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Discrete & Continuous Dynamical Systems
March 2013 , Volume 33 , Issue 3
Special Issue
Dedicated to Ernesto A. Lacomba
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2013, 33(3): i-i
doi: 10.3934/dcds.2013.33.3i
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Abstract:
The material of this special issue of DCDS-A was originally dedicated in honor of the 65-th birthday of Prof. Ernesto A. Lacomba. Some of the papers in this issue reflect the joyful spirit surrounding this celebration. Sadly, shortly after the preparation of this volume was completed, Prof. Ernesto A. Lacomba passed away on June 26, 2012. Therefore this special issue is also paying a tribute to his long standing mathematical legacy.
The work of Prof. Lacomba comprised research on geometric theory of ordinary differential equations, dynamical systems, and symplectic geometry, with applications to celestial mechanics, classical mechanics, vortex theory, thermodynamics and electrical circuits. Prof.~Lacomba was the leader of a strong research group working in these areas. In 1991 he started organizing, jointly with some members of his group and with other collaborators, the International Symposium on Hamiltonian Systems and Celestial Mechanics (HAMSYS), which became a great success over the next several years. These symposia brought together top researches from several countries, working in the aforementioned topics, as well as many graduate students who had the opportunity to learn from and connect with the experts in the field, and often get inspiration and motivation to improve and finalize their doctoral theses.
The framework for the celebration of the 65-th birthday of Prof. Lacomba was the VI-th edition of HAMSYS, which was held in México D.F. between November 29 -- December 3, 2010.
This symposium assembled an impressive number of highly respected researches who generated important discussions among the participants, presented new problems, and identified future research directions. The emphasis of the talks was on Hamiltonian dynamics and its relationship to several aspects of mechanics, geometric mechanics, and dynamical systems in general. The papers in this volume are an outgrowth of the themes of the symposium. All papers that were submitted to this special issue underwent a through refereeing process typical to any top mathematical journal. The accepted papers form the present issue of DCDS-A.
The symposium received generous support from CONACYT México and UAM-I. Special thanks are due to Universidad Autónoma Metropolitana for hosting the symposium in the beautiful colonial building Casa de la primera imprenta de América. Last but not least, we thank all participants for contributing to a week-long intense and highly productive mathematical experience.
The material of this special issue of DCDS-A was originally dedicated in honor of the 65-th birthday of Prof. Ernesto A. Lacomba. Some of the papers in this issue reflect the joyful spirit surrounding this celebration. Sadly, shortly after the preparation of this volume was completed, Prof. Ernesto A. Lacomba passed away on June 26, 2012. Therefore this special issue is also paying a tribute to his long standing mathematical legacy.
The work of Prof. Lacomba comprised research on geometric theory of ordinary differential equations, dynamical systems, and symplectic geometry, with applications to celestial mechanics, classical mechanics, vortex theory, thermodynamics and electrical circuits. Prof.~Lacomba was the leader of a strong research group working in these areas. In 1991 he started organizing, jointly with some members of his group and with other collaborators, the International Symposium on Hamiltonian Systems and Celestial Mechanics (HAMSYS), which became a great success over the next several years. These symposia brought together top researches from several countries, working in the aforementioned topics, as well as many graduate students who had the opportunity to learn from and connect with the experts in the field, and often get inspiration and motivation to improve and finalize their doctoral theses.
The framework for the celebration of the 65-th birthday of Prof. Lacomba was the VI-th edition of HAMSYS, which was held in México D.F. between November 29 -- December 3, 2010.
This symposium assembled an impressive number of highly respected researches who generated important discussions among the participants, presented new problems, and identified future research directions. The emphasis of the talks was on Hamiltonian dynamics and its relationship to several aspects of mechanics, geometric mechanics, and dynamical systems in general. The papers in this volume are an outgrowth of the themes of the symposium. All papers that were submitted to this special issue underwent a through refereeing process typical to any top mathematical journal. The accepted papers form the present issue of DCDS-A.
The symposium received generous support from CONACYT México and UAM-I. Special thanks are due to Universidad Autónoma Metropolitana for hosting the symposium in the beautiful colonial building Casa de la primera imprenta de América. Last but not least, we thank all participants for contributing to a week-long intense and highly productive mathematical experience.
2013, 33(3): 965-986
doi: 10.3934/dcds.2013.33.965
+[Abstract](3199)
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Abstract:
In this paper we analyze the non-integrability of the Wilbeforce spring-pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.
In this paper we analyze the non-integrability of the Wilbeforce spring-pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.
2013, 33(3): 987-1008
doi: 10.3934/dcds.2013.33.987
+[Abstract](3275)
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Abstract:
Using collinear reversible configurations and some properties of symmetry we obtain horseshoe periodic orbits in the general planar three-body problem with masses $m_1\gg m_2 \geq m_3$, which usually represents a system formed by a planet and two small satellites; for instance, the system Saturn-Janus-Epimetheus. For the numerical analysis we have taken the values $m_2/m_1 = 3.5 \times 10^{-4}$ and $m_3/m_1 = 9.7 \times 10^{-5}$ corresponding to $10^5$ times the mass ratios of Saturn-Janus and Saturn-Epimetheus,
Using collinear reversible configurations and some properties of symmetry we obtain horseshoe periodic orbits in the general planar three-body problem with masses $m_1\gg m_2 \geq m_3$, which usually represents a system formed by a planet and two small satellites; for instance, the system Saturn-Janus-Epimetheus. For the numerical analysis we have taken the values $m_2/m_1 = 3.5 \times 10^{-4}$ and $m_3/m_1 = 9.7 \times 10^{-5}$ corresponding to $10^5$ times the mass ratios of Saturn-Janus and Saturn-Epimetheus,
2013, 33(3): 1009-1032
doi: 10.3934/dcds.2013.33.1009
+[Abstract](2996)
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Abstract:
We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincaré second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincaré only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.
We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincaré second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincaré only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.
2013, 33(3): 1033-1047
doi: 10.3934/dcds.2013.33.1033
+[Abstract](2783)
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There are two main reasons why relative equilibria of $N$ point masses under the influence of Newton attraction are mathematically more interesting to study when space dimension is at least 4:
1) in a higher dimensional space, a relative equilibrium is determined not only by the initial configuration but also by the choice of a hermitian structure on the space where the motion takes place (see [3]); in particu\-lar, its angular momentum depends on this choice;
2) relative equilibria are not necessarily periodic: if the configuration is balanced but not central (see [3,2,7]), the motion is in general quasi-periodic.
In this exploratory paper we address the following question, which touches both aspects: what are the possible frequencies of the angular momentum of a given central (or balanced) configuration and at what values of these frequencies bifurcations from periodic to quasi-periodic relative equilibria do occur? We give a full answer for relative equilibrium motions in $R^4$ and conjecture that an analogous situation holds true for higher dimensions. A refinement of Horn's problem given in [12] plays an important role.
There are two main reasons why relative equilibria of $N$ point masses under the influence of Newton attraction are mathematically more interesting to study when space dimension is at least 4:
1) in a higher dimensional space, a relative equilibrium is determined not only by the initial configuration but also by the choice of a hermitian structure on the space where the motion takes place (see [3]); in particu\-lar, its angular momentum depends on this choice;
2) relative equilibria are not necessarily periodic: if the configuration is balanced but not central (see [3,2,7]), the motion is in general quasi-periodic.
In this exploratory paper we address the following question, which touches both aspects: what are the possible frequencies of the angular momentum of a given central (or balanced) configuration and at what values of these frequencies bifurcations from periodic to quasi-periodic relative equilibria do occur? We give a full answer for relative equilibrium motions in $R^4$ and conjecture that an analogous situation holds true for higher dimensions. A refinement of Horn's problem given in [12] plays an important role.
2013, 33(3): 1049-1060
doi: 10.3934/dcds.2013.33.1049
+[Abstract](2571)
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Abstract:
In this paper we prove the existence of central configurations of the $n+2$--body problem where $n$ equal masses are located at the vertices of a regular $n$--gon and the remaining $2$ masses, which are not necessarily equal, are located on the straight line orthogonal to the plane containing the $n$--gon passing through its center. Here this kind of central configurations is called bi--pyramidal central configurations. In particular, we prove that if the masses $m_{n+1}$ and $m_{n+2}$ and their positions satisfy convenient relations, then the configuration is central. We give explicitly those relations.
In this paper we prove the existence of central configurations of the $n+2$--body problem where $n$ equal masses are located at the vertices of a regular $n$--gon and the remaining $2$ masses, which are not necessarily equal, are located on the straight line orthogonal to the plane containing the $n$--gon passing through its center. Here this kind of central configurations is called bi--pyramidal central configurations. In particular, we prove that if the masses $m_{n+1}$ and $m_{n+2}$ and their positions satisfy convenient relations, then the configuration is central. We give explicitly those relations.
2013, 33(3): 1061-1076
doi: 10.3934/dcds.2013.33.1061
+[Abstract](2418)
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Abstract:
Let $H=-\Delta +V$ be a Schrödinger hamiltonian acting on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $V$ a potential of order zero plus a short-range perturbation. In this work we investigate the behavior of the resolvent $R(z)=(H-z)^{-1}$ of $H$ as Im$\,z \downarrow 0$, at high energies and in the framework of Besov spaces $B(\mathbb{R}^n)$. For $\lambda_0>0$ sufficiently large and $\lambda\geq\lambda_0$, we show that there exists a linear operator $R(\lambda+i0)$ such that $R(\lambda+i\epsilon)$ converges to $R(\lambda+i0)$ as $\epsilon\downarrow 0$, strongly in $\mathcal{L}(L^{2, s}(\mathbb{R}^n),L^{2,-s}(\mathbb{R}^n))$, $s>1/2$, and weakly in $\mathcal{L}(B(\mathbb{R}^n),B^*(\mathbb{R}^n))$. We achieve this through a Mourre-estimate strategy.
Let $H=-\Delta +V$ be a Schrödinger hamiltonian acting on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $V$ a potential of order zero plus a short-range perturbation. In this work we investigate the behavior of the resolvent $R(z)=(H-z)^{-1}$ of $H$ as Im$\,z \downarrow 0$, at high energies and in the framework of Besov spaces $B(\mathbb{R}^n)$. For $\lambda_0>0$ sufficiently large and $\lambda\geq\lambda_0$, we show that there exists a linear operator $R(\lambda+i0)$ such that $R(\lambda+i\epsilon)$ converges to $R(\lambda+i0)$ as $\epsilon\downarrow 0$, strongly in $\mathcal{L}(L^{2, s}(\mathbb{R}^n),L^{2,-s}(\mathbb{R}^n))$, $s>1/2$, and weakly in $\mathcal{L}(B(\mathbb{R}^n),B^*(\mathbb{R}^n))$. We achieve this through a Mourre-estimate strategy.
2013, 33(3): 1077-1088
doi: 10.3934/dcds.2013.33.1077
+[Abstract](2826)
+[PDF](386.6KB)
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The Hamiltonization problem for projectable vector fields on general symplectic fiber bundles is studied. Necessary and sufficient conditions for the existence of Hamiltonian structures in the class of compatible symplectic structures are derived in terms of invariant symplectic connections. In the case of a flat symplectic bundle, we show that this criterion leads to the study of the solvability of homological type equations.
The Hamiltonization problem for projectable vector fields on general symplectic fiber bundles is studied. Necessary and sufficient conditions for the existence of Hamiltonian structures in the class of compatible symplectic structures are derived in terms of invariant symplectic connections. In the case of a flat symplectic bundle, we show that this criterion leads to the study of the solvability of homological type equations.
2013, 33(3): 1089-1112
doi: 10.3934/dcds.2013.33.1089
+[Abstract](2921)
+[PDF](487.1KB)
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For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in a Hamiltonian system. This consists in the following steps: (i) computation of the scattering map and of the transition map for the Hamiltonian flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.
For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in a Hamiltonian system. This consists in the following steps: (i) computation of the scattering map and of the transition map for the Hamiltonian flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.
2013, 33(3): 1113-1116
doi: 10.3934/dcds.2013.33.1113
+[Abstract](2241)
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First and second laws of thermodynamics are naturally associated, respectively, to contact and Hessian geometries. In this paper we seek for a unique geometric setting that might account for both thermodynamic laws. Using Riemannian metrics that are compatible with the contact structure, we prove that the Hessian manifold of thermodynamic states cannot isometrically be embedded as Legendre submanifold of a contact manifold. Well known fibrations suggest the nature of the obstruction for such embedding.
First and second laws of thermodynamics are naturally associated, respectively, to contact and Hessian geometries. In this paper we seek for a unique geometric setting that might account for both thermodynamic laws. Using Riemannian metrics that are compatible with the contact structure, we prove that the Hessian manifold of thermodynamic states cannot isometrically be embedded as Legendre submanifold of a contact manifold. Well known fibrations suggest the nature of the obstruction for such embedding.
2013, 33(3): 1117-1135
doi: 10.3934/dcds.2013.33.1117
+[Abstract](2615)
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A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid $G$ may be described in terms of Lagrangian implicit difference equations of the corresponding cotangent groupoid $T^*G$. Other situations include finite difference methods for time-dependent linear differential-algebraic equations and discrete nonholonomic Lagrangian systems, as parti-cular examples.
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid $G$ may be described in terms of Lagrangian implicit difference equations of the corresponding cotangent groupoid $T^*G$. Other situations include finite difference methods for time-dependent linear differential-algebraic equations and discrete nonholonomic Lagrangian systems, as parti-cular examples.
2013, 33(3): 1137-1155
doi: 10.3934/dcds.2013.33.1137
+[Abstract](3455)
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The hip-hop orbit is an interesting symmetric periodic family of orbits whereby the global existence methods of variational analysis applied to the N-body problem result in a collision free solution of (1). Perturbation techniques have been applied to study families of hip-hop like orbits bifurcating from a uniformly rotating planar 2N-gon [4] with equal masses, or a uniformly rotating planar 2N+1 body relative equilibrium with a large central mass [18]. We study the question of stability or instability for symmetric periodic solutions of the equal mass $2N$-body problem without perturbation methods. The hip-hop family is a family of $\mathbb{Z}_2$-symmetric action minimizing solutions, investigated by [7,23], and is shown to be generically hyperbolic on its reduced energy-momentum surface. We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing hip-hop orbit to develop conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle.
The hip-hop orbit is an interesting symmetric periodic family of orbits whereby the global existence methods of variational analysis applied to the N-body problem result in a collision free solution of (1). Perturbation techniques have been applied to study families of hip-hop like orbits bifurcating from a uniformly rotating planar 2N-gon [4] with equal masses, or a uniformly rotating planar 2N+1 body relative equilibrium with a large central mass [18]. We study the question of stability or instability for symmetric periodic solutions of the equal mass $2N$-body problem without perturbation methods. The hip-hop family is a family of $\mathbb{Z}_2$-symmetric action minimizing solutions, investigated by [7,23], and is shown to be generically hyperbolic on its reduced energy-momentum surface. We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing hip-hop orbit to develop conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle.
2013, 33(3): 1157-1175
doi: 10.3934/dcds.2013.33.1157
+[Abstract](2334)
+[PDF](1376.8KB)
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The problem of three bodies with equal masses in $\mathbb{S}^2$ is known to have Lagrangian homographic orbits. We study the linear stability and also a "practical'' (or effective) stability of these orbits on the unit sphere.
The problem of three bodies with equal masses in $\mathbb{S}^2$ is known to have Lagrangian homographic orbits. We study the linear stability and also a "practical'' (or effective) stability of these orbits on the unit sphere.
2013, 33(3): 1177-1199
doi: 10.3934/dcds.2013.33.1177
+[Abstract](2947)
+[PDF](441.2KB)
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We study the dynamics near an equilibrium point of a $2$-parameter family of a reversible system in $\mathbb{R}^6$. In particular, we exhibit conditions for the existence of periodic orbits near the equilibrium of systems having the form $x^{(vi)}+ \lambda_1 x^{(iv)} + \lambda_2 x'' +x = f(x,x',x'',x''',x^{(iv)},x^{(v)})$. The techniques used are Belitskii normal form combined with Lyapunov-Schmidt reduction.
We study the dynamics near an equilibrium point of a $2$-parameter family of a reversible system in $\mathbb{R}^6$. In particular, we exhibit conditions for the existence of periodic orbits near the equilibrium of systems having the form $x^{(vi)}+ \lambda_1 x^{(iv)} + \lambda_2 x'' +x = f(x,x',x'',x''',x^{(iv)},x^{(v)})$. The techniques used are Belitskii normal form combined with Lyapunov-Schmidt reduction.
2013, 33(3): 1201-1214
doi: 10.3934/dcds.2013.33.1201
+[Abstract](3065)
+[PDF](374.4KB)
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We study the stability of an equilibrium point of a Hamiltonian system with $n$ degrees of freedom. A new concept of stability called normal stability is given which applies to a system in normal form and relies on the existence of a formal integral whose quadratic part is positive definite. We give a necessary and sufficient condition for normal stability. This condition depends only on the quadratic terms of the Hamiltonian. We relate normal stability with formal stability and Liapunov stability. An application to the stability of the $L_4$ and $L_5$ equilibrium points of the spatial circular restricted three body problem is given.
We study the stability of an equilibrium point of a Hamiltonian system with $n$ degrees of freedom. A new concept of stability called normal stability is given which applies to a system in normal form and relies on the existence of a formal integral whose quadratic part is positive definite. We give a necessary and sufficient condition for normal stability. This condition depends only on the quadratic terms of the Hamiltonian. We relate normal stability with formal stability and Liapunov stability. An application to the stability of the $L_4$ and $L_5$ equilibrium points of the spatial circular restricted three body problem is given.
2013, 33(3): 1215-1230
doi: 10.3934/dcds.2013.33.1215
+[Abstract](3307)
+[PDF](204.0KB)
Abstract:
An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented by using an orthocentric tetrahedron, which edge lengths are function of given masses. Each mass is placed at the corresponding vertex of the tetrahedron. The center of mass (and orthocenter) of the tetrahedron is at the origin of coordinates. The initial position of the tetrahedron is placed with two pairs of vertices each in a coordinate plan, the lines joining any pair of them parallel to a coordinate axis, the center of masses of each and the center of mass of the four on one coordinate axis. From this original position the tetrahedron is rotated by two angles around the center of mass until the direction of configuration coincides with one axis of coordinates. The four coordinates of the vertices of the tetrahedron along this direction determine the central configuration by finding the two angles corresponding to it. The twelve possible configurations predicted by Moulton's theorem are computed for a particular mass choice.
An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented by using an orthocentric tetrahedron, which edge lengths are function of given masses. Each mass is placed at the corresponding vertex of the tetrahedron. The center of mass (and orthocenter) of the tetrahedron is at the origin of coordinates. The initial position of the tetrahedron is placed with two pairs of vertices each in a coordinate plan, the lines joining any pair of them parallel to a coordinate axis, the center of masses of each and the center of mass of the four on one coordinate axis. From this original position the tetrahedron is rotated by two angles around the center of mass until the direction of configuration coincides with one axis of coordinates. The four coordinates of the vertices of the tetrahedron along this direction determine the central configuration by finding the two angles corresponding to it. The twelve possible configurations predicted by Moulton's theorem are computed for a particular mass choice.
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