Symplectic Invariants and Hamiltonian Dynamics

Hofer, Helmut; Zehnder, Eduard

doi:10.1007/978-3-0348-0104-1

Cited by 246 publications

(514 citation statements)

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“…The infimum is taken over all smooth paths in Ham c R 2n from ϕ to ψ. The following proposition holds [12]:…”

Section: Hofer's Metricmentioning

confidence: 99%

“…There is a lot of information scattered in the literature on the connectedness of the group of symplectic and Hamiltonian maps [1,15,12]. There is also some confusion regarding certain aspects of this topic.…”

Section: Connectedness Of the Group Of Hamiltonian Symplectic Mapsmentioning

confidence: 99%

“…It is also known that the action spectrum of compactly supported Hamiltonian maps is compact and nowhere dense, and in general does not depend continuously with respect to M. For details we refer the reader to [12]. On the other hand,…”

Section: Theoremmentioning

confidence: 99%

“…To this end, the conditions of theorem (12) are satisfied, and the Hamilton-Jacobi equation provides a method to compute the distance between symplectic maps. Using (73) in (71), taking norms on both sides of (71), and using the invariance of the oscillation norm under the adjoint action, we obtain…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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Local Calculation of Hofer's Metric and Applications to Beam Physics

Erdélyi¹

2012

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Hofer's metric is a very interesting way of measuring distances between compactly supported Hamiltonian symplectic maps. Unfortunately, it is not known yet how to compute it in general, for example for symplectic maps far away from each other. It is known that Hofer's metric is locally flat, and it can be computed by the so-called oscillation norm of the difference between the Poincare generating functions of symplectic maps close to identity. It is shown here that the same result holds for arbitrary extended generating function types and symplectic maps, as long as the respective generating functions are well defined for the given symplectic maps. This result plays a crucial role is formulating and solving the optimal symplectic approximation problem in Hamiltonian nonlinear dynamics. Applications to beam physics are oulined.

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“…The infimum is taken over all smooth paths in Ham c R 2n from ϕ to ψ. The following proposition holds [12]:…”

Section: Hofer's Metricmentioning

confidence: 99%

Section: Connectedness Of the Group Of Hamiltonian Symplectic Mapsmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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Local Calculation of Hofer's Metric and Applications to Beam Physics

Erdélyi¹

2012

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“…It is well-known ( [HZ94]) that if α is non-resonant up to order K, that is k ∈ Z n , 0 < |k 1 | + · · · + |k n | ≤ K =⇒ k · α = 0 then there exists a symplectic transformation Φ K , well-defined in a smaller neighborhood of the origin, such that…”

Section: Introductionmentioning

confidence: 99%

Nekhoroshev estimates for steep real-analytic elliptic equilibrium points

Bounemoura

Fayad²,

Niederman³

2019

Nonlinearity

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We prove that steep real-analytic elliptic equilibrium points are exponentially stable, generalizing results which were known only under a convexity assumption. This proves the general case of a conjecture of Nekhoroshev. This result is also an important step in our proof that generically, both in a topological and measure-theoretical sense, equilibrium points are super-exponentially stable.From (A.12) and (A.13) we getand this ends the proof.Comment. The preprint "Double exponential stability for generic real-analytic elliptic equilibrium points" was first submitted to the Arxiv in August 2015; in order to make it more accessible, we decided to withdraw this preprint and split it into two parts. This corresponds to the first part, the second part being [BFN19].

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Heat Transfer and Entanglement–Non‐Equilibrium Correlation Spectra of Two Quantum Oscillators

Henkel

2021

Annalen der Physik

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The non‐equilibrium state of two oscillators with a mutual interaction and coupled to separate heat baths is discussed. Bosonic baths are considered, and an exact spectral representation for the elements of the covariance matrix is provided analytically. A wide class of spectral densities for the relevant bath modes is allowed for. The validity of the fluctuation‐dissipation relation is established for global equilibrium (both baths at the same temperature) in the stationary state. Spectral measures of entanglement are suggested by comparing to the equilibrium spectrum of zero‐point fluctuations. No rotating‐wave approximation is applied, and anomalous heat transport from cold to hot bath, as reported in earlier work, is demonstrated not to occur.

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Symplectic Invariants and Hamiltonian Dynamics

Cited by 246 publications

References 0 publications

Local Calculation of Hofer's Metric and Applications to Beam Physics

Local Calculation of Hofer's Metric and Applications to Beam Physics

Nekhoroshev estimates for steep real-analytic elliptic equilibrium points

Heat Transfer and Entanglement–Non‐Equilibrium Correlation Spectra of Two Quantum Oscillators

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