D. Mladenov - On Integrable Systems in Cosmology

On Integrable Systems in Cosmology Dimitar Mladenov Theoretical Physics Department, Faculty of Physics, Soﬁa University• Dynamics on Lie groups• Geodesic ﬂow on GL(n, R) group manifold• Homogeneous cosmological modelsFaculty of Science and Mathematics, University of Niˇ, sNiˇ, Serbia, 7 December 2011 s 2 1/95 ¹

CMS modelsMore than thirty ﬁve years after the famous paper ofFrancesco CalogeroSolution of the one-dimensional n-body problem with quadraticand/or inversely quadratic pair potentialsJournal of Mathematical Physics 12 (1971) 419-436Bill SutherlandPhysical Review A 1971, 1972J¨ rgen Moser uAdv. Math. 1975 ´ 2/95 ¹

CMS modelsThe Calogero-Sutherland-Moser type system which con-sists of n-particles on a line interacting with pairwise po-tential V (x) admits the following spin generalization 1 n 2 1 n HECM = p + fab fba V (xa − xb ) 2 a=1 a 2 a=b ´ 3/95 ¹

CMS modelsThe Euler-Calogero-Sutherland-Moser type systems areintegrable in the following cases I. V(z) = z −2 Calogero II. V(z) = a2 sinh−2 az III. V(z) = a2 sin−2 az Sutherland IV. V(z) = a2 ℘(az) V. V(z) = z −2 + ωz 2 CalogeroThe nonvanishing Poisson bracket relations {xi , pj } = δij , {fab , fcd } = δbc fad − δad fcb ´ 4/95 ¹

Dynamics on Lie groupsLet G be a real Lie group and A(G) be its Lie algebra. Theaction of the Lie group G in its group manifold is given bythe following diﬀeomorphisms Lg : G −→ G Lg h = gh Rg : G −→ G Rg h = hg h, g ∈ GThe adjoint and coadjoint actions are Adg : A(G) −→ A(G) Adgh = Adg Adh Ad∗ : A(G) −→ A(G) g Ad∗ = Ad∗ Ad∗ gh h g ´ 5/95 ¹

Dynamics on Lie groupsThe Lie-Poisson brackets are {F, H} =< x, [ F (x), H(x)] > , ∂F ∂H k {F, H} = Cij xk , x ∈ A∗ (G) ∂xi ∂xjThe left and right invariant 1-forms obey the First structureCartan equation dω L + ω L ∧ ω L = 0 , dω R − ω R ∧ ω R = 0 ´ 6/95 ¹

Dynamics on Lie groupsThe geodesic ﬂow on the matrix Lie group G is describedby the map t −→ g(t) ∈ Gand by the Lagrangian 1 L= < ω, ω >, ω ∈ A(G) 2In local coordinates we have 1 L= Gµν X µ X ν 2 ´ 7/95 ¹

Dynamics on Lie groupsThe phase space is the cotangent bundle T ∗ (G) withcanonical symplectic structure Ω = d(Pµ dX µ ) {X µ , X ν } = 0 , {Pµ , Pν } = 0 , {X µ , Pν } = δν µand Hamiltonian 1 µν H= G Pµ Pν 2 ´ 8/95 ¹

Dynamics on Lie groupsLet us deﬁne the functions ξµ = ΩL,R L,R ν µ Pνsuch that ξµ = Adν ξν R µ Lwith Poisson bracket relations R R γ R {ξα , ξβ } = − Cαβ ξγ , L L γ R {ξα , ξβ } = Cαβ ξγ , R R {ξα , ξβ } = 0 ´ 9/95 ¹

Geodesic ﬂow on GL(n, R) group manifoldLet us consider the Lagrangian 1 L= < ω, ω >, ω ∈ gl(n, R) 2 which is deﬁned by the following one parameter family ofnon-singular metrics on GL(n, R) group manifold α < X , Y >= TrXY − TrX TrY , αn − 2where X, Y ∈ gl(n, R) and α ∈ [0, 1] ´ 10/95 ¹

Geodesic ﬂow on GL(n, R)Thus we have 1 2 α L= tr g −1 g ˙ − tr2 g −1 g = ˙ 2 2(αn − 2) 1 −1 = G gab gcd ˙ ˙ 2 ab,cdHere g ∈ GL(n, R) and α Gab,cd = gad gcb − gab gcd , 2 α G−1 ab,cd −1 −1 = gda gbc − g −1 g −1 αn − 2 ba dc ´ 11/95 ¹

Geodesic ﬂow on GL(n, R)The invariance of the Lagrangian under the left and righttranslations leads to the possibility of explicit integrationof the dynamical equations d −1 g g = 0 ⇒ g(t) = g(0) exp (tJ) ˙ dtThe canonical Hamiltonian corresponding to the bi-invariantLagrangian L is 1 2 α 2 T 1 H= tr π T g − tr π g = Gab,cd πab πcd 2 4 2 ´ 12/95 ¹

Geodesic ﬂow on GL(n, R)The nonvanishing Poisson brackets between the fundamen-tal phase space variables are {gab , πcd } = δab δcdAt ﬁrst we would like to analyze the following symmetryaction on SO(n, R) in GL(n, R) g → g = RgIt is convenient to use the polar decomposition for an arbi-trary element of GL(n, R) g = OS ´ 13/95 ¹

Geodesic ﬂow on GL(n, R)The orthogonal matrix O(φ) is parameterized by the Eulerangles (φ1 , · · · , φ n(n−1) ) and S is a positive deﬁnite symmetric 2matrixWe can treat the polar decomposition as 1 − 1 transforma-tion from n2 variables g to a new set of Lagrangian variables: g −→ (Sab , φa ) , n(n−1) n(n+1)the 2 Euler angles and 2 symmetric matrix variablesSab ´ 14/95 ¹

Geodesic ﬂow on GL(n, R)In terms of these new variables the Lagrangian can berewritten as 1 ˙ 2 LGL = tr ΘL + S S −1 2The polar decomposition induces a canonical transforma-tion (g, π) −→ (Sab , Pab ; φa , Pa )The canonical pairs obey the Poisson bracket relations 1 {Sab , Pcd } = (δac δbd + δad δbc ) , {φa , Pb } = δab 2 ´ 15/95 ¹

Geodesic ﬂow on GL(n, R)In the case of GL(3, R) the orthogonal matrix is given byO(φ1 , φ2 , φ3 ) = eφ1 J3 eφ2 J1 eφ3 J3 ∈ SO(3, R). The canonical trans-formation is given by (g, π) −→ (Sab , Pab ; φa , Pa ) ,where π = O (P − ka Ja ) , −1 L ka = γab ηb − εbmn (SP )mnand γik = Sik − δik trS ´ 16/95 ¹

Geodesic ﬂow on GL(n, R) LHere ηa are three left-invariant vector ﬁelds on SO(3, R) L sin φ3 η1 = − P1 − cos φ3 P2 + cot φ2 sin φ3 P3 , sin φ2 L cos φ3 η2 = − P1 + sin φ3 P2 + cot φ2 cos φ3 P3 , sin φ2 L η3 = −P3 ´ 17/95 ¹

Geodesic ﬂow on GL(n, R)The right-invariant vector ﬁelds are R sin φ1 η1 = − sin φ1 cot φ2 P1 + cos φ1 P2 + P3 , sin φ2 R cos φ1 η2 = cos φ1 cot φ2 P1 + sin φ1 P2 − P3 , sin φ2 R η 3 = P1 ´ 18/95 ¹

Geodesic ﬂow on GL(n, R)In terms of the new variables the canonical Hamiltoniantakes the form 1 α 1 H = tr (P S)2 − tr2 (P S) + tr (Ja SJb S) ka kb 2 4 2The canonical variables (Sab , Pab ) are invariant while (φa , Pa )undergo changes ´ 19/95 ¹

Geodesic flow on GL(n, R)The configuration space S of the real symmetric 3 × 3 ma-trices can be endowed with the flat Riemannian metric ds2 =< dQ , dQ >= Tr dQ2 ,whose group of isometry is formed by orthogonal trans-formations Q = RT Q R. The system is invariant under theorthogonal transformations S = RT S R. The orbit space isgiven as a quotient space S/SO(3, R) which is a stratifiedmanifold ´ 20/95 ¹

Geodesic ﬂow on GL(n, R)(1) Principal orbit-type stratum, when all eigenvalues are unequal x1 < x2 < x3 with the smallest isotropy group Z2 ⊗ Z2(2) Singular orbit-type strata forming the boundaries of orbit space with (a) two coinciding eigenvalues (e.g. x1 = x2 ) , when the isotropy group is SO(2) ⊗ Z2 (b) all three eigenvalues are equal (x1 = x2 = x3 ) , here the isotropy group coincides with the isometry group SO(3, R) ´ 21/95 ¹

Geodesic ﬂow on GL(n, R) Hamiltonian on the Principal orbitTo write down the Hamiltonian describing the motion onthe principal orbit stratum we use the main-axes decompo-sition in the form S = RT (χ)e2X R(χ) ,where R(χ) ∈ SO(3, R) is parametrized by three Euler anglesχ = (χ1 , χ2 , χ3 ) and e2X is a diagonal e2X = diag e2x1 , e2x2 , e2x3 ´ 22/95 ¹

Geodesic ﬂow on GL(n, R)The original physical momenta Pab are expressed in termsof the new canonical pairs (Sab , Pab ) −→ (xa , pa ; χa , Pχa ) ,where 3 3 P = RT e−X ¯ ¯ Pa αa + Pa αa e−X R a=1 a=1 ´ 23/95 ¹

Geodesic ﬂow on GL(n, R)with ¯ 1 Pa = pa , 2 R 1 ξa Pa = − , (cyclic a = b = c) 4 sinh(xb − xc )In the representation we introduce the orthogonal basis forthe symmetric 3×3 matrices αA = (αi , αi ), i = 1, 2, 3 with thescalar product tr(¯ a αb ) = δab , α ¯ tr(αa αb ) = 2δab , tr(¯ a αb ) = 0 α ´ 24/95 ¹

Geodesic ﬂow on GL(n, R)In this case the SO(3, R) right-invariant Killing vector ﬁeldsare R ξ1 = −pχ1 , R sin χ1 ξ2 = sin χ1 cot χ2 pχ1 − cos χ1 pχ2 − pχ , sin χ2 3 R cos χ1 ξ3 = − cos χ1 cot χ2 pχ1 − sin χ1 pχ2 + p χ3 sin χ2 ´ 25/95 ¹

Geodesic ﬂow on GL(n, R)After passing to these main-axes variables the canonicalHamiltonian reads 2 1 3 2 α 3 H= pa − pa 8 a=1 16 a=1 2 1 (ξa )2 R 1 Rab ηb + 1 ξa L 2 R + 2 − 16 (abc) sinh (xb − xc ) 4 (abc) cosh2 (xb − xc )The integrable dynamical system describing the free motionon principal orbits represents theGeneralized Euler-Calogero-Moser-Sutherland model ´ 26/95 ¹

Geodesic ﬂow on GL(n, R) Reduction to Pseudo-Euclidean Euler-Calogero-Moser modelWe demonstrate how IIA2 Euler-Calogero-Moser-Sutherlandtype model arises from the Hamiltonian after projectiononto a certain invariant submanifold determined by dis-crete symmetries. Let us impose the condition of symmetryof the matrices g ∈ GL(n, R) (1) ψa = εabc gbc = 0 ´ 27/95 ¹

Geodesic ﬂow on GL(n, R)One can check that the invariant submanifold of T ∗ (GL(n, R))is deﬁned by (1) (2) ΨA = (ψa , ψa )and the dynamics of the corresponding induced system isgoverned by the reduced Hamiltonian 1 α H|ΨA =0 = tr (P S)2 − tr2 (P S) 2 4 ´ 28/95 ¹

Geodesic ﬂow on GL(n, R)The matrices S and P are now symmetric and nondegener-ate.It can be shown that 3 α 1 (ξa )2 R HP ECM = p2 − a (p1 + p2 + p3 )2 + a=1 2 2 (abc) sinh2 (xb − xc )which is a Pseudo-Euclidean version of the Euler-Calogero-Moser-Sutherland model. ´ 29/95 ¹

Geodesic ﬂow on GL(n, R)After the projection the Poisson structure is changed ac-cording to the Dirac prescription −1 {F, G}D = {F, G}P B − {F, ψA }CAB {ψB , G}To verify this statement it is necessary to note that the (1) (2)Poisson matrix CAB = {ψa , ψb } is not degenerate. Theresulting fundamental Dirac brackets between the main-axes variables are 1 1 {xa , pb }D = δab , {χa , pχb }D = δab 2 2 ´ 30/95 ¹

Geodesic flow on GL(n, R)The Dirac bracket algebra for the right-invariant vectorfields on SO(3, R) reduces to R R 1 R {ξa , ξb }D = εabc ξc . 2All angular variables are gathered in the Hamiltonian in Lthree left-invariant vector fields ηa . The corresponding right- R Linvariant fields ηa = Oab ηb are the integrals of motion R {ηa , HP ECM } = 0 ´ 31/95 ¹

Geodesic flow on GL(n, R)Thus the surface on the phase space, determined by theconstraints R ηa = 0defines the invariant submanifold. Using the relation be- R Ltween left and right-invariant Killing fields ηa = Oab ηb wefind out that after projection to the constraint surface theHamiltonian reduces to 3 α (ξa )2 R 4 HP ECM = p2 − a (p1 + p2 + p3 )2 + 2 (abc) sinh 2(xb − xc ) a 2 ´ 32/95 ¹

Geodesic ﬂow on GL(n, R)Apart from the integrals η R the system possesses other inte-grals. The integrals of motion corresponding to the geodesicmotion with respect to the bi-invariant metric on GL(n, R)group are Jab = (π T g)ab The algebra of this integrals realizes on the symplectic levelthe GL(n, R) algebra {Jab , Jcd } = δbc Jad − δad Jcb ´ 33/95 ¹

Geodesic ﬂow on GL(n, R)After transformation to the scalar and rotational variablesthe expressions for the current J reads 3 1 J= RT (pa αa − ia αa − ja Ja ) R , ¯ 2 a=1where 1 R L 1 R ia = ξa coth(xb − xc ) + Ram ηm + ξa tanh(xb − xc ) (abc) 2 2and L R ja = Ram ηm + ξa ´ 34/95 ¹

Geodesic ﬂow on GL(n, R)After performing the reduction to the surface deﬁned bythe vanishing integrals ja = 0we again arrive at the same Pseudo-Euclidean version ofEuler-Calogero-Moser-Sutherland system. ´ 35/95 ¹

Geodesic ﬂow on GL(n, R) Lax representation for the Pseudo-Euclidean Euler-Calogero-Moser modelThe Hamiltonian equations of motion can be written in aLax form ˙ L = [A, L] ,where the 3 × 3 matrices are given explicitly as ´ 36/95 ¹

Geodesic ﬂow on GL(n, R)   p1 − α 2 (p1 + p2 + p3 ) L+ , 3 L− 2  L= L− , p2 − α L+   3 2 (p1 + p2 + p3 ) 1    L+ , 2 L− , 1 p3 − α 2 (p1 + p2 + p3 )   0 −A3 , A2 1  A=  A3 , 0, −A1    4  −A2 , A1 , 0 ´ 37/95 ¹

Geodesic ﬂow on GL(n, R)Entries Aa and L± are given as a 1 R 1 R L± = − ξa coth(xb − xc ) − Ram ηm + ξa tanh(xb − xc ) a L 2 2 L R ± Ram ηm + ξa , 1 R ξa Ram ηm + 1 ξa L 2 R Aa = − 2 sinh2 (xb − xc ) cosh2 (xb − xc ) ´ 38/95 ¹

Geodesic ﬂow on GL(n, R)Singular orbits. The case of GL(3, R) and GL(4, R)The motion on the Singular orbit is modiﬁed due to thecontinuous isotropy group. This symmetry of dynamicalsystem is encoded in constraints on phase space variables 1 1 ψ1 = √ (x2 − x3 ) , ψ2 = √ (p2 − p3 ) , 2 2 R R R ψ3 = ξ2 − ξ3 , ψ4 = ξ1 ´ 39/95 ¹

Geodesic ﬂow on GL(n, R)One can check that this surface represents the invariantsubmanifold. These constraints are the second class in theDirac terminology hence we have to replace the Poissonbrackets by the Dirac ones 1 {x1 , p1 }D = 1 , {xi , pj }D = , i, j = 2, 3 , 2 R R {ξa , ξb }D = 0 , a, b = 1, 2, 3 ´ 40/95 ¹

Geodesic ﬂow on GL(n, R)As a result we obtain (2) 1 2 1 2 g2 HGL(3,R) = p1 + p2 + 2 4 sinh2 (x1 − x2 )This is an integrable mass deformation of theIIA1 Calogero-Moser-Sutherland model.The Lax pair for system can be obtained from the L andA matrices letting η R = 0 and projecting on the constraintshell ´ 41/95 ¹

Geodesic ﬂow on GL(n, R)   1  p, 2 1 R −ξ2 L+ , R −ξ2 L+    (2)  R − 1 LGL(3,R) =  ξ2 L , p 2 2 0       ξ2 L− , R 0, 1 p 2 2    0 −1, 1  R (2) ξ2  AGL(3,R) =  1, 0, 0    2 sinh (x1 − x2 )     −1, 0, 0 ´ 42/95 ¹

Geodesic ﬂow on GL(n, R)Here L− := (1 − coth(x1 − x2 )) , L+ := (1 + coth(x1 − x2 )) ´ 43/95 ¹

Geodesic flow on GL(n, R)Consider the case of GL(4, R) group restricted on the Sin-gular orbit with equal eigenvalues x3 = x4 . The invariantsubmanifold is fixed by the five constraints 1 ψ1 := √ (x3 − x4 ) , 2 1 ψ2 := √ (p3 − p4 ) , 2 R ψ3 := l34 , R R ψ4 := l13 − l14 , R R ψ5 := l23 − l24 ´ 44/95 ¹

Geodesic flow on GL(n, R)The Poisson matrix {ψm , ψn } , m, n = 1, . . . , 5 is degeneratewith rank {ψm , ψn } = 4. To find out the proper gauge andsimplify the constraints it is useful to pass R lab = ya πb − yb πa , {ya , πb } = δab , a, b = 1, . . . , 4and choose the following gauge-fixing condition 1 ψ := √ (y3 − y4 ) = 0 2 ´ 45/95 ¹

Geodesic ﬂow on GL(n, R)Finally the reduced phase space corresponding to the Sin-gular orbit is deﬁned by the set of four second class con-straints {ψ, Π, ψ, Π} 1 1 ψ := √ (x3 − x4 ) = 0 , Π := √ (p3 − p4 ) = 0 , 2 2 1 1 ψ := √ (y3 − y4 ) = 0 , Π := √ (π3 − π4 ) = 0 , 2 2 ´ 46/95 ¹

Geodesic ﬂow on GL(n, R)This set of constraints form an invariant submanifold of thephase space under the action of the discrete permutationgroup S2          xi   xS(i)   yi   yS(i)    →  ,   →  , pi pS(i) πi πS(i)where i = 3, 4. ´ 47/95 ¹

Geodesic ﬂow on GL(n, R)These constraints form the canonical set of second classconstraints with non-vanishing Poisson brackets {ψ, Π} = 1 , ¯ {ψ, Π} = 1and thus the fundamental Dirac brackets for canonical vari-ables are 1 1 {xi , pj }D = , {yi , πj }D = , i, j = 1, 2 2 2 ´ 48/95 ¹

Geodesic ﬂow on GL(n, R)The system reduces to the following one (3) 1 2 1 2 1 2 HGL(4,R) = p + p + p 2 1 2 2 4 3 (l12 )2 R (l13 )2 R (l23 )2 R + + + 16 sinh2 (x1 − x2 ) 8 sinh2 (x1 − x3 ) 8 sinh2 (x2 − x3 )with the Poisson bracket algebra {xr , ps } = δrs , R R R R R R {lpq , lrs } = δps lqr − δpr lqs + δqs lpr − δqr lps , p, q, r, s = 1, 2, 3 ´ 49/95 ¹

Geodesic ﬂow on GL(n, R)In this case L and A matrices are the well-known matricesfor the spin Calogero-Sutherland model 1 R Lij = pi δij − (1 − δij )lij Φ(xi − xj ) , 2 R Aij = −(1 − δij )lij V (xi − xj ) ,where 1 Φ(x) = coth(x) + 1 , V (x) = 2 sinh2 (x) ´ 50/95 ¹

Geodesic ﬂow on GL(n, R)After projection to the invariant submanifold correspond-ing to the motion on the Singular orbits we arrive at theLax matrices   1 R R R  p, 2 1 −l12 Φ12 , −l13 Φ13 , −l13 Φ13    R 1 R R  −l12 Φ12 , p, −l23 Φ23 , −l23 Φ23   (3) 2 2  LGL(4,R) =    −lR Φ , R 1    13 13 −l23 Φ23 , p, 2 3 0     R R 1 −l13 Φ13 , −l23 Φ23 , 0, p 2 3 ´ 51/95 ¹

Geodesic ﬂow on GL(n, R)   R R R  0, l12 V12 , l13 V13 , l13 V13    R R R  −l12 V12 0, l23 V23 , l23 V23   (3) AGL(4,R) =    −lR V , R    13 13 −l23 V23 , 0, 0     R R −l13 V13 , −l23 V23 , 0, 0 ´ 52/95 ¹

Geodesic ﬂow on gl(n, R)Consider the Lagrangian represented by the left-invariantmetric 1 ˙ ˙ L = < A , A >, A ∈ GL(n, R) 2which is deﬁned on gl(n, R) algebra < X , Y >= Tr X T Y , X, Y ∈ gl(n, R) ´ 53/95 ¹

Geodesic ﬂow on gl(n, R)The obviously conserved angular momentum ˙ µ = [A , A]leads to the possibility to integrate the equations of motion ¨A = 0 in the form A = at + b. The canonical Hamiltonian cor-responding to L is 1 2 H= tr P T P 2 ´ 54/95 ¹

Geodesic ﬂow on gl(n, R)The nonvanishing Poisson brackets between the fundamen-tal phase space variables are {Aab , Pcd } = δab δcdUsing the same machinery as in the previous case we obtainthe Hamiltonian 2 L 1 R 1 3 2 1 (ξa )2 R Rab ηb + 2 ξa H= pa + + , 2 a=1 4 (abc) (xb − xc )2 (abc) (xb − xc )2which is a generalization of therational Euler-Calogero-Moser-Sutherland model ´ 55/95 ¹

Geodesic ﬂow on gl(n, R)After reduction on the invariant submanifold deﬁned by theequations η R = 0 we obtain the Hamiltonian 1 3 2 1 1 1 H= pa + (ξa )2 R 2 + 2 a=1 4 (abc) (xb − xc ) (xb − xc )2which coincides with the Hamiltonian of theID3 Euler-Calogero-Moser-Sutherland model ´ 56/95 ¹

Geodesic motion on GL(n, R)Forty years after the famous papers ofM. TodaJournal of Physical Society of Japan 20 (1967) 431Journal of Physical Society of Japan 20 (1967) 2095The Hamiltonian of the non-periodic Toda lattice is 1 n 2 n−1 HN P T = p + g2 exp [2(xa − xa+1 )] , 2 a=1 a a=1and the Poisson bracket relations are {xi , pj } = δij . ´ 57/95 ¹

Geodesic motion on GL(n, R)The equations of motion for the non-periodic Toda latticeare xa = pa , a = 1, 2, . . . , n , ˙ pa = −2 exp [2(xa − xa+1 )] + 2 exp [2(xa−1 − xa )] , ˙ p1 = −2 exp [2(x1 − x2 )] , ˙ pn = −2 exp [2(xn−1 − xn )] , ˙and for the periodic Toda lattice xa = p a , ˙ pa = 2 exp [2(xa−1 − xa )] − exp [2(xa − xa+1 )] . ˙ ´ 58/95 ¹

Geodesic motion on GL(n, R)Further for simplicity we put kab = 0 and α = 0.We use the Gauss decomposition for the positive-deﬁnitesymmetric matrix S S = Z D ZT . ´ 59/95 ¹

Geodesic motion on GL(n, R)Here D = diag x1 , x2 , . . . xn is a diagonal matrix with posi-tive elements and Z is an upper triangular matrix    1 z12 . . . z1n    0 1 . . . z2n    Z= . .  .. .  .  . . . .  . . .    0 0 ... 1 ´ 60/95 ¹

Geodesic motion on GL(n, R)The diﬀerential of the symmetric matrix S is given by dS = Z dD + DΩ + (DΩ)T Z T , where Ω is a matrix-valued right-invariant 1-form deﬁnedby Ω := dZZ −1 .In the Lie algebra gl(n, R) of n×n real matrices we introduceWeyl basis with elements (eab )ij = δai δbj , ´ 61/95 ¹

Geodesic motion on GL(n, R)which are n × n matrices in the form  . .   .    . . . ... ... 1 . . .      . .  eab =   .   . .     . .      . .  .The scalar product in gl(n, R) is deﬁned by (eab , ecd ) = tr(eT , ecd ) = δac δbd . ab ´ 62/95 ¹

Geodesic motion on GL(n, R)Let us introduce also the matrices   0 ... ... ... 0 . . . ... . . .   . . αa = ¯ . . . 1 . . .. .    . . . . .   0 ... ... ... 0 ´ 63/95 ¹

Geodesic motion on GL(n, R)and  . . . .   . .    . . . ... ... 1 . . .      . . . .  αab =   . .   . .   ... .   . . . 1 . . . .    . . . .  . . ´ 64/95 ¹

Geodesic motion on GL(n, R)with scalar product is deﬁned by (¯ a , αb ) = tr(¯ a αb ) = δab , α ¯ α ¯ (αab , αcd ) = tr(αab αcd ) = 2 (δad δbc + δac δbd ) , (¯ a , αbc ) = tr(¯ a αbc ) = 0 . α α ´ 65/95 ¹

Geodesic motion on GL(n, R)Now we can write the diﬀerential of the matrix S in theform   n n dS = Z dxa αa + ¯ xa Ωab αab  Z T . a=1 a<b=1Here Ωab are the coeﬃcients of the matrix Ω Ω= Ωab eab . a<b ´ 66/95 ¹

Geodesic motion on GL(n, R)In the case of Gauss decomposition of the symmetric matrixS = ZDZ T the corresponding momenta we seek in the form   n n P = (Z T )−1  ¯ ¯ Pa αa + Pab αab  Z −1 . a=1 a<b ´ 67/95 ¹

Geodesic motion on GL(n, R)From the condition of invariance of the symplectic 1-form n n tr (P dS) = pa dxa + pab dzab , i=1 a<b=1 where the new canonical variables (xa , pa ) , (zab , pab ) obeythe Poisson bracket relations {xa , pb } = δab , {zab , pcd } = δac δbd ,we obtain ¯ lab Pa = pa , Pab = . 2xa ´ 68/95 ¹

Geodesic motion on GL(n, R)Here lab are right-invariant vector ﬁelds on the group ofupper triangular matrices with unities on the diagonal T lab = Ω−1 pcd , ab,cd where Ωab,cd are coeﬃcients in the decompositions of the1-forms Ωab in coordinate basis Ωab = Ωab,cd dzcd , ´ 69/95 ¹

Geodesic motion on GL(n, R)The canonical Hamiltonian in the new variables takes theform 1 n 2 2 1 n 2 xa H= p x + l . 2 a=1 a a 4 a<b=1 ab xb ´ 70/95 ¹

Geodesic motion on GL(n, R)After the canonical transformation xa = eya , pa = πa e−ya we arrive to Hamiltonian in the form ofspin nonperiodic Toda chain n n 1 1 H= 2 πa + lab eya −yb . 2 2 a=1 4 a<b=1 ´ 71/95 ¹

Geodesic motion on GL(n, R)The explicit form of the generators of the group of uppertriangular matrices is n n l12 = p12 + z2k p1k , l13 = p13 + z2k p1k . k=2 k=2The internal variables satisfy the Poison bracket relations {lab , lcd } = C efab,cd lef (1)with structure coeﬃcients C efab,cd = δbc δae δdf . (2) ´ 72/95 ¹

1. Homogeneous cosmological models1.1. Spacetime decompositionLet (M, g) be a smooth four-dimensional paracompact andHausdorf manifold. In each point of the open set U ⊂ M wepropose that are deﬁned the local basis of 1-forms and itsdual basis {ω µ , µ = 0, 1, 2, 3} {eν , ν = 0, 1, 2, 3} ´ 73/95 ¹

such that [eα , eβ ] = C µαβ eµ ,where C µαβ are the basis structure functions. The symmetricmetric is g = gµν ω µ ⊗ ω ν , g −1 = g µν eµ ⊗ eνWe suppose that on the manifold (M, g) is defined affineconnection Γµν = Γµνα ω α ⇐⇒ Γµβα = ω µ , eα eβIn manifold with affine connection we can construct the ´ 74/95 ¹

bilinear antisymmetric mapping T (X, Y ) = X Y − Y X − [X , Y ]The First Cartan structure equation is 1 µ dω µ + Γµα ∧ ω α = T 2The tensor type (1, 3) deﬁned by R(ω, Z, X, Y ) = ω, R( X, Y )Zis called the curvature tensor, where R( X, Y ) = X Y − Y X −[ X, Y] ´ 75/95 ¹

If we define the curvature 2-form Ωµν by Ωµν = dΓµν + Γµα ∧ Γανone can obtain the Second structure Cartan equation 1 µ Ωµν = R ωα ∧ ωβ 2 ναβIn the Riemannian geometry the commutator of two vectorfields is given by [X, Y ] = X Y − Y Xand for every vector field X we have the condition Xg = 0. ´ 76/95 ¹

The connection and the Riemann tensor in the noncoordi-nate basis {eν } are given with 1 Γµαβ = g µσ (eα gβσ + eβ gασ − eσ gαβ ) 2 1 µσ 1 − g (gαρ C ρβσ + gβρ C ρασ ) − C µαβ , 2 2 Rσαµν = eµ Γσαν − eν Γσαµ + Γραν Γσρµ − Γραµ Γσρν − Γσαρ C ρµν ,where C µαβ = Γµβα − Γµαβ .Hereinafter we suppose that M = T1 × Σt ´ 77/95 ¹

Let us introduce the noncoordinate basis of vector ﬁelds{e⊥ , ea } [e⊥ , ea ] = C ⊥ e⊥ + C d⊥a ed , ⊥a d [ea , eb ] = C ab ed ,with the structure functions 1 C ⊥ = ea lnN, ⊥a C d⊥a = (N b C dab + ea N d ) NThe metric in the corresponding dual basis {θ⊥ , θa } g = −θ⊥ ⊗ θ⊥ + γab θa ⊗ θb , ´ 78/95 ¹

where γ is the induced metric on the submanifold Σt . Thecomponents of the vector ﬁeld X0 in this basis X0 = N e⊥ + N a eaare the Lagrange multipliers in ADM scheme which can be ∂ ∂obtained if we pass to the coordinate basis {X0 = ∂t , ∂xa } g = − N 2 − N a Na dt ⊗ dt + 2Na dt ⊗ dxa + γab dxa ⊗ dxbThe second fundamental form characterizes the embedingof ´ 79/95 ¹

Σt in (M, g) K(X, Y ) : Tx M × Tx M → R 1 K(X, Y ) = (Y. X e⊥ − X. Y e⊥ ) , X, Y ∈ T Σt 2The another representation for the extrinsic curvature is 1 Kab = − Le⊥ γab 2To ﬁnd the 3 + 1 decomposition we deﬁne the induced onthe Σt connection 3 c Γ ba := Γc ba = 3 c 3 θ, ea e b , ´ 80/95 ¹

where 3 X is the covariant derivative with respect to γ andcorresponding curvature tensor is 3 R(X, Y ) = 3 X 3 Y −3 Y 3 X − [3 X, 3 Y]In the basis {e⊥ , ea } we ﬁnd the componets of the connection Γ⊥⊥⊥ = 0, Γ⊥⊥i = 0, Γj ⊥⊥ = cj , Γj ⊥i = −K ji , Γ⊥i⊥ = ci , Γ⊥ji = −Kji , Γj i⊥ = −K ji + 3 j θ , Le⊥ ei , Γkij = 3 Γkij ´ 81/95 ¹

and of the Riemann tensor 1 js3 Rj ⊥k⊥ = K js Ksk + γ js Le⊥ Kks + γ ek 3 es N , N R⊥ijk = 3 ek Kij − 3 ej Kik , Rs ijk = 3Rs ijk + Kik Kj s − Kij Kk sFinally the scalar curvature can be obtained in the form 2 ab3 R = 3R + Ka a Kb b − 3Kab K ab − γ ea 3 eb N − 2γ ab Le⊥ Kab NThe classical behavior of the dynamical variables N , N a , γ ´ 82/95 ¹

is determined by the Hilbert-Einstein action t2 . . . L[N, Na , γab , N , Na , γ ab ] = dt 3σN 3 R + Ka a Kb b − 3Kab K ab t1S t2 2 ab − dt 3 σN γ 3 ea 3 eb N − 2γ ab Le⊥ Kab N t1S √ 1where 3σ = γ θ ∧ θ2 ∧ θ3 is the volume 3-form on Σt . ´ 83/95 ¹

1.2. Bianchi modelsL. Bianchi 1897Sugli spazii atre dimensioni du ammettono un gruppo continuo dimovimentiSoc. Ital. della Sci. Mem. di Mat.(Dei. XL) (3) 3 267By deﬁnition, Bianchi models are manifolds with producttopology M = R × G3On the three dimensional Riemannnian manifold Σt γ there ´ 84/95 ¹

exist left-invariant 1-forms {χa } such that 1 dχa = − C abc χb ∧ χc 2The dual vector ﬁelds {ξa } form a basis in the Lie algebraof the group G3 [ξa , ξb ] = C dab ξdwith structure constants C dab = 2 dχd (ea , eb ). In the invariantbasis [e⊥ , ea ] = C d⊥a ed , [ea , eb ] = −C dab ed ,with C d⊥a = N −1 N b C dab the metric takes the form g = − θ⊥ ⊗ θ⊥ + γab θa ⊗ θb ´ 85/95 ¹

The preferable role of this choice for a coframe is that thefunctions N, N a and γab depend on the time parameter tonly. Due to this simpliﬁcation the initial variational prob-lem for Bianchi A models is restricted to a variational prob-lem of the “mechanical” system t2 √ 3 L (N, Na , γab , γab ) = ˙ dt γN R − Ka a Kb b + Kab K ab , t1where 3R is the curvature scalar formed from the spatialmetric γ 1 1 3 R = − γ ab C cda C dcb − γ ab γ cd γij C i ac C jbd , 2 ´ 86 4 ¹/95

and 1 Kab = − (γad C dbc + γbd C dac )N c + γab ˙ 2Nis the extrinsic curvature of the slice Σt deﬁned by the re-lation 1 Kab = − Le⊥ γab 2In the theory we have four primary constraints δL δL π := = 0, π a := =0 ˙ δN ˙ δ Na δL √ π ab := = γ(γab K i i − K ab ) δ γab ˙ ´ 87 /95 ¹

The symplectic structure on the phase space is deﬁned bythe following non-vanishing Poisson brachets 1 r s {N, π} = 1, {Na , π b } = δa , b {γcd , π rs } = s r δ δ + δc δd 2 c dDue to the reparametrization symmetry of inherited fromthe diﬀeomorphism invariance of the initial Hilbert-Einsteinaction, the evolution of the system is unambiguous and itis governed by the total Hamiltonian HT = N H + N a Ha + u0 P 0 + ua P awith four arbitrary functions ua (t) and u0 (t). One can verify ´ 88/95 ¹

that the secondary constraints are ﬁrst class and obey thealgebra {H, Hb } = 0, {Ha , Hb } = −C dab HdFrom the condition of time conservation of the primaryconstaraints folllows four secondary constaraints 1 1 √ H = √ π ab πab − π aa π b b − γ 3R, Ha = 2 C dab π bc γcd , γ 2which obey the algebra {H, Ha } = 0, {Ha , Hb } = −C dab HdThe Hamiltonian form of the action for the Bianchi A mod- ´ 89/95 ¹

els can be obtained in the form t2 a ab L[N, Na , γab , π, π , π ] = π ab dγab − HC dt t1where the canonical Hamiltonian is a linear combination ofthe secondary constraints HC = N H + N a Ha . ´ 90/95 ¹

1.3. Hamiltonian reduction of Bianchi I cosmol- ogyIn the case of Bianchi I model the group which acts on(Σt , γ) is T3 and the action takes the form t2 a ab L[N, Na , γab , π, π , π ] = π ab dγab − N Hdt . t1Using the decomposition for arbitrary symmetric non-singularmatrix γ = RT (χ)e2X R(χ) , ´ 91/95 ¹

where X = diag x1 , x2 , x3 is diagonal matrix and R(ψ, θ, φ) = eψJ3 eθJ1 eφJ3we can pass to the new canonical variables (γab , π ab ) =⇒ (χa , pchia ; xa , pa )The corresponding momenta are 3 3 T π=R P s αs + Ps αs R s=1 s=1 ´ 92/95 ¹

where P a = pa , 1 ξa Pa = − , (cyclic permutations a = b = c) 4 sinh(xb − xc )and the left-invariant basis of the action of the SO(3, R) inthe phase space with three dimensional orbits is given by sin ψ ξ1 = pφ + cos ψ pθ − sin ψ cot θ pψ , sin θ cos ψ ξ2 = − pφ + sin ψ pθ + cos ψ cot θ pψ , sin θ ξ3 = pψ ´ 93/95 ¹

In the new variables the Hamiltonian constraint reads 3 2 1 1 ξc H= p2 − a p a pb + 2 a=1 a<b 2 (abc) sinh2 (xa − xb ) ´ 94/95 ¹

Many thanks TO THE ORGANIZERS for the nice Conference !!! Also many thanks to all of you !!! I hope we will see each otherin the next edition of the Conference . . . ´ 95/95 2

D. Mladenov - On Integrable Systems in Cosmology

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