Francesco Calogero
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199535286
- eISBN:
- 9780191715853
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535286.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
In Chapter 5 various tricks are introduced, transforming Hamiltonian systems into isochronous Hamiltonian systems. The possibility to apply this procedure to large classes of Hamiltonian systems ...
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In Chapter 5 various tricks are introduced, transforming Hamiltonian systems into isochronous Hamiltonian systems. The possibility to apply this procedure to large classes of Hamiltonian systems justifies the statement that isochronous Hamiltonian systems are not rare. Several examples are discussed. A (very recent) variant of this approach transforms a very large class of (real, autonomous) Hamiltonian systems into Ω-modified isochronous (also real and autonomous)Hamiltonian systems: for instance, it transforms the most general nonrelativistic many-body problem with arbitrary potentials (only restricted to be translation-invariant) into an entirely isochronous many-body problem—completely periodic (in all degrees of freedom) with (arbitrary) period T = 2π/Ω for arbitrary initial data—but behaving, in its center-of-mass frame, essentially as the original system (which might be chaotic) over time intervals short with respect to the period T.Less
In Chapter 5 various tricks are introduced, transforming Hamiltonian systems into isochronous Hamiltonian systems. The possibility to apply this procedure to large classes of Hamiltonian systems justifies the statement that isochronous Hamiltonian systems are not rare. Several examples are discussed. A (very recent) variant of this approach transforms a very large class of (real, autonomous) Hamiltonian systems into Ω-modified isochronous (also real and autonomous)Hamiltonian systems: for instance, it transforms the most general nonrelativistic many-body problem with arbitrary potentials (only restricted to be translation-invariant) into an entirely isochronous many-body problem—completely periodic (in all degrees of freedom) with (arbitrary) period T = 2π/Ω for arbitrary initial data—but behaving, in its center-of-mass frame, essentially as the original system (which might be chaotic) over time intervals short with respect to the period T.
Robert C. Hilborn
- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198507239
- eISBN:
- 9780191709340
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507239.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
When a system involves no dissipative mechanisms such as friction, we say that the system is conservative because its total energy is conserved and the behaviour is described by a time-independent ...
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When a system involves no dissipative mechanisms such as friction, we say that the system is conservative because its total energy is conserved and the behaviour is described by a time-independent Hamiltonian function. In that case, the notion of attractor no longer applies. Different initial conditions (starting points in state space) can lead to dramatically different behaviour and there is no convergence to an attractor. Nevertheless, the state space is still highly organized. The famous Kolmogorov–Arnold–Moser (KAM) Theorem describes how the state space structure changes as the system behaviour changes from integrable to non-integrable. The Henon–Heiles model is used to illustrate many of these features. Area-preserving iterated map systems, such as the Chirikov Standard Map and the Arnold Cat Map, share many of the features of Hamiltonian systems and these are explored through a series of numerically generated diagrams. Applications of Hamiltonian dynamics in astronomy, particle accelerator dynamics, superconductivity and optics are described briefly.Less
When a system involves no dissipative mechanisms such as friction, we say that the system is conservative because its total energy is conserved and the behaviour is described by a time-independent Hamiltonian function. In that case, the notion of attractor no longer applies. Different initial conditions (starting points in state space) can lead to dramatically different behaviour and there is no convergence to an attractor. Nevertheless, the state space is still highly organized. The famous Kolmogorov–Arnold–Moser (KAM) Theorem describes how the state space structure changes as the system behaviour changes from integrable to non-integrable. The Henon–Heiles model is used to illustrate many of these features. Area-preserving iterated map systems, such as the Chirikov Standard Map and the Arnold Cat Map, share many of the features of Hamiltonian systems and these are explored through a series of numerically generated diagrams. Applications of Hamiltonian dynamics in astronomy, particle accelerator dynamics, superconductivity and optics are described briefly.
Francesco Calogero
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199535286
- eISBN:
- 9780191715853
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535286.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
A classical dynamical system is called isochronous if it features in its phase space an open, fully dimensional sector where all its solutions are periodic in all their degrees of freedom with the ...
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A classical dynamical system is called isochronous if it features in its phase space an open, fully dimensional sector where all its solutions are periodic in all their degrees of freedom with the same, fixed period. Recently, a simple transformation has been introduced, featuring a real parameter ω and reducing to the identity for ω=0. This transformation is applicable to a quite large class of dynamical systems and it yields ω-modified autonomous systems which are isochronous, with period T = 2π/ω. This justifies the notion that isochronous systems are not rare. In this monograph—which covers work done over the last decade by its author and several collaborators—this technology to manufacture isochronous systems is reviewed. Many examples of such systems are provided, including many-body problems characterized by Newtonian equations of motion in spaces of one or more dimensions, Hamiltonian systems, and also nonlinear evolution equations (PDEs: Partial Differential Equations). This monograph shall be of interest to researchers working on dynamical systems, including integrable and nonintegrable models, with a finite or infinite number of degrees of freedom. It shall also appeal to experimenters and practitioners interested in isochronous phenomena. It might be used as basic or complementary textbook for an undergraduate or graduate course.Less
A classical dynamical system is called isochronous if it features in its phase space an open, fully dimensional sector where all its solutions are periodic in all their degrees of freedom with the same, fixed period. Recently, a simple transformation has been introduced, featuring a real parameter ω and reducing to the identity for ω=0. This transformation is applicable to a quite large class of dynamical systems and it yields ω-modified autonomous systems which are isochronous, with period T = 2π/ω. This justifies the notion that isochronous systems are not rare. In this monograph—which covers work done over the last decade by its author and several collaborators—this technology to manufacture isochronous systems is reviewed. Many examples of such systems are provided, including many-body problems characterized by Newtonian equations of motion in spaces of one or more dimensions, Hamiltonian systems, and also nonlinear evolution equations (PDEs: Partial Differential Equations). This monograph shall be of interest to researchers working on dynamical systems, including integrable and nonintegrable models, with a finite or infinite number of degrees of freedom. It shall also appeal to experimenters and practitioners interested in isochronous phenomena. It might be used as basic or complementary textbook for an undergraduate or graduate course.
Francesco Calogero
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199535286
- eISBN:
- 9780191715853
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535286.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
In Chapter 4—the longer one in this book—a lemma is first introduced and several isochronous systems of ODEs encompassed by it are treated. One-, two-, three- and multi-dimensional isochronous ...
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In Chapter 4—the longer one in this book—a lemma is first introduced and several isochronous systems of ODEs encompassed by it are treated. One-, two-, three- and multi-dimensional isochronous systems of ODEs—many of them interpretable as many-body models—are then discussed, including several integrable and solvable variants of the “goldfish” many-body problem, nonlinear oscillators models, and two Hamiltonian systems.Less
In Chapter 4—the longer one in this book—a lemma is first introduced and several isochronous systems of ODEs encompassed by it are treated. One-, two-, three- and multi-dimensional isochronous systems of ODEs—many of them interpretable as many-body models—are then discussed, including several integrable and solvable variants of the “goldfish” many-body problem, nonlinear oscillators models, and two Hamiltonian systems.
Francesco Calogero
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199535286
- eISBN:
- 9780191715853
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535286.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
In the introductory Chapter 1 a few representative instances of isochronous dynamical systems are tersely reviewed.
In the introductory Chapter 1 a few representative instances of isochronous dynamical systems are tersely reviewed.
Vadim Kaloshin and Ke Zhang
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was ...
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Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. This book provides the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom). This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. The book follows Mather's strategy but emphasizes a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, the book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.Less
Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. This book provides the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom). This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. The book follows Mather's strategy but emphasizes a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, the book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.
Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on ...
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This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface, all but a set of initial conditions of zero measure have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems, there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Vladimir Arnold. The chapter explains that the book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.Less
This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface, all but a set of initial conditions of zero measure have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems, there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Vladimir Arnold. The chapter explains that the book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.
Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter formulates and proves the jump mechanism. It constructs a variational problem which proves forcing equivalence for the original Hamiltonian using Definition 6.18. It first constructs a ...
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This chapter formulates and proves the jump mechanism. It constructs a variational problem which proves forcing equivalence for the original Hamiltonian using Definition 6.18. It first constructs a special variational problem for the slow mechanical system. A solution of this variational problem is an orbit “jumping” from one homology class to the other. The chapter then modifies this variational problem for the fast time-periodic perturbation of the slow mechanical system. This is achieved by applying the perturbative results established in Chapter 7. Recall the original Hamiltonian system near a double resonance can be brought to a normal form and this normal form, in turn, is related to the perturbed slow system through coordinate change and energy reduction. The variational problem for the perturbed slow system can then be converted to a variational problem for the original.Less
This chapter formulates and proves the jump mechanism. It constructs a variational problem which proves forcing equivalence for the original Hamiltonian using Definition 6.18. It first constructs a special variational problem for the slow mechanical system. A solution of this variational problem is an orbit “jumping” from one homology class to the other. The chapter then modifies this variational problem for the fast time-periodic perturbation of the slow mechanical system. This is achieved by applying the perturbative results established in Chapter 7. Recall the original Hamiltonian system near a double resonance can be brought to a normal form and this normal form, in turn, is related to the perturbed slow system through coordinate change and energy reduction. The variational problem for the perturbed slow system can then be converted to a variational problem for the original.
