Reports de recerca
http://hdl.handle.net/2117/85178
2024-08-03T13:41:52ZBreakdown of homoclinic orbits to L3 in the RPC3BP (II). An asymptotic formula
http://hdl.handle.net/2117/368821
Breakdown of homoclinic orbits to L3 in the RPC3BP (II). An asymptotic formula
Baldomá Barraca, Inmaculada; Giralt Miron, Mar; Guàrdia Munarriz, Marcel
The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1,.., L5.
The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio 0<µ«1. In particular we show that L3 cannot have (one round) homoclinic orbits.
If the ratio between the masses of the primaries µ is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order µ--v, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to µ--v. Thus, classical perturbative methods (i.e the Melnikov-Poincaré method) can not be applied.
The obtention of this asymptotic formula relies on the results obtained in the prequel paper on the complex singularities of the homoclinic of a certain averaged equation and on the associated inner equation.
In this second paper, we relate the solutions of the inner equation to the analytic continuation of the parameterizations of the invariant manifolds of L3 via complex matching techniques. We complete the proof of the asymptotic formula for their distance showing that its dominant term is the one given by the analysis of the inner equation.
Preprint
2022-06-21T16:03:50ZBaldomá Barraca, InmaculadaGiralt Miron, MarGuàrdia Munarriz, MarcelThe Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1,.., L5.
The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio 0<µ«1. In particular we show that L3 cannot have (one round) homoclinic orbits.
If the ratio between the masses of the primaries µ is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order µ--v, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to µ--v. Thus, classical perturbative methods (i.e the Melnikov-Poincaré method) can not be applied.
The obtention of this asymptotic formula relies on the results obtained in the prequel paper on the complex singularities of the homoclinic of a certain averaged equation and on the associated inner equation.
In this second paper, we relate the solutions of the inner equation to the analytic continuation of the parameterizations of the invariant manifolds of L3 via complex matching techniques. We complete the proof of the asymptotic formula for their distance showing that its dominant term is the one given by the analysis of the inner equation.Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
http://hdl.handle.net/2117/135375
Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors<k, ¿>,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequencies
2019-06-26T08:42:03ZDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, PereWe study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors<k, ¿>,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequenciesAn invitation to singular symplectic geometry
http://hdl.handle.net/2117/106301
An invitation to singular symplectic geometry
Miranda Galcerán, Eva; Delshams Valdés, Amadeu; Planas Bahí, Arnau; Oms, Cedric; Dempsey Bradell, Roisin Mary
In this paper we analyze in detail a collection of motivating examples to consider bm-
symplectic forms and folded-type symplectic structures. In particular, we provide models in
Celestial Mechanics for every bm-symplectic structure. At the end of the paper, we introduce
the odd-dimensional analogue to b-symplectic manifolds: b-contact manifolds.
2017-07-10T09:18:33ZMiranda Galcerán, EvaDelshams Valdés, AmadeuPlanas Bahí, ArnauOms, CedricDempsey Bradell, Roisin MaryIn this paper we analyze in detail a collection of motivating examples to consider bm-
symplectic forms and folded-type symplectic structures. In particular, we provide models in
Celestial Mechanics for every bm-symplectic structure. At the end of the paper, we introduce
the odd-dimensional analogue to b-symplectic manifolds: b-contact manifolds.Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio
http://hdl.handle.net/2117/85283
Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly-integrable
Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied.
2016-04-06T10:18:30ZDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, PereThe splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly-integrable
Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied.Examples of integrable and non-integrable systems on singular symplectic manifolds
http://hdl.handle.net/2117/85177
Examples of integrable and non-integrable systems on singular symplectic manifolds
Delshams Valdés, Amadeu; Miranda Galcerán, Eva; Kiesenhofer, Anna
We present a collection of examples borrowed from celes- tial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization trans- formations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lower- ing its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with sin- gularities which are mainly of two types: b m -symplectic and m -folded symplectic structures. These examples comprise the three body prob- lem as non-integrable exponent and some integrable reincarnations such as the two xed-center problem. Given that the geometrical and dy- namical properties of b m -symplectic manifolds and folded symplectic manifolds are well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL, GLPR, MO, S, GMW], we envisage that this new point of view in this collection of examples can shed some light on classical long-standing problems concerning the study of dynamical properties of these systems seen from the Poisson viewpoint.
2016-04-05T09:58:51ZDelshams Valdés, AmadeuMiranda Galcerán, EvaKiesenhofer, AnnaWe present a collection of examples borrowed from celes- tial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization trans- formations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lower- ing its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with sin- gularities which are mainly of two types: b m -symplectic and m -folded symplectic structures. These examples comprise the three body prob- lem as non-integrable exponent and some integrable reincarnations such as the two xed-center problem. Given that the geometrical and dy- namical properties of b m -symplectic manifolds and folded symplectic manifolds are well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL, GLPR, MO, S, GMW], we envisage that this new point of view in this collection of examples can shed some light on classical long-standing problems concerning the study of dynamical properties of these systems seen from the Poisson viewpoint.