A la recerca de l'estabilitat efectiva dels asteroids trojans : algunes idees constructives de tècniques KAM sense ssar coordenades acció-angle

THE QUEST FOR THE LONG TIME STABILITY OF THE TROJAN ASTEROIDS: SOME CONSTRUCTIVE IDEAS OF KAM METHODS WITHOUT ACTION{ANGLE COORDINATES  Angel Jorba, Rafael de la Llave and Jordi Villanueva Novemb er 9, 2000 { The RTBP Sun{Jupiter -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 11.5 L 1 L 2 L 3 L 5 L 4 SJ      { Trojan Asteroids  They have orbits placed close to the p oint L 5 of the Sun-Jupiter system (dened geometrically).  If we integrate numerically the orbits of these asteroids in the full solar system (JPL ephemeris), we observe that some of them are \stable" forlong spans of time: they seem to lie on invariant tori. 1 -1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 Aquiles JPL x-y 2 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Aquiles JPL z-dz 3 -1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 Aquiles RTBP x-y 4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Aquiles RTBP z-dz 5 -1.1 -1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 Hector JPL x-y 6 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Hector JPL z-dz 7 -1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 Hector RTBP x-y 8  What can be an alternative approach ? If Trojans are not close enough to L 5 , we have to rep eat normal form computations around an invariant object close enough to (a given) one of them.  As Trojans seems to be close to invariant tori, it is natural to use as a \reference" an invariant torus close to the \apparent" torus describ ed by the Trojan. 15 { Description of this dierent approach 1. We take an initial condition of a Trojan, and we write it in the co ordinates of the RTBP Sun-Jupiter. 2. We perform a numerical integration of this trajectory, and we apply a metho d of analysis of frequencies in order to obtain the approximated frequencies of the quasi-torus describ ed by the trajectory,as well a trigonometric p olynomial that describ es this torus. 3. We replace the set of computed frequencies by another one, close as p ossible, but for which a standard Diophantine condition is fullled. 4. We prove that there exists an invariant torus of the system, close to the predicted one (with the mo died frequencies). 16 5. We p erform \normal form computations" around this torus, to obtain a domain of long time stability, which is exp ected to contain the initial condition of the chosen Trojan. The classical set-up for KAM metho ds for maximal dimensional tori of Hamiltonian systems  Quasi-integrable Hamiltonian system with ` degrees of freedom, given in actionangle co ordinates: H (  I ) = h ( I ) + "f (  I )  with (  I ) 2 T `  R ` , 2  -p erio dic in  ,and " 2 R a small parameter.  If " = 0 (integrable case): any Torus T ` f I 0 g is invariant, carrying linear quasip erio dic ow, with vector of basic frequencies given by ! = @ I h ( I 0 ).  We said that ! 2 R ` is Diophantine, if there is c > 0 and   ` ; 1, such that: jh k ! ij  c j k j  1  for any k 2 Z ` n f 0 g  with j k j 1 = j k 1 j + j k ` j . 17 { Kolmogorov Theorem  Given a Diophantine ! 2 R ` we can ask for the p ersistence of the corresp onding invariant torus of the case " = 0 for values of " 6 = 0 small enough: Kolmogorov Theorem .  Standard metho dology: it lo oks for a sequence of canonical transformations such that applied on H leads to an equivalent Hamiltonian forwhich the existence of the torus is explicit. We will refer to such constructions as \ transformation theory ". 18 Theorem 1 We consider a quasi-integrable Hamiltonian system with ` degrees of freedom, given in action-angle co ordinates: H (  I ) = h ( I ) + "f (  I ) : We assume:  Analyticity: For any " 2 R small enough, H is real analytic for (  I ) 2 T `  U , with U  R ` .  Non-resonance: For a given I 0 2 U , ! = @ I h ( I 0 ) is Diophantine.  Non-degeneracy: det @ 2 II h ( I 0 ) 6 = 0 . Then, for small " , there exists a real analytic canonical transformation  ,with  ; Id 2  - p erio dic in  and small ( O ( " ) ), such that H  (  I ) = const. + h ! I ; I 0 i + O 2 ( I ; I 0 ) : So, T `  f I 0 g is an invariant torus of H   , carrying linear quasi-p erio dic ow, with vector of basic frequencies given by ! . The invariant torus of H is ( !t +  I 0 ) . 19 { Some basic ideas ab out the pro of: a generic step of the iterative metho d  We assume I 0 =0, and we consider a Hamiltonian of the form H = a + A (  ) + h ! + B (  ) I i + 1 2 h I C (  ) I i + H (  I )  1. A and B have size of O 1 ( M ), with M \small", and H = O 3 ( I ). 2. A = 0 and det C 6 =0.  Bar means average with resp ect to the angular variables  : A = Z 0  2  ] ` A (  ) d : 20  We lo ok for a generating function G (  I ) = h   i + X (  ) + h Y (  ) I i  with  2 R ` , X (  ) and Y (  ) 2  -p erio dic in  , such that if we apply the canonical transformation given by the ow time one of the Hamiltonian system G , then the new A (1) and B (1) corresp onding to H (1) = H   G t =1 have size basically controlled by M 2 (Newton-like metho d).  We lo ok for an iterative pro cess with quadratic sp eed of convergence in order to control the \bad eect" of the small divisors.  H (1) = H   G t =1 can be computed using the Lie series metho d : H   G t =1 = H + 1 1! f H G g + 1 2! ff H G g G g +   with f  g the Poisson bracket asso ciated to (  I ): f f (  I ) g (  I ) g = h @  f @ I g i ; h @ I f @  g i : 21 It suces to lo ok for G such that:  h ! I i + 1 2 h I CI i G  + A + h B I i = O 2 ( I ) : It leads to the following (homological) equations: ( eq 1 ) L ! X = A (  )  ( eq 2 ) L ! Y j = D j (  )  B j ; P ` l =1 C jl (  l + @  l X )  where L ! is the linear dierential op erator dened as L ! X = h ! @  X i : 22 ( eq 1 )As A = 0, then X = L ; 1 ! A ,with X = 0, can be computed expanding the  - functions in Fourier series: L ; 1 ! A (  ) = X k 2 Z ` nf 0 g ^ A k i h k ! i e i h k i  which is convergent if ! is Diophantine. ( eq 2 )  is taken as  = ( C ) ; 1 ( B ; C@  X )  in such a way D =0. Then, Y j = L ; 1 ! D j  with Y j = 0. 23 Remark 2 Working with action-angle variables, if we consider the iterative expression of the Hamiltonian in the pro of of the Kolmogorov theorem: H = a + A (  ) + h ! + B (  ) I i + 1 2 h I C (  ) I i + H (  I )  and recalling that the approximate torus is (  0) , then JD 2 H (  0) =  0 C (  ) 0 0 ! + O 1 ( M ) : Due to this triangular structure, in this context R can be reduced to constant co e- cients.  How we can recoverer this structure in the non-action-angle context ? 30 { Denition (an imp ortant one): T (  ) is a Lagrangian torus if T  $ =0, i.e. if S  T >  J T  = 0.  Lo ok at the Kolmogorov theorem: ( T `  f I 0 g )is a Lagrangian torus.  What can we said in our context ? Lemma 1 With the non-action angle setting, we have S = 0 (as $ = d )and S jl = ; ( @  j c l ; @  l c j )  where c l = L ; 1 ! ( $ ( @  l T R )) : Remark 3 If R  0 ,then S  0 , else, we have that S  O 1 ( R ) . 31 Lemma 2 With the non-action angle setting, we have: R ( T  ) = R   R ( J T  " ; 1 + T  C ) = T  E + E (  )  where " = T >  T  A = " ; 1 T >  R ( J T  " ; 1 )  E = A C = ; L ; 1 ! ( A ; A )  E  O 1 ( R ) : Remark 4 If R  0 , then R is reducible if we express it in the symplectic basis given by the columns of fT  J T  " ; 1 + T  C g ,else, in the quasi-invariant case, R is quasi-reducible expressed in the same basis, which is quasisymplectic. 32 { \Solving" the equation of the Newton metho d We lo ok for ! T = T  a + ( J T  " ; 1 + T  C ) b , which leads to the following equations (skipping O 2 ( R ) terms): ( eq 1 ) L ! b = T >  JR , ( eq 2 ) L ! a = ; " ; 1 T >  R ; C T >  JR ; Eb . 1. T >  JR = 0. So, ( eq 1 )can be solved with an indeterminate value for b . 2. If det E 6 = 0 (non-degeneracy condition), we can dene b from ( eq 2 ) in such a way the average of the right-hand side vanishes. So, we can compute a ( a plays no role). One can check that R (! T ) + R = O 2 ( R ). 33 Theorem 2 Let T (  ) : T ` 7! R 2 ` ( 2  -p erio dic in  )be a ! -quasi-torus of a Hamiltonian H (  ) with ` degrees of freedom, that is L ! T (  ) = J r H ( T (  )) + R (  ) : We assume:  Analyticity: H is real analytic for  2 U ,with U  R ` ,and T is real analytic in T ` ,with T ( T ` )  U .  Non-resonance: ! 2 R ` is Diophantine.  Non-degeneracy: 1. T is a non-degenerate manifold: if " = T >  T  , then det " 6 = 0 in T ` . 2. T is non-degenerate as a quasi-torus of H : if A = " ; 1 T >  R ( J T  " ; 1 ) and E = A , then det E 6 = 0 . Then, if k R k is small enough, there exists ^ T a true ! -invariant torus of H ,that it is k R k - close to T . 34 { Eective stability: the standard actionangle setting  We consider H (  I ) a real analytic Hamiltonian system with ` degrees of freedom taking the form: H (  I ) = h ! I i + H 2 (  I )  with H 2 (  I )  O 2 ( I ). So H has an ! - invariant torus at I = 0. We assume ! Diophantine.  Finite Normal form pro cess: we lo ok for a canonical transformation  N such that: H   N (  I ) = h ! I i + N ( I ) + H N (  I )  with H N  O N ( I ). So, if j I j is small, k H N k is also small.  As _ I = ; @ I H N , if H N  0then I are rst integrals of the transformed system ( I ( t ) = const.), else I ( t ) moves slow. 35 { Eective stability: the non-action-angle setting: Lindstedt series  We consider T (  ), real analytic 2  -p erio dic in  , an ! -invariant torus of areal analytic Hamiltonian H (  ) with ` degrees of freedom. We assume ! Diophantine.  Lindstedt series: We lo ok for a formal expansion    (  I ) = X j 2 N `  ( j ) (  ) I j  with  (0) (  ) = T (  )and  ( j ) 2  -p erio dic in  , such that the dynamics of H expressed in variables (  I ) takes the form: _ I = 0  _  = h ( I ) = P l 2 N ` h ( l ) I l  with h (0) = ! and h ( l ) 2 R ` . 36 Remark 5 I are formal rst integrals of H around T . Remark 6 We do no ask I  I (  ) to be in involution, and hence (  I ) 7!  =  (  I ) is not a formal canonical change. We only need D I  (  0) to be non-degenerate. 37 { Computation of  ( j ) (  ) and h ( j ) X j ( L h ( I )  ( j ) ) I j = J r H ( X j  ( j ) I j ) : We equate powers of I m :  For m = 0: L !  (0) = J r H (  (0) )  true.  For any m 2 N ` , j m j 1  1: L h ( m )  (0) + R (  ( m ) ) = Z ( m )  where Z ( m ) (  ) = # J r H (#  ] < j m j 1 )] m ; X j 2 N ` nf 0 m g m ; j  0 L h ( j )  ( m ; j ) : 38  We use the notations: #  ] <N (  I ) = X j j j 1 <N  ( j ) (  ) I j  and #  ] m the co ecient of I m of the expression inside.  We lo ok for  ( m ) = T  V ( m ) + ( J T  " ; 1 + T  C ) W ( m )  obtaining: ( eq 1 ) L ! W ( m ) = ;T >  JZ ( m ) , ( eq 2 ) h ( m ) + L ! V ( m ) + EW ( m ) ; C T >  J Z ( m ) = " ; 1 T >  Z ( m ) . Remark 7 We recall that if T is an ! -invariant torus, then T is Lagrangian and R is reducible. 39