Conservation laws for the geodesic equations of the canonical connection on Lie groups in dimensions two and three

Transcription

1 Appl Math Inf Sci 7 No (013) 311 Applied Mathematics & Information Sciences An International Journal Conservation laws for the geodesic equations of the canonical connection on Lie groups in dimensions two and three Ryad A Ghanam 1 and Ahmad Y Al-Dweik 1 Department of Mathematics University of Pittsburgh Greensburg PA USA Department of Mathematics King Fahd University of Petroleum & Minerals Dhahran 3161 KSA Received: 18 Jul 01; Revised 19 Oct 01; Accepted 3 Oct 01 Published online: 1 Jan 013 Abstract: In order to characterize the systems of second-order ODEs which admit a regular Lagrangian function Noether symmetries for the geodesic equations of the canonical linear connection on Lie groups of dimension three or less are obtained so the characterization of these geodesic equations through their Noether s symmetries Lie Algebras is investigated The corresponding conservation laws and the first integral for each geodesics are constructed Keywords: Systems of second-order ODEs inverse problem of Lagrangian dynamics conservations laws geodesic equations canonical linear connection on Lie groups 1 Introduction Second order ordinary differential equations (ODEs) on finite dimensional manifolds appear in a wide variety of applications in mathematics physics and engineering In classical mechanics they are Newtons equations of motion and the Euler-Lagrange equations of a mechanical Lagrangian The inverse problem of Lagrangian dynamics consists of finding necessary and sufficient conditions for a system of second order ODEs to be the EulerLagrange equations of a regular Lagrangian function and in case they are to describe all possible such Lagrangians The inverse problem of Lagrangian dynamics for the geodesic spray associated to the canonical symmetric linear connection on a Lie group of dimension three or less was solved in [1] This connection was first introduced by Cartan and JA Schouten [] and its properties were studied in details in [3] In [1] it was proved that the geodesic equations of this connection are variational for all Lie groups in dimensions two and three Moreover an explicit Lagrangians for each group in these dimensions were given For more details on the inverse problem and the canonical connection we refer the reader to [4513] Our need to characterize those systems of second-order ODEs which admit a regular Lagrangian function indicates the importance of studying these geodesics that were constructed In this paper the characterization of these geodesics through their Noether s symmetries Lie Algebras is investigated The corresponding conservation laws and the first integral for each geodesics are obtained The outline of the paper is as follows In Section we investigate the Noether symmetries and Noether theorem Lastly in Sections 3 and 4 the Noether symmetries for all the geodesic equations were derived with the integrals of motion The classification of the Lie algebras of the Noether symmetries is studied Concluding and remarks are given in the last section Noether symmetries and Noether theorem let us consider the k-th order system of partial differential equations (PDEs) of n independent variables x = (x 1 x x n ) and m dependent variables u = (u 1 u u m ) E α (x u u (1) u (k) ) = 0 α = 1 m (1) Corresponding author: ghanam@pittedu

2 31 R A Ghanam A Y Al-Dweik: Geodesic equations of the canonical connection on where u (1) u () u (k) denote the collections of all first second k-order partial derivatives ie u α i = D i (u α ) u α ij = D jd i (u α )respectively with the total differentiation operator with respect to x i given by D i = x i + u α i u α + u α ij u α j + i = 1 n () in which the summation convention is used The following definitions are well-known (see eg [7 10]) Definition 1(The conserved vector) The n-tuple vector T = (T 1 T T n ) T j A j = 1 n is a conserved vector of (1) if T i satisfies D i T i (1) = 0 (3) where A is the space of differential functions Definition (The Lie-Bäcklund operator) The Lie-Bäcklund operator is X = ξ i x + η α i u ξ i η α A (4) α where A is the space of differential functions The operator (4) is an abbreviated form of the infinite formal sum X = ξ i x + η α i u α + s 1 ζi α 1i i s u α i 1 i is (5) where the additional coefficients are determined uniquely by the prolongation formulae ζ α i = D i (W α ) + ξ j u α ij ζ α i 1i s = D i1 D is (W α ) + ξ j u α ji 1i s s > 1 in which W α is the Lie characteristic function (6) W α = η α ξ j u α j (7) Definition 3(The Euler-Lagrange operator) The Euler- Lagrange operator for each α is δ δu = α u + ( 1) s D α i1 D is s 1 u α i 1 i is α = 1 m (8) Definition 4(Lagrangian and Euler-Lagrange equations) If there exists a function L = L(x u u (1) u (l) ) A l k such that the system (1) can be written as δl/δu α = 0 then L is called a Lagrangian of the system (1) and the differential equations of the form δl δu α = 0 α = 1 m (9) are called Euler-Lagrange equations Definition 5(The Action Integral) The action integral of a Lagrangian L is the following functional J[u] = L(x u u (1) u (k) )dx (10) Ω where L is defined on a domain Ω in the space x = (x 1 x x n ) Definition 6The functional (10) is said to be invariant with respect to the group G r if for all transformations of the group and all functions u = u(x) the following equality is fulfilled irrespective of the choice of the domain of integration L(x u u (1) u (k) )dx Ω = Ω L( x ū ū (1) ū (k) )d x (11) where ū and Ω are the images of u and Ω respectively under the group G r Lemma 1[10] The functional (10) is invariant with respect to the group G r with the Lie-Bäcklund operator X of the form (5) if and only if the following equalities hold W α δl/δu α + D i (N i L) XL + L D i ξ i = D i B i (1) where δ N i = ξ i + W α δu α i + D i1 D is (W α δ ) δu i = 1 n α ii s 1 1 i is (13) Theorem 1(Noether s Theorem) [10] Let the functional (10) be invariant with respect to the group G r with the Lie-Bäcklund operator X of the form (5) Then the Euler- Lagrange equations (9) have r linearly independent conservation laws D i T i = 0 where T i = B i N i L i = 1 n (14) 3 The two-dimensional Lie algebra In this section there are two Lie algebras to consider namely the non-abelian g non abelian and the abelian one In both cases the geodesic equations of the canonical connection and their corresponding non-singular Lagrangians are given The Lie algebras of the Noether s symmetries for both cases are classified and the integrals of motions are obtained For simplicity we shall denote the derivatives (ẋ ẏ ż) of x y and z wrt the independent variable t by (x 1 ) As a final remark since the calculations are very lengthy we will only do the non-abelian -dimensional case (31) in details and for the rest of the cases we will list the Noether s symmetries

3 Appl Math Inf Sci 7 No (013) / wwwnaturalspublishingcom/journalsasp [e 1 e ] = e 1 (The Lie algebra of the affine group on the real line g non abelian ) In local coordinates (x y) the geodesic equations are given by ẍ = ẋẏ (31) ÿ = 0 System (31) has a Lagrangian [1] L = e y x 1 + y 1 satisfies (1) viz (3) X = ξ 1 t + η 1 x + η y (33) η x 1 + t η x 1 x 1 t η 1 x 1 x η 1 x 1 y η 1 + y η x y B 1e y + t B 1 e y t η 3 e y 4 y η e y + 4 t ξ 1e y + 5 y ξ 1e y + x 1 x B 1 e y 3 +x 1 x η x 1 x η y 3 1 e y + y 4 1 x 1 x ξ 1e y = 0 (34) Now comparing coefficients of the derivatives of x and y we obtain the following overdetermined linear system t η 1 = 0 η x η 1 + y η = 0 t η = 0 y η t ξ 1 = 0 x η = 0 y η 1 x B 1e y = 0 x η = 0 t η y B 1 = 0 x ξ 1 = 0 t B 1 = 0 y ξ 1 = 0 Solving this system gives rise to ξ 1 = c 1 η 1 = 1 c 3 x + c 4 e y + c η = c 3 B 1 = c 4 x + c 5 (35) (36) At this stage we construct the Noether symmetries corresponding to each of the constant involved These are total of four generators X 1 = t X 3 = x x + y X = x X 4 = e y x (37) the 4 Noether symmetries (37) are [X X 3 ] = X [X 3 X 4 ] = X 4 (38) e 1 = X e = X 3 e 3 = X 4 (39) g = e 1 e e 3 R (310) where e 1 e e 3 is the three-dimensional solvable Lie algebra the non-zero brackets [e 1 e ] = e 1 [e e 3 ] = e 3 To find the conservation laws corresponding to the above Noether symmetries we use the formula (14) T 1 = T = e y x 1 T 3 = (x1y1x x1 + 3 e y )e y T 4 = x x 1 3 [e 1 e ] = 0 (Abelian Lie algebra) (311) In local coordinates (x y) the geodesic equations are given by ẍ = 0 (31) ÿ = 0 System (31) has a Lagrangian [1] L = x 1 + (313) X = ξ 1 t + η 1 x + η y (314) satisfies (1) can be estimated easily as X 1 = t X 3 = y X 5 = t y X 7 = t t + x x + y y X = x X 4 = t x X 6 = y x x X 8 = t t + tx x + ty y y (315) with the gauge term B 1 (t x y) = c 8 ( x + y ) +c 4 x+ c 5 y + c 9 the 8 Noether symmetries (315) are [X 1 X 4 ] = X [X 1 X 5 ] = X 3 [X 1 X 7 ] = X 1 [X 1 X 8 ] = X 7 [X X 6 ] = X 3 [X X 7 ] = X [X X 8 ] = X 4 [X 3 X 6 ] = X [X 3 X 7 ] = X 3 [X 3 X 8 ] = X 5 [X 4 X 6 ] = X 5 [X 4 X 7 ] = X 4 [X 5 X 6 ] = X 4 [X 5 X 7 ] = X 5 [X 7 X 8 ] = X 8 (316) e 1 = X e = X 3 e 3 = X 4 e 4 = X 5 e 5 = X 6 e 6 = X 1 e 7 = X 8 e 8 = X 7 (317) The Lie algebra g of the Noether symmetries is a semidirect sum of the form g = e 1 e e 3 e 4 e 5 sl( R) (318) where e 1 e e 3 e 4 e 5 is the five-dimensional solvable Lie algebra the non-zero brackets [e 1 e 5 ] = e [e e 5 ] = e 1 [e 3 e 5 ] = e 4 [e 4 e 5 ] = e 3

4 314 R A Ghanam A Y Al-Dweik: Geodesic equations of the canonical connection on T 1 = x 1 + T = x 1 T 3 = T 4 = x x 1 t T 5 = y t T 6 = x 1 y x T 7 = ( x 1 + ) t x 1 x y T 8 = ( x 1 + ) t ( y + x 1 x) t + x + y (319) 4 The three-dimensional Lie algebra In this section we consider the list of the three-dimensional Lie algebras described in Jacobson s classification [6] Following the convention in [1] the brackets of the Lie algebras (41-46) are [e 1 e ] = 0 [e 1 e 3 ] = a e 1 + c e [e e 3 ] = b e 1 + d e (41) and their corresponding geodesic equations take the form ẍ = (aẋ + bẏ)ż ÿ = (cẋ + dẏ)ż (4) For the algebra 47 it is a direct sum of the two-dimensional non-abelian algebra with the reals and so we only add to the system of the geodesic equation for the twodimensional case and for the algebras in 48 the connection is flat and we consider a quadratic Lagrangian For all cases we only list the Noether symmetries classify their Lie algebras and obtain the integrals of motion 41 [e 1 e 3 ] = a e 1 [e e 3 ] = d e where (a + d) 0 ad(a d) 0 In local coordinates (x y z) the geodesic equations are ẍ = aẋż ÿ = dẏż (43) System (43) has a Lagrangian [1] L = e a z x 1 + e d z + (44) with the gauge term B 1 (t x y z) = c 5 a x + c 6 d y + c 7 the 6 Noether symmetries (46) are [X X 4 ] = a X [X 3 X 4 ] = d X 3 [X 4 X 5 ] = a X 5 [X 4 X 6 ] = d X 6 (47) e 1 = X e = X 3 e 3 = X 4 e 4 = X 5 e 5 = X 6 e 6 = X 1 (48) g = e 1 e e 3 e 4 e 5 R (49) where e 1 e e 3 e 4 e 5 is the five-dimensional solvable Lie algebra the non-zero brackets [e 1 e 3 ] = ae 1 [e e 3 ] = de [e 3 e 4 ] = ae 4 [e 3 e 5 ] = d e 5 T 1 = z 1 T = e a z x 1 T 3 = e d z T 4 = e z(a+d) ( z 3 1 e z(a+d) + (y d )e a z +x 1 e d z (x a x 1 )) T 5 = xa x 1 a T 6 = yd d (410) 4 [e 1 e 3 ] = a e 1 b e [e e 3 ] = b e 1 + a e where a 0 b 0 a + b = 1 In local coordinates (x y z) the geodesic equations are ẍ = (aẋ + bẏ)ż ÿ = ( bẋ + aẏ)ż (411) System (411) has a Lagrangian [1] L = e a z (( x 1 ) cos (bz) + x 1 sin (bz) ) + 3 (41) X = ξ 1 t + η 1 x + η y + η 3 z (413) satisfies (1) can be estimated easily as X = ξ 1 t + η 1 x + η y + η 3 z (45) satisfies (1) can be estimated easily as X 1 = t X 4 = ax x + dy y + z X = x X 5 = e az x (46) X 3 = y X 6 = e dz y X 1 = t X = x X 3 = y X 4 = (a x + b y) x + (a y b x) y + z X 5 = (b cos (b z) a sin (b z)) e a z x (a cos (b z) + b sin (b z)) e a z y X 6 = (a cos (b z) + b sin (b z)) e a z x + (b cos (b z) a sin (b z)) e a z y (414)

5 Appl Math Inf Sci 7 No (013) / wwwnaturalspublishingcom/journalsasp 315 with the gauge term B 1 (t x y z) = c 6 x + c 5 y + c 7 the 6 Noether symmetries (414) are [X X 4 ] = a X b X 3 [X 3 X 4 ] = b X + a X 3 [X 4 X 5 ] = a X 5 b X 6 [X 4 X 6 ] = b X 5 + a X 6 (415) e 1 = X e = X 3 e 3 = X 4 e 4 = X 5 e 5 = X 6 e 6 = X 1 (416) g = e 1 e e 3 e 4 e 5 R (417) where e 1 e e 3 e 4 e 5 is the five-dimensional solvable Lie algebra the non-zero brackets [e 1 e 3 ] = ae 1 be [e e 3 ] = be 1 + ae [e 3 e 4 ] = ae 4 be 5 [e 3 e 5 ] = be 4 + ae 5 T 1 = 3 T = z (x1 cos(b z) y1 sin(b z))e a z (y1 cos(b z)+x1 sin(b z))e a T 3 = T 4 = ( ( y + xx 1 ) a x 1 + y ) 1 cos (b z) e a z + (( x 1 y x) a + x 1 ) sin (b z) e a z + (x 1 y + x) b cos (b z) e a z z ( y + xx 1 ) b sin (b z) e a z z 1 1 T 5 = y b x 1 a T 6 = x+b a x 1 43 [e 1 e 3 ] = e 1 [e e 3 ] = e 1 + e (418) In local coordinates (x y z) the geodesic equations are ẍ = (ẋ + ẏ)ż ÿ = ẏż (419) System (419) has a Lagrangian [1] L = x 1 (ln ln z)+ e z +y + (40) X = ξ 1 t + η 1 x + η y + η 3 z (41) satisfies (1) can be estimated easily as X 1 = t X = x (4) X 3 = y with the gauge term B 1 (t x y z) = c 3 z + c 4 The commutators for the Lie algebra arising from the 3 Noether symmetries (4) are all zeros so the Lie algebra g of the Noether symmetries is the three-dimensional abelian Lie algebra T 1 = T = ln ln z T 3 = z x 1 e z (43) 44 [e 1 e 3 ] = e [e e 3 ] = e 1 (The Euclidean Group on the plane E()) In local coordinates (x y z) the geodesic equations are ẍ = ẏż ÿ = ẋż (44) System (44) has a Lagrangian [1] L = x yx x 1 + (45) X = ξ 1 t + η 1 x + η y + η 3 z (46) satisfies (1) can be estimated easily as X 1 = t X 5 = y x x y X = x X 6 = sin z x + cos z y y X 3 = y X 7 = cos z x sin z X 4 = z (47) with the gauge term B 1 (t x y z) = (xc 6 yc 7 ) cos (z)+ ( xc 7 yc 6 ) sin (z) xc 3 + c y + c 8 the 7 Noether symmetries (47) are [X X 5 ] = X 3 [X 3 X 5 ] = X [X 4 X 6 ] = X 7 [X 4 X 7 ] = X 6 [X 5 X 6 ] = X 7 [X 5 X 7 ] = X 6 (48) e 1 = X e = X 3 e 3 = X 4 + X 5 e 4 = X 4 e 5 = X 6 e 6 = X 7 e 7 = X 1 (49) The Lie algebra of the Noether symmetries is decomposable as a direct sum of two copies of the Euclidean algebra and the reals (E() E() R) T 1 = z 1 T = y x 1 T 3 = x + T 4 = y1 z 1 + x1 z 1 T 5 = y + x + xy1 T 6 = sin(z)x 1+cos(z) T 7 = cos(z)x 1 sin(z) yx1 (430)

6 316 R A Ghanam A Y Al-Dweik: Geodesic equations of the canonical connection on 45 [e 1 e 3 ] = e 1 [e e 3 ] = e In local coordinates (x y z) the geodesic equations are ẍ = ẋż ÿ = ẏż (431) System (431) has a Lagrangian [1] L = e z x 1 + (43) X = ξ 1 t + η 1 x + η y + η 3 z (433) satisfies (1) can be estimated easily as X 1 = t X 4 = y y + z X 6 = e z y X = x X 5 = x x y y X 7 = e z x (434) X 3 = y with the gauge term B 1 (t x y z) = c 6 x + c 7 y + c 8 the 7 Noether symmetries (434) are [X X 5 ] = X [X 3 X 4 ] = X 3 [X 3 X 5 ] = X 3 [X 4 X 7 ] = X 7 [X 5 X 6 ] = X 6 [X 5 X 7 ] = X 7 (435) e 1 = X e = X 4 + X 5 e 3 = X 6 e 4 = X 3 e 5 = X 4 e 6 = X 7 e 7 = X 1 (436) g = e 1 e e 3 e 4 e 5 e 6 R (437) where e 4 e 5 e 6 is a copy of e 1 e e 3 the non-zero brackets [e 1 e 3 ] = e 1 [e e 3 ] = e T 1 = T = e z T 3 = x1 T 4 = + x 1( y ) T 6 = x x 1 T 7 = y e z e z T 5 = y1x x1y e z 46 [e 1 e 3 ] = e 1 [e e 3 ] = e (438) In local coordinates (x y z) the geodesic equations are ẍ = ẋż ÿ = ẏż (439) System (439) has a Lagrangian [1] L = e z x 1 + e z + (440) X = ξ 1 t + η 1 x + η y + η 3 z (441) satisfies (1) can be estimated easily as X 1 = t X 4 = e z x X 6 = ye z x xe z y X = x X 5 = e z y X 7 = x x y y + z X 3 = y (44) with the gauge term B 1 (t x y z) = (c 6 y + c 4 ) x c 5 y + c 8 the 7 Noether symmetries (44) are [X X 6 ] = X 5 [X X 7 ] = X [X 3 X 6 ] = X 4 [X 3 X 7 ] = X 3 [X 4 X 6 ] = X 3 [X 4 X 7 ] = X 4 [X 5 X 6 ] = X [X 5 X 7 ] = X 5 (443) e 1 = X i X 5 e = X 3 i X 4 e 3 = (X 7+i X 6 ) (X7 i X6) e 4 = X + i X 5 e 5 = X 3 + i X 4 e 6 = e 7 = X 1 (444) over the complex numbers g = e 1 e e 3 e 4 e 5 e 6 R (445) where e 4 e 5 e 6 is a copy of e 1 e e 3 the non-zero brackets [e 1 e 3 ] = e 1 [e e 3 ] = e T 1 = z 1 T = x 1 e z T 3 = e z T 4 = x x 1 T 5 = y + T 6 = xy x 1y T 7 = ( y+ ) e z + x 1(x 1 x) e z 47 [e 1 e ] = e 1 ( g non abelian R ) + x (446) In local coordinates (x y z) the geodesic equations are ẍ = ẋẏ ÿ = 0 (447) System (447) has a Lagrangian [1] L = e y x 1 + y 1 + (448) X = ξ 1 t + η 1 x + η y + η 3 z (449)

7 Appl Math Inf Sci 7 No (013) / wwwnaturalspublishingcom/journalsasp 317 satisfies (1) can be estimated easily as X 1 = t X 4 = e y x X = x X 5 = t z (450) X 3 = z X 6 = x x + y with the gauge term B 1 (t x y) = c 4 x + c 5 z + c 7 the 6 Noether symmetries (450) are [X 1 X 5 ] = X 3 [X X 6 ] = X [X 4 X 6 ] = X 4 (451) e 1 = X e = X 6 e 3 = X 4 e 4 = X 1 e 5 = X 3 e 6 = X 5 (45) g = e 1 e e 3 H 3 (453) where e 1 e e 3 is the three-dimensional solvable Lie algebra the non-zero brackets [e 1 e ] = e 1 [e e 3 ] = e 3 and H 3 is the three dimensional Heisenberg Lie algebra T 1 = y 1 + z 1 T = x1 e y T 3 = T 4 = x x 1 T 5 = z t T 6 = x 1(x x 1 ) y 1 e y + (454) 48 [e 1 e ] = 0 [e 1 e 3 ] = 0 [e e 3 ] = 0 (Abelian Lie algebra) OR [e 1 e ] = e 3 (Heisenberg Lie algebra) OR [e 1 e ] = e 3 [e 1 e 3 ] = e [e e 3 ] = e 1 (so(3)) OR [e 1 e ] = e 3 [e 1 e 3 ] = e 1 [e e 3 ] = e (sl(r)) In local coordinates (x y z) the geodesic equations are ẍ = 0 ÿ = 0 (455) System (455) has a Lagrangian [1] L = x (456) X = ξ 1 t + η 1 x + η y + η 3 z (457) satisfies (1) can be estimated easily as X 1 = t X 6 = t y X = x X 7 = t z X 3 = y X 8 = y x x y X 4 = z X 9 = z x x z X 5 = t x X 10 = z y y z X 11 = t t + x x + y y + z z X 1 = t t + tx x + ty y + tz z (458) with the gauge term B 1 (t x y) = c 1 ( x + y + z ) + c 5 x + c 6 y + c 7 z + c 13 the 1 Noether symmetries (458) are [X 1 X 5 ] = X [X 1 X 6 ] = X 3 [X 1 X 7 ] = X 4 [X 1 X 11 ] = X 1 [X 1 X 1 ] = X 11 [X X 8 ] = X 3 [X X 9 ] X 4 [X X 11 ] = X [X X 1 ] = X 5 [X 3 X 8 ] = X [X 3 X 10 ] = X 4 [X 3 X 11 ] = X 3 [X 3 X 1 ] = X 6 [X 4 X 9 ] = X [X 4 X 10 ] = X 3 [X 4 X 11 ] = X 4 [X 4 X 1 ] = X 7 [X 5 X 8 ] = X 6 [X 5 X 9 ] = X 7 [X 5 X 11 ] = X 5 [X 6 X 8 ] = X 5 [X 6 X 10 ] = X 7 [X 6 X 11 ] = X 6 [X 7 X 9 ] = X 5 [X 7 X 10 ] = X 6 [X 7 X 11 ] = X 7 [X 8 X 9 ] = X 10 [X 8 X 10 ] = X 9 [X 9 X 10 ] = X 8 [X 11 X 1 ] = X 1 (459) e 1 = X e = X 3 e 3 = X 4 e 4 = X 5 e 5 = X 6 e 6 = X 7 e 7 = X 1 e 8 = X 11 e 9 = X 1 e 10 = X 8 e 11 = X 9 e 1 = X 10 (460) the Lie algebra g of the Noether symmetries is a semi-direct sum of the six-dimensional abelian solvable Lie algebra e 1 e e 3 e 4 e 5 e 6 and a six-dimensional semi-simple Lie algebra e 7 e 8 e 9 e 10 e 11 e 1 where the latter is a direct sum of sl( R) and so(3) Finally the conservation laws corresponding to the above T 1 = x T = x 1 T 3 = T 4 = T 5 = x x 1 t T 6 = y t T 7 = z t T 8 = x 1 y x T 9 = x 1 z x T 10 = z y T 11 = ( x ) t (x 1 x + y + z) T 1 = ( x ) t ( x 1 x + y + z) t + x + y + z (461) 5 Conclusion The geodesic equations of the canonical connection on Lie groups in dimensions two and three admit the invariance of a variational principle under time translation which gives rise to the conservation of energy and invariance under translations in the x- t direction which implies conservation of linear momentum x Additional conservation laws for each case are given A summary of the Noether s symmetries Lie Algebras for the geodesic equations of the canonical connection on Lie groups in dimensions two and three is given below: I In dimensions two: 1 Direct sum of three-dimensional solvable Lie algebra and R

8 318 R A Ghanam A Y Al-Dweik: Geodesic equations of the canonical connection on Semi-direct sum of five-dimensional solvable Lie algebra and sl( R) II In dimensions three: 1 The three-dimensional abelian Lie algebra Direct sum of five-dimensional solvable Lie algebra and R 3 Direct sum of three-dimensional solvable Lie algebra and the three dimensional Heisenberg Lie algebra 4 Direct sum of two copies of the Euclidean algebra and the reals g = E() E() R 5 Direct sum of the form g = e 1 e e 3 e 4 e 5 e 6 R where e 4 e 5 e 6 is a copy of e 1 e e 3 the non-zero brackets [e 1 e 3 ] = e 1 [e e 3 ] = e 6 Semi-direct sum of the six-dimensional abelian solvable Lie algebra and a six-dimensional semi-simple Lie algebra where the latter is a direct sum of sl( R) and so(3) Acknowledgement The authors would like to thank the King Fahd University of Petroleum and Minerals for providing research support through FT10100 References [1] G Thompson Variational connection on Lie groups Differential Geometry and its Applications 18 (003) [] E Cartan and JA Schouten On the geometry of the groupmanifold of simple and semi-simple groups Proc Akad Wekensch Amsterdam 9 (196) [3] R Ghanam G Thompson and E Miller Variationallity of four-dimensional Lie Group ConnectionsJournal of Lie Theor4(004) [4] J Douglas Solution of the inverse problem of the calculus of variations Trans Amer Math Soc 50 (1941) [5] I Anderson and G Thompson The inverse problem of the calculus of variations for ordinary differential equations Mem Amer Math Soc 98 (473)(199) [6] N Jacobson Lie Algebras Interscience Publishers (196) [7] G W Bluman and S Kumei Symmetries and Differential Equations Springer-Verlag New York 1989 [8] G W Bluman and S C Anco Symmetry and Integration Methods for Differential Equations Springer 00 [9] N H Ibragimov A H Kara and F M Mahomed Lie- Bäcklund and Noether symmetries with applications Nonlin Dynam 15(1998) [10] N H Ibragimov selected works(volume iii) ALGA Publications 008 Ryad Ghanam: I am currently as associate professor of Mathematics at the University of Pittsburgh at Greensburg I have finished my undergraduate degree in Mathematics from the University of Jordan in 1995 I received my PhD in Mathematics from the University of Toledo Ohio USA in 000 My areas of expertise are applied differential geometry and Lie groups I am also interested in numerical analysis and scientific computing Ahmad Al-Dweik has finished his undergraduate degree in Mathematics in 001 from the University of Cairo Egypt He received the MSc degree in Mathematics in 007 from University of Jordan Jordan and the PhD degree in Mathematics in 010 from King Fahd University of Petroleum and Minerals Saudi Arabia In 010 he joined the Mathematics Department of the King Fahd University of Petroleum and Minerals as assistant professor His research interest currently is Symmetry Analysis of Differential Equations