Proceedings of 10th International Conference in MOdern GRoup ANalysis 2005, 58–63The Lie Derivative and Lie SymmetriesAlexander DAVISONSchool of Mathematics, University of the Witwatersrand, Johannesburg,Private Bag 3, WITS 2050, South AfricaE-mail: adavison@maths.wits.ac.zaOrdinarily Lie point symmetries, Noether symmetries, potential symmetriesand so on, of differential equations have been calculated by determining theaction of a vector field on solutions of the equation. An alternative method,devised by B. Kent Harrison and Frank Estabrook [4], calculates Lie symmetriesof differential equations by calculating the Lie derivative of differential formsassociated with the differential equation. In this paper the original method ismodified slightly and then extended to incorporate potential symmetries andapproximate symmetries. Examples are given.1 IntroductionA method for writing a differential equation or system of differential equationsin terms of differential forms is described. The method was devised by B. KentHarrison and F. Estabrook and the reader is referred to [4] for a more completedescription. For the properties of differential forms and their products see, e.g.,Do Carmo [3].A modification to the method is demonstrated on a wave equation with variablespeed and the modified method is extended to calculate approximate and potentialsymmetries, and finally, Noether symmetries.2 A ModificationIn Harrison and Estabrook’s original method differential equations were expressedin terms of differential forms and one requirement was that the differential formsshould form a differential ideal. Another requirement was that the Lie derivativeof these forms should remain in the ideal. Here, rather than ensuring that the Liederivative of our forms stays within an ideal, we ensure that the Lie derivativeof the forms is zero when the forms themselves are zero. There are some advantagesto this method, one of which being that it is easy to extend the methodto approximate symmetries. A wave equation with variable speed is used as anexample.Consider the equationu tt = e 2x u xx . (1)

The Lie Derivative and Lie Symmetries 59Let u t = w and u x = z. Then (1) becomes w t = e 2x z x , which is now a first orderequation. Letandα = du − wdt − zdx = u x dx + u t dt − wdt − zdx,α = 0 =⇒ u t = w ; u x = z, (annulling)(sectioning)β = dwdx + e 2x dzdt = (w x dx + w t dt)dx + e 2x (z x dx + z t dt)dt= w t dtdx + e 2x z x dxdt, (sectioning)β = 0 =⇒ w t = e 2x z x .(annulling)We do not worry about dα or dβ because it is not necessary that they are membersof an ideal. In fact, it turns out that imposing the condition dα = 0 (whichcorresponds to the equation u tx = u xt ) actually limits the number of symmetries.This is because L X β = 0 does not imply that L X dβ = 0.Now we calculate the Lie derivatives of α and β on solutions of (1), i.e. whenα = β = 0:L X α| α=0, β=0 = X⌋(dα) + d(X⌋α)| α=0, β=0 = −X z dx − X w dt + η x dx+η t dt + η u (zdx + wdt) − zξ x dx − zξ t dt − zξ u (zdx + wdt)−wτ x dx − wτ t dt − wτ u (zdx + wdt),where α = 0 implies du = wdt + zdx, and β = 0 has no effect because β isa 2-form. Separating the coefficients of dt and dx shows thatL X α| α=0, β=0 = 0 =⇒dx : X z = η x + z(η u − ξ x ) − z 2 ξ u − wτ x − wzτ u ,dt : X w = η t + w(η u − τ t ) − zξ t − zwξ u − w 2 τ u .In other words, L X α| α=0, β=0 = 0 gives us the prolongation coefficients of X.Now, our wave equation (1) is a second-order equation, but with this technique,we need only calculate prolongation coefficients up to first order.Next we turn our attention to β.L X β| α=0, β=0 = X⌋(dβ) + d(X⌋β)| α=0, β=0= (X w t − e 2x X z x − ze 2x X z u + wX w u )dtdx − (ξ t + e 2x X z w + wξ u )dtdw− e 2x (X z z + τ t + wτ u − X w w − ξ x − zξ u + 2ξ)dtdz− (X w z+ e 2x τ x + e 2x zτ u )dxdz,where the substitutions du = wdt + zdx and dwdx = e 2x dtdz were made. Separatingcoefficients of dtdx, etc, givesdtdx : X w t − e 2x X z x − ze 2x X z u + wX w u = 0,dtdw : ξ t + e 2x X z w + wξ u = 0,dtdz : X z z + τ t + wτ u − X w w − ξ x − zξ u + 2ξ = 0,dxdz : X w z + e 2x τ x + e 2x zτ u = 0.

60 A. **Davison**We substitute the prolongation formulae, so that all equations are in terms of w,z and functions independent of w and z. The coefficients of w and z are split,to get the usual determining equations, and, eventually, the symmetriesX 1 = u ∂∂u , X 2 = ∂ ∂t , X 3 = ∂X 4 = t ∂∂x − 1 2 (t2 + e −2x ) ∂ ∂t + tu 2∂x − t ∂ ∂t + u 2∂∂u ,∂∂u ,X ∞ = B(x, t) ∂∂u ,where B tt = e 2x B xx . We note that the algebra is complete, with commutators asfollows:[X 1 , X 2 ] = 0, [X 1 , X 3 ] = 0, [X 1 , X 4 ] = 0, [X 1 , X ∞ ] = −X ∞ ,∂[X 2 , X 3 ] = −X 2 , [X 2 , X 4 ] = X 3 , [X 2 , X ∞ ] = B t∂u = X ∞,([X 3 , X 4 ] = −X 4 , [X 3 , X ∞ ] = B x − tB t − 1 ) ∂2 B ∂u = X ∞,([X 4 , X ∞ ] = tB x − 1 (t 2 + e −2x) B t − t ) ∂22 B ∂u = X ∞.3 ExtensionsThe ideas above may also be used to calculate potential symmetries, approximatesymmetries and Noether symmetries.3.1 Potential SymmetriesThe method is demonstrated in the following example.Consider Burgers’ equation u xx − uu x − u t = 0 which has the associated auxilliarysystemv x = 2u, v t = 2u x − u 2 . (2)We introduce the 2-formsα = dvdt − 2udxdt = v x dxdt − 2udxdt,β = dvdx + 2dudt + u 2 dtdx = v t dtdx − 2u x dtdx + u 2 dtdx,which return the system (2) when annulled. It is unnecessary to introduce newvariables since the equations are already first-order. This means that no prolongationcoefficients need be calculated. To calculate a symmetry X = τ∂ t + ξ∂ x +φ∂ u + η∂ v of (2) we calculate the Lie derivatives of these forms. FirstlyL X α = X⌋dα + d(X⌋α) = (2φ − η x + 2uξ x + 2uτ t )dtdx+(2uξ u − η u )dtdu + (2uξ v − η v − τ t )dtdv − 2uτ u dxdu+(−τ x − 2uτ v )dxdv − τ u dudv.

The Lie Derivative and Lie Symmetries 63Next h 2 = ɛ −1 L X0 γ| I=0 = (F x − tF t + (uF u + 3wF w + zF z − 5F )/2) dtdx. Thenext step in our algorithm is to find X 1 (which we call Y to avoid confusion withsubscripts) such that L Y β| α=β=0 +h 2 = 0, where we recall that β = γ when ɛ = 0.L Y β| α=β=0 + h 2 = {Y ⌋dβ + d(Y ⌋β)} | α=β=0 + h 2 = ( Y wt− e 2x Y zx−ze 2x Y z u + wY w u +F x − tF t + u 2 F u + 3w 2 F w + z 2 F z − 5 2 F )dtdx+ ( −2ξ t − e 2x Y z w − wξ u)dtdw + e 2x (−ξ − Y zz − τ t − wτ u + Y w w+ξ x + zξ u ) dtdz + ( −Y wzThus L Y β| α=β=0 + h 2 = 0 implies that− e 2x τ x − ze 2x τ u)dxdz.Y wt− e 2x Y zx − ze 2x Y z u + wY w u + F x − tF t + u 2 F u + 3w 2 F w + z 2 F z − 5 2 F = 0,ξ t + e 2x Y z w + wξ u = 0,Y wz + e 2x τ x + ze 2x τ u = 0,2ξ + Y zz + τ t + wτ u − Y w w − ξ x − zξ u = 0,which is exactly the same set of determining equations that the ordinary methodgives and so from here on the calculations are identical.4 ConclusionWe see that the Lie derivative offers, in some ways, a more natural way of calculatingsymmetries of differential equations; with this method, fewer prolongationcoefficients need be calculated than with the traditional method.There are possible insights to be gained from the way the independent variablesare handled using these methods. For example, we see that the exact role played bythe equality (or lack of equality) of mixed derivatives of the dependent variables,and the way that this seems to limit the number of symmetries available, couldperhaps be made clearer.[1] V.A. Baikov, R.K. Gazizov and N.H. Ibragimov, Approximate Transformation Groupsand Deformations of Symmetry Lie Algebras, CRC Handbook of Lie Group Analysis ofDifferential Equations. Vol. 3 (ed. N.H. Ibragimov), CRC Press, Boca Raton, Florida,1996.[2] V.A. Baikov, R.K. Gazizov, N.H. Ibragimov, Perturbation Methods in Group Analysis,J. Sov. Math. 55, no. 1 (1991), 1450–1490.[3] M.P. Do Carmo, Differential Forms and Applications, Springer-Verlag, Berlin, 1994.[4] B.K. Harrison and F.B. Estabrook, Geometric Approach to Invariance Groups and Solutionof Partial Differential Systems, J. Math. Phys. 12 (1971), 653–666.