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DifferentialGeometry

 InfinitesimalTransformation
 compute the Lie algebra of infinitesimal generators for an action of a Lie group on a manifold

 Calling Sequence InfinitesimalTransformation(phi, par, initialpoint)

Parameters

 phi - a transformation from a manifold M to M, depending upon 1 or more parameters [a, b, ...]. par - a list of the group parameters appearing in the transformation phi id - (optional) a list of equations initialpoint = [a = a0, b = b0, ... ] specifying the values of the parameters a, b, ... which give the identity transformation on M; the default value each parameter value is 0.

Description

 • Let mu: G x M -> M define a (right) action of an r-dimensional Lie group G on a manifold M.
 • Let R_i, i = 1 ... r denote a basis for the right invariant vector fields on G.  Then the vector fields X_i(x) = mu_*(e, x)(R_i(e)) (where mu_* is the Jacobian of mu, e is the identity element of G, and x is a point of M) define a Lie algebra of vector fields on M whose structure constants coincide with the structure constants of the Lie algebra of right invariant vector fields R_i. The vector fields X_i are called the infinitesimal generators for the action mu.
 • For convenience, the command InfinitesimalTransformation treats the action mu as a parameterized family of transformations phi: M -> M.  The infinitesimal transformations are then computed by taking the derivatives of the components of phi with respect to the group parameters and evaluating the result at the identity.  A list of vector fields is returned, one vector field for each group parameter in par.
 • This command is part of the DifferentialGeometry package, and so can be used in the form InfinitesimalTransformation(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-InfinitesimalTransformation.

Examples

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

Example 1.

The group of Euclidean motions in the plane, consisting of translations in the coordinate directions and rotations about the origin. We initialize the coordinates on the plane and define a 3-parameter transformation consisting of all the Euclidean motions.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$
 > $\mathrm{Φ1}≔\mathrm{Transformation}\left(M,M,\left[x=\mathrm{cos}\left(\mathrm{\theta }\right)x+\mathrm{sin}\left(\mathrm{\theta }\right)y+a,y=-\mathrm{sin}\left(\mathrm{\theta }\right)x+\mathrm{cos}\left(\mathrm{\theta }\right)y+b\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\mathrm{cos}{}\left(\mathrm{θ}\right)& \mathrm{sin}{}\left(\mathrm{θ}\right)\\ -\mathrm{sin}{}\left(\mathrm{θ}\right)& \mathrm{cos}{}\left(\mathrm{θ}\right)\end{array}\right]\right]\right]{,}\left[\left[{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{x}{+}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{y}{+}{a}{,}{x}\right]{,}\left[{-}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{x}{+}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{y}{+}{b}{,}{y}\right]\right]\right]\right)$ (1)
 > $\mathrm{Γ1}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{Φ1},\left[a,b,\mathrm{\theta }\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{y}\right]{,}\left[\left[{2}\right]{,}{-}{x}\right]\right]\right]\right)\right]$ (2)

To calculate the structure equations for this Lie algebra of vector fields, use the LieAlgebraData command from the LieAlgebras package.  Here [e1, e2, e3] denote the vectors in Gamma and only the non-trivial brackets are displayed.

 > $\mathrm{LieAlgebraData}\left(\mathrm{Γ1}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (3)

Example 2.

The group of fractional linear transformations on the line.

 > $\mathrm{DGsetup}\left(\left[x\right],R\right):$
 > $\mathrm{Φ2}≔\mathrm{Transformation}\left(M,R,\left[x=\frac{ax+b}{cx+d}\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{R}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\frac{a}{c{}x+d}-\frac{\left(a{}x+b\right){}c}{{\left(c{}x+d\right)}^{2}}& 0\end{array}\right]\right]\right]{,}\left[\left[\frac{{a}{}{x}{+}{b}}{{c}{}{x}{+}{d}}{,}{x}\right]\right]\right]\right)$ (4)

The identity transformation is given by a = 1, b = 0, c = 0, d = 1.  Only the non-zero parameter values need to be specified.

 > $\mathrm{Γ2}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{Φ2},\left[a,b,c,d\right],\mathrm{initialpoint}=\left[a=1,d=1\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{{x}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{x}\right]\right]\right]\right)\right]$ (5)

Note that these vectors fields are not linearly independent over the real numbers (Gamma2 = - Gamma2).  This is because the parameter values [a = t, b = 0, c = 0, d = 1] and [a = 1, b = 0, c = 0, d = 1/t] generate the same 1-parameter group of transformations, that is, the action is not effective.

We can remove the linearly dependent elements of Gamma2 a with the DGbasis command.

 > $\mathrm{Γ2}≔\mathrm{DGbasis}\left(\mathrm{Γ2},\mathrm{method}="real"\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{{x}}^{{2}}\right]\right]\right]\right)\right]$ (6)

Alternatively, we can make the action effective by normalizing the parameters to a*b - c*d = 1.  (Now the group is SL2, the set of all 2 x 2 matrices with a determinant of 1.)

 > $G≔\mathrm{solve}\left(ad-bc=1,\left\{d\right\}\right)$
 ${G}{≔}\left\{{d}{=}\frac{{b}{}{c}{+}{1}}{{a}}\right\}$ (7)
 > $\mathrm{Φ3}≔\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{Φ2},G\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{R}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\frac{{a}^{2}}{{\left(a{}c{}x+b{}c+1\right)}^{2}}& 0\end{array}\right]\right]\right]{,}\left[\left[\frac{\left({a}{}{x}{+}{b}\right){}{a}}{{1}{+}\left({a}{}{x}{+}{b}\right){}{c}}{,}{x}\right]\right]\right]\right)$ (8)
 > $\mathrm{Γ3}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{Φ3},\left[a,b,c\right],\mathrm{initialpoint}=\left[a=1\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{2}{}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{R}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{{x}}^{{2}}\right]\right]\right]\right)\right]$ (9)

Example 3.

The group of fractional linear transformations in the plane.

 > $\mathrm{Φ4}≔\mathrm{Transformation}\left(M,M,\left[x=\frac{ax+by+c}{rx+sy+t},y=\frac{dx+ey+f}{rx+sy+t}\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\frac{a}{r{}x+s{}y+t}-\frac{\left(a{}x+b{}y+c\right){}r}{{\left(r{}x+s{}y+t\right)}^{2}}& \frac{b}{r{}x+s{}y+t}-\frac{\left(a{}x+b{}y+c\right){}s}{{\left(r{}x+s{}y+t\right)}^{2}}\\ \frac{d}{r{}x+s{}y+t}-\frac{\left(d{}x+e{}y+f\right){}r}{{\left(r{}x+s{}y+t\right)}^{2}}& \frac{e}{r{}x+s{}y+t}-\frac{\left(d{}x+e{}y+f\right){}s}{{\left(r{}x+s{}y+t\right)}^{2}}\end{array}\right]\right]\right]{,}\left[\left[\frac{{a}{}{x}{+}{b}{}{y}{+}{c}}{{r}{}{x}{+}{s}{}{y}{+}{t}}{,}{x}\right]{,}\left[\frac{{d}{}{x}{+}{e}{}{y}{+}{f}}{{r}{}{x}{+}{s}{}{y}{+}{t}}{,}{y}\right]\right]\right]\right)$ (10)
 > $\mathrm{Γ4}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{Φ4},\left[a,b,c,d,e,f,r,s,t\right],\mathrm{initialpoint}=\left[a=1,e=1,t=1\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{{x}}^{{2}}\right]{,}\left[\left[{2}\right]{,}{-}{y}{}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{y}{}{x}\right]{,}\left[\left[{2}\right]{,}{-}{{y}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{x}\right]{,}\left[\left[{2}\right]{,}{-}{y}\right]\right]\right]\right)\right]$ (11)

Again we have to remove linearly dependent vectors:

 > $\mathrm{Γ5}≔\mathrm{DGbasis}\left(\mathrm{Γ4},\mathrm{method}="real"\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{{x}}^{{2}}\right]{,}\left[\left[{2}\right]{,}{-}{y}{}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{y}{}{x}\right]{,}\left[\left[{2}\right]{,}{-}{{y}}^{{2}}\right]\right]\right]\right)\right]$ (12)
 > $\mathrm{LieAlgebraData}\left(\mathrm{Γ5}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e1}}{-}{2}{}{\mathrm{e5}}\right]$ (13)
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