Type II hidden symmetries of the Laplace equation

Transcription

1 Type II hidden symmetries of the Laplace equation Andronikos Paliathanasis Larnaka, 2014 A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

2 Plan of the Talk 1 The generic symmetry vector of second order PDE s 2 De nition of Type II hidden symmetries 3 Reduction of the Laplace equation in Riemannian spaces and the origin of Type II hidden symmetries 4 Applications A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

3 Lie symmetries of a general second order PDE Lie symmetry conditions Consider the second order PDE H : A ij x k, u u ij B k (x, u)u k f (x, u) = 0. where A ij is a non degenerate tensor and X = ξ i x k, u i + η x k, u u is the generator of a in nitesimal transformation. We shall say that the vector eld X will be a Lie symmetry of the PDE H if there exist a function λ such as X [2] H = λh where X [2] is the second prologation vector of X. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

4 Lie symmetries of a general second order PDE Lie symmetry conditions From the Lie symmetry condition we have the following determining system A ij (a ij u + b ij ) (a,i u + b,i )B i ξ k f,k auf,u bf,u + λf = 0 A ij ξ k,ij 2A ik a,i + ab k + aub k,u ξ k,i Bi + ξ i B k,i λb k + bb k,u = 0 L X A ij = (λ a)a ij (1) η = a(x i )u + b(x i ), ξ k,u = 0, ξk (x i ). Eq. (1) implies that the symmetry vector X is a CKV of the second order tensor A ij. with conformal factor ψ = 1 2 (a λ). A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

5 Lie symmetries Poisson equation Theorem The Lie symmetries of the Poisson equation u f x i, u = 0, = p 1 pjgjg ij are generated from the jg j x i x j CKVs of the metric g ij de ning the Laplace operator. The generic Lie symmetry vector is 2 n X = ξ i x k i + ψ x k u + a 0 u + b x k u 2 where ξ i x k is a CKV with conformal factor ψ x k and the following condition holds 2 n 2 ψu + g ij b i;j ξ k 2 n f,k ψuf,u n ψf bf,u = 0. (2) 2 A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

6 Lie symmetries Laplace equations In the case of the Laplace equation u = 0, the symmetry condition becomes 2 n 2 ψ = 0. That means that if n > 2 the CKVs generate a Lie point symmetry for the Laplace equation if and only if the conformal factor ψ x k satis es the Laplace equation. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

7 Type II hidden Symmetries De nition Consider an ODE which admits the Lie point symmetries X 1, X 2 which are such that [X 1, X 2 ] = C12 1 X 1. Then, the reduction by X 1 leads to a reduced equation which admits X 2 as a Lie symmetry whereas reduction of the ODE by X 2 leads to a reduced equation, which does not admit X 1 as a Lie symmetry. In case the generators X 1, X 2 form an Abelian Lie algebra, i.e. [X 1, X 2 ] = 0, then, the reduction preserves the Lie symmetries. Since the reduced equation is di erent from the original equation, it is possible the reduced equation to admit extra Lie symmetries, which are not Lie symmetries of the original equation. These new Lie symmetries have been named Type II hidden symmetries. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

8 Type II hidden Symmetries Problem Study the origin of Type II hidden symmetries of the Laplace equation in certain Riemannian spaces A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

9 Laplace eqn. in Riemannian spaces In a general Riemannian space, Laplace eqn. u = 0, admits the Lie symmetries X u = u u, X b = b (t, x) u where b (t, x) is a solution of Laplace equation and X u existed since the Laplace eqn. is linear. We restrict our considerations to spaces which admit a conformal algebra (proper or not) in order to have extra Lie symmetries and to apply the zero order invariants to reduce the Laplace eqn. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

10 Laplace eqn. in Riemannian spaces a. If a (1 + n) d R.s. admits a gradient KV, the S i = z the metric is ds 2 = dz 2 + h AB y A y B, h AB = h AB y C b. If a (1 + n) d R.s. admits a gradient HV, the H i = r r the metric is ds 2 = dr 2 + r 2 h AB dy A dy B, h AB = h AB y C c. If a (1 + n) d R.s. admits a sp.ckv then admits a gradient KV and a gradient HV and the metric can be written in the generic form ds 2 = dz 2 + dr 2 + R 2 f AB y C dy A dy B while the sp.ckv is C S = z 2 +R 2 2 z + zr R with conformal factor ψ CS = z. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

11 Reduction with a gr. KV The Laplace equation in a Riemannian space which admit the gr. KV z can be written as follows u zz + h AB y C u AB Γ A y C u B = 0. (3) The Laplace equation admits as extra Lie symmetry the gradient KV z. By using the zeroth order invariants y A, w = u of z the reduced equation is h w = 0. We recall the following geometrical results A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

12 Reduction with a gr. KV The KVs of the n metric are identical with those of the n + 1 metric The 1 + n metric admits a HV if and only if the n metric admits one and if n H A is the HV of the n metric then the HV of the 1 + n metric is given by the expression 1+nH µ = zδ µ z + n H A δ µ A µ = x, 1,..., n. (4) The 1 + n metric admits CKVs if and only if the n metric h AB admits a gradient CKV. Therefore Type II hidden symmetries exist if the n metric h AB admits more symmetries. Speci cally, the sp.ckvs of the h AB metric as well as the proper CKVs whose conformal function is a solution of Laplace equation generate Type II hidden symmetries. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

13 Reduction with a gr. HV In Riemannian spaces which admit a gradient HV, H = r r, the Laplace equation becomes u rr + 1 r 2 hab y C u AB + (n 1) 1 u r r r 2 ΓA y C u A = 0 (5) and admits the extra Lie symmetry H. The zeroth order invariants of H are y A, w y A and using them it follows easily that the reduced equation is h u = 0 (6) that is, the Laplacian de ned by the metric h AB. It is easy to establish the following results concerning the conformal algebras of the metrics g ij and h AB. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

14 Reduction with a gr. HV The KVs of h AB are also KVs of g ij. The HV of g ij is independent from that of h AB. The metric g ij admits proper CKVs if and only if the n metric h AB admits gradient CKVs. This is because g ij is conformally related with the decomposable metric ds 20 = dr 2 + h AB y C dy A dy B. (7) The above imply, that Type II hidden symmetries we shall have from the HV of the metric h AB, the sp.ckvs and nally from the proper CKVs of h AB whose conformal factor is a solution of Laplace equation h u = 0. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

15 Reduction with a sp.ckv In a Riemannian space which admit a sp. CKV the Laplace equation is u zz + u RR + 1 R 2 hab u AB + and admits the Lie point symmetries where 2p = 1 m (m 1) 1 u R R R 2 ΓA u A = 0. (8) X 1 = K G, X 2 = H, X 3 = C S + 2pzu u 2 and the non zero commutators are X 1, X 2 = X 1, X 2, X 3 = X 3 X 1, X 3 = X 2 + 2pX u. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

16 Reduction with a sp.ckv Under the coordinate transformation z = q R (xr 1) x invariants of the Lie symmetry X 3 are x, y A, w = ur 2p. The reduced equation is the zero order x 2 w xx + f AB w AB Γ A w A 2p (2p + 1) w = 0 If dim (f AB ) = 2, the reduced equation becomes (m=3) w = 0 which is the Laplacian in the three dimensional space with metric d s 2 (m=3) = 1 x 4 dx2 + 1 x 2 f AB dy A dy B. (9) By making the new transformation x = 1 r, the metric admits the gradient HV r r which gives an inherited symmetry. We conclude that Type II hidden symmetries of (9) will be generated from the proper CKVs of the metric (9). A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

17 Reduction with a sp.ckv If dim (f AB ) 3, if we apply the transformation x = (m 2) 2 m φ m 2 the reduced equation becomes where V (x) = V (φ) = (m4) w 2p (2p + 1) V (φ) w = 0 (10) (2 m)2 φ 2 (m4) is the Laplace operator with metric is the well known Ermakov potential and d s 2 (m4) = dφ2 + φ 2 f AB dy A dy B (11) The reduced equation is a KG equation. It is easy to see that the gradient HV φ φ of (11) is a Lie symmetry of (??) which is the Lie symmetry X 2. Therefore, if the metric f AB admits proper CKVs then these vectors generate Type II hidden symmetries. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

18 Type II hidden Symmetries Laplace equation: Main Results If we reduce the Laplace equation with a gradient KV the reduced equation is a Laplace equation in the non-decomposable space. In this case, the Type II hidden symmetries are generated from the special and the proper CKVs of the non-decomposable space. If we reduce the Laplace equation with a gradient HV the reduced equation is a Laplace equation for an appropriate metric. In this case, the Type II hidden symmetries are generated from the HV and the special/proper CKVs. If we reduce the Laplace equation with the symmetry generated by a sp.ckv, the reduced equation is the Klein Gordon equation for an appropriate metric that inherits the Lie point symmetry generated by the gradient HV. In this case, the Type II hidden symmetries are generated from the proper CKVs of the appropriate metric. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

19 1+3 Wave equation Consider the Laplace equation in the four dimensional Minkowski spacetime M 4 u tt u xx u yy u zz = 0. The Lie point symmetries are: K 1 G, K A G, X 1A R, X AB R, H, X 1 C tu u, X A C y A u u where y A = (x, y, z). The nonzero commutators are h i h i i KG I, X R IJ = KG J, KG I, H = KG hk I, G I, X C I = H X u h i i h i KG I, X C J = XR hh, IJ, XC I = XC I, XR IJ, X C I = XC J. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

20 1+3 Wave equation Reduction with a gr. KV Reduction with the gradient KV K z G = z. The reduced equation is w tt w xx w yy = 0 which is Laplace equation in the space M 3. The Lie symmetries are KG 1, K G x, K y G, X C tu u, X C x X 1a R, XR ab (12) 1 2 yu u (13) 1 2 xu u, X y C K, X R, H are inherited symmetries the other vectors are Type II hidden symmetries which are generated from the sp.ckvs of M 3. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

21 1+3 Wave equation Reduction with a gr HV In spherical coordinates the HV is H = r r. The reduced equation is w θθ + w φφ cosh 2 θ + w ζζ cosh 2 θ cosh φ 2 + 2tanh θ r 2 w θ + tanh φ r 2 cosh 2 θ w φ = 0. which is the Laplace in a space of constant curvature. The Lie symmetries of the reduced equation are the elements of the so (3) Lie algebra and are inherits symmetries. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

22 Type II hidden Symmetries Reduction with a sp.ckv In the reduction with a sp.ckv we found that the reduce equation is the 3d Laplace equation in M 3. The M 3 does not admits proper (non special) CKVs here there are not any type II hidden symmetries. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

23 1+2 Wave equation The wave equation in E 2 is u tt u xx u yy = 0. (14) Reduction with the gradient KV K y G = y. The reduced equation is w tt w xx which is the one dimensional wave equation. The 2d space (t, x) has an in nite number of CKVs therefore the reduced equations has in nite Lie point symmetries. From these symmetries the KVs and the HV are inherited symmetries and the CKVs are Type II symmetries. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

24 1+2 Wave equation In order to do the reduction with the gradient HV we introduce spherical coordinates (r, θ, φ). The Laplace equation is 1 u rr r 2 u θθ + u φφ cosh θ r u r tanh θ r 2 u θ = 0. (15) Therefore by applying the zero order invariants of the HV the reduced equation is w θθ + w φφ cosh 2 θ + tanh θw θ = 0. (16) By making the transformation θ = ln tan ρ 2 equation (16) becomes sin (x) 2 w ρρ + w φφ = 0 which is the wave equation (??) with t = ρ, x = iφ. A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25

25 Conclusion Thank you! A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, / 25