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184
Relative equilibria of Hamiltonian systems with symmetry: linearization, smoothness
, 1995
"... We show that, given a certain transversality condition, the set of relative equilibria E near pe ∈ E of a Hamiltonian system with symmetry is locally Whitneystratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentumgenerat ..."
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Cited by 62 (9 self)
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We show that, given a certain transversality condition, the set of relative equilibria E near pe ∈ E of a Hamiltonian system with symmetry is locally Whitneystratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentumgenerator pairs (µ, ξ) of the relative equilibria. The dimension of the stratum of the conjugacy class (K) is dimG+2dim Z(K) −dim K, where Z(K) is the center of K, and transverse to this stratum E is locally diffeomorphic to the commuting pairs of the Lie algebra of K/Z(K). The stratum E(K) is a symplectic submanifold of P near pe ∈ E if and only if pe is nondegenerate and K is a maximal torus of G. We also show that there is a dense subset of Ginvariant Hamiltonians on P for which all the relative equilibria are transversal. Thus, generically, the types of singularities that can be found in the set of relative equilibria of a Hamiltonian system with symmetry are those types found amongst the singularities at zero of the sets of commuting pairs of certain Lie subalgebras of the symmetry group.
A Proof of the Gutzwiller Semiclassical Trace Formula using Coherent Sates Decomposition
 Commun. in Math. Phys
, 1999
"... The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral ..."
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Cited by 50 (7 self)
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The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of H. Later, using the theory of Fourier integral operators, mathematicians gave rigorous proofs of the formula in various settings. Here we show how the use of coherent states allows us to give a simple and direct proof. 1
On the vertical families of twodimensional tori near the triangular points of the Bicircular problem
, 1999
"... This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the EarthMoon system. The model for the motion of the particle is the socalled Bicircular problem (BCP), that includes the eect of Earth and Moon as in the spatial Restricted Three Body Proble ..."
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Cited by 33 (8 self)
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This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the EarthMoon system. The model for the motion of the particle is the socalled Bicircular problem (BCP), that includes the eect of Earth and Moon as in the spatial Restricted Three Body Problem (RTBP), plus the eect of the Sun as a periodic timedependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we rst discuss some numerical methods for the accurate computation of quasiperiodic solutions, and then we apply them to the BCP to obtain families of 2D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in th...
A methodology for the numerical computation of normal forms, centre manifolds and first integrals of Hamiltonian systems
 Experiment. Math
, 1999
"... This paper deals with the e ective computation of normal forms, centre manifolds and rst integrals in Hamiltonian mechanics. These kind of calculations are very useful since they allow, for instance, to give explicit estimates on the di usion time or to compute invariant tori. The approach presente ..."
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Cited by 33 (6 self)
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This paper deals with the e ective computation of normal forms, centre manifolds and rst integrals in Hamiltonian mechanics. These kind of calculations are very useful since they allow, for instance, to give explicit estimates on the di usion time or to compute invariant tori. The approach presented here is based on using algebraic manipulation for the formal series but taking numerical coe cients for them. This, jointly with a very e cient implementation of the software, allows big savings in both memory and execution time of the algorithms if we compare with the use of commercial algebraic manipulators. The algorithms are presented jointly with their C/C++ implementations, and they are applied to some concrete examples coming from celestial mechanics. rst integrals, algebraic manipulators, in
Hamiltonian Systems Near Relative Equilibria
 J. Dierential Equations
, 1999
"... We give explicit differential equations for the dynamics near relative equilibria of Hamiltonian systems. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic ..."
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Cited by 30 (8 self)
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We give explicit differential equations for the dynamics near relative equilibria of Hamiltonian systems. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic structure and symmetry properties of the Hamiltonian system is analysed and the effects of time reversing symmetries are included. The results will be applicable to the stability and bifurcation theories of relative equilibria of Hamiltonian systems.
Hamiltonian Square Roots of SkewHamiltonian Matrices
, 1997
"... We present a constructive existence proof that every real skewHamiltonian matrix W has a real Hamiltonian square root. The key step in this construction shows how one may bring any such W into a real quasiJordan canonical form via symplectic similarity. We show further that every W has infinitely ..."
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Cited by 27 (9 self)
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We present a constructive existence proof that every real skewHamiltonian matrix W has a real Hamiltonian square root. The key step in this construction shows how one may bring any such W into a real quasiJordan canonical form via symplectic similarity. We show further that every W has infinitely many real Hamiltonian square roots, and give a lower bound on the dimension of the set of all such square roots. AMS subject classification. 65F15 1 Introduction Any matrix X such that X 2 = A is said to be a square root of the matrix A. For general complex matrices A 2 C n\Thetan there exists a welldeveloped although somewhat complicated theory of matrix square roots [7, 14], and a number of algorithms for their effective computation [2, 11]. Similarly for the theory and computation of real square roots for real matrices [10, 14]. By contrast structured square root problems, where both the matrix A and its square root X are required to have some extra (not necessarily the same) spe...
A Restricted FourBody Model For The Dynamics Near The Lagrangian Points Of The SunJupiter System
 Discrete Contin. Dynam. Systems  Series B
, 2001
"... We focus on the dynamics of a small particle near the Lagrangian points of the SunJupiter system. To try to account for the eect of Saturn, we develop a specic model based on the computation of a true solution of the planar threebody problem for Sun, Jupiter and Saturn, close to the real motion of ..."
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Cited by 24 (12 self)
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We focus on the dynamics of a small particle near the Lagrangian points of the SunJupiter system. To try to account for the eect of Saturn, we develop a specic model based on the computation of a true solution of the planar threebody problem for Sun, Jupiter and Saturn, close to the real motion of these three bodies. Then, we can write the equations of motion of a fourth innitesimal particle moving under the attraction of these three masses. Using suitable coordinates, the model is written as a timedependent perturbation of the wellknown spatial Restricted ThreeBody Problem. Then, we study the dynamics of this model near the triangular points. The tools are based on computing, up to high order, suitable normal forms and rst integrals. From these expansions, it is not dicult to derive approximations to invariant tori (of dimensions 2, 3 and 4) as well as bounds on the speed of diusion on suitable domains. We have also included some comparisons with the motion of a few Trojan asteroids in the real Solar system.
Structured factorizations in scalar product spaces
 SIAM J. Matrix Anal. Appl
"... Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. Fo ..."
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Cited by 20 (7 self)
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Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors and singular values that persists across a wide range of scalar products. A key feature of our analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of different characterizations of these scalar product classes is given.
Wigner’s dynamical transition state theory in phase space: Classical and quantum
 Nonlinearity
, 2008
"... We develop Wigner’s approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our development is the construction of a normal form for describing the dynamics locally in the neighborhood of a specific type of saddle point that go ..."
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Cited by 18 (6 self)
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We develop Wigner’s approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our development is the construction of a normal form for describing the dynamics locally in the neighborhood of a specific type of saddle point that governs the evolution from reactants to products in high dimensional systems. In the classical case this is just the standard PoincaréBirkhoff normal form. In the quantum case we develop a version of the PoincaréBirkhoff normal form for quantum systems and a new algorithm for computing this quantum normal form that follows the same steps as the algorithm for computing the classical normal form. The classical normal form allows us to discover and compute phase space structures that govern reaction dynamics. From this knowledge we are able to provide a direct construction of an energy dependent dividing surface in phase space having the properties that trajectories do not locally “recross ” the surface and the directional flux across the surface is minimal. Using this, we are able to give a formula for the directional flux that goes beyond the harmonic approximation. We relate this construction to the fluxflux autocorrelation function which is a standard ingredient in the expression for the reaction rate in the chemistry community. We also give a classical mechanical interpretation of the activated complex as a normally hyperbolic invariant manifold (NHIM), and further describe the NHIM in terms of a foliation by invariant tori. The quantum normal form allows us to understand the quantum mechanical significance of the classical phase space structures and quantities governing reaction dynamics. In particular,