-2\mathrm{D\_u}\+2v\mathrm{D\_v}\+x\mathrm{D\_x}\,-\mathrm{D\_v}\,2\mathrm{D\_u}\+{\ⅇ}^{-u}\mathrm{D\_v}\+y\mathrm{D\_y}\right]$

$\mathrm{LieAlgebras}:-\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-\mathrm{e6}\,\left[\mathrm{e1}\,\mathrm{e7}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=2\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e7}\right]\=-\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=-\mathrm{e6}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e3}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=2\mathrm{e3}\+\mathrm{e4}\,\left[\mathrm{e4}\,\mathrm{e7}\right]\=-2\mathrm{e3}\,\left[\mathrm{e5}\,\mathrm{e6}\right]\=-2\mathrm{e6}\right]$

$\mathrm{LieBracket}\left(\mathrm{Gamma}\left[1\right]\,\mathrm{Gamma}\left[7\right]\right)$

$\mathrm{D\_y}$

$X≔\mathrm{evalDG}\left(\mathrm{D\_u}+a\left(u\,v\right)\mathrm{D\_v}+b\left(u\,v\right)y\mathrm{D\_y}\right)$

$X:=\mathrm{D\_u}\+a\left(u\,v\right)\mathrm{D\_v}\+b\left(u\,v\right)y\mathrm{D\_y}$

$\mathrm{HomothetyVectors}\left(g\,\mathrm{ansatz}=X\,\mathrm{unknowns}=\left[a\left(u\,v\right)\,b\left(u\,v\right)\right]\right)$

$\left[\mathrm{D\_u}\+\left(\frac{1}{2}{\ⅇ}^{-u}\+1\right)\mathrm{D\_v}\+\frac{1}{2}y\mathrm{D\_y}\right]\,\left[\right]$

$\mathrm{HomothetyVectors}\left(g\,\mathrm{output}=''general''\right)$

$\left(2\mathrm{\_C3}-2\mathrm{\_C6}\right)\mathrm{D\_u}\+\left({\ⅇ}^{-u}\mathrm{\_C3}\+2\mathrm{\_C6}v\+\mathrm{\_C5}x\+\mathrm{\_C2}y-\mathrm{\_C7}\right)\mathrm{D\_v}\+\left(\mathrm{\_C6}x\+\mathrm{\_C4}\+\mathrm{\_C5}u\right)\mathrm{D\_x}\+\left(\mathrm{\_C3}y\+\mathrm{\_C1}\+\mathrm{\_C2}{\ⅇ}^u\right)\mathrm{D\_y}$

$g≔\mathrm{evalDG}\left(\left(x^2+y^2+bx\right)\mathrm{du}\&t\mathrm{du}+\mathrm{dv}\&t\mathrm{dv}+\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)$

$g:=\left(x^2\+y^2\+bx\right)\mathrm{du}\mathrm{du}\+\mathrm{dv}\mathrm{dv}\+\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}$

$\mathrm{HomothetyVectors}\left(g\,\mathrm{parameters}=\left\{b\right\}\right)$

$\left[\left[v\mathrm{D\_v}\+x\mathrm{D\_x}\+y\mathrm{D\_y}\right]\,\left[\arctan \left(\frac{x}{y}\right)\mathrm{D\_u}-uy\mathrm{D\_x}\+ux\mathrm{D\_y}\,-y\mathrm{D\_x}\+x\mathrm{D\_y}\,\mathrm{D\_v}\,\mathrm{D\_u}\right]\right]\,\left[\left[\right]\,\left[-2y\mathrm{D\_x}\+\left(2x\+b\right)\mathrm{D\_y}\,\mathrm{D\_v}\,\mathrm{D\_u}\right]\right]\,\left[\left\{b\=0\right\}\,\left\{b\=b\right\}\right]$