Vasily E. Tarasov
We consider an invariant skew-symmetric phase-space metric for non-Hamiltonian systems. We say that the metric is an invariant if the metric tensor field is an integral of motion. We derive the time-dependent skew-symmetric phase-space metric that satisfies the Jacobi identity. The example of non-Hamiltonian systems with linear friction term is considered.
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In this paper, the leading term of the asymptotics of the number of possible final positions of a random walk on a directed Hamiltonian metric graph is found. Consideration of such dynamical systems could be motivated by problems of propagation of narrow wave packets on metric graphs.
Bumpy metric theorem in the sense of Man{é} for non-convex Hamiltonians
本文在Mañe意义上证明了非凸哈密顿量的崎岖度量定理，针对满足特定几何条件的哈密顿量。
We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all the related matrix differential-geometric objects, namely, the matrix V^i_j(u) of this system of hydrodynamic type, the metrics g^{ij}_1(u) and g^{ij}_2(u) and the affinors (w_{1, n})^i_j(u) and (w_{2,n})^i_j(u) of the non-…
Let $Σ$ be a compact surface equipped with an area form. There is an long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the set of integrable diffeomorphisms. A natural generalization of this question is to ask to what extent one family of `simple' Hamiltonian diffeomorphisms of $Σ$ can be approximated by the other. In this paper we show that the set of autonomous Hamiltonian diffeomorphisms is not Hofer-dense in the set of integrable Hamiltonians. We construct explicit examples of int…
We construct an embedding $Φ$ of $[0,1]^{\infty}$ into $Ham(M, ω)$, the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold $(M, ω)$. We then prove that $Φ$ is in fact a quasi-isometry. After imposing further assumptions on $(M, ω)$, we adapt our methods to construct a similar embedding of $\mathbb{R} \oplus [0,1]^{\infty}$ into either $Ham(M, ω)$ or $\widetilde{Ham}(M, ω)$, the universal cover of $Ham(M, ω)$. Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on a continuity result…
Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group
本研究在闭辛流形的哈密顿微分同胚群及自治哈密顿微分同胚空间上构造了具备全支集等优良性质的概率测度，生成对应群上的随机游走，并证明极限情形下可在哈密顿同胚空间及相关度量完备化上生成概率测度。
In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltoni-ans by adding potentials (functions depending only on the base point). This led to the definition of Ma{ñ}{é} genericity: a property is generic if, given a Hamiltonian H, the set of potentials u such that H + u satisfies the property is generic. This notion is mostly used in the context of Hamiltonians which are convex in p, in the sense that $\partial$ 2 pp H is positive definite at each points. We will also restrict our study to this situation. There is a close relation between perturbations of Hamil…
Bifurcation of closed orbits of Hamiltonian systems with application to geodesics of the Schwarzschild metric
该研究结合Moser不变曲线定理与Poincaré-Birkhoff不动点定理，证明两自由度近可积哈密顿系统中周期非退化不变环面会分岔出无穷多闭轨道，并给出非退化条件的拱线角导数表示，将其应用于中心力问题与Schwarzschild度规赤道测地线扰动研究。
Invariant sets and spectral rigidity of Hamiltonian diffeomorphisms
该研究解决了Polterovich2002年关于二维哈密顿微分同胚Hofer度量刚性的问题，在高维自治哈密顿流等情形取得进展，相关结果适用于Viterbo谱度量等，运用定量拉格朗日Floer理论完成论证。
An adequate characterization of the dynamics of Hamiltonian systems at physically relevant scales has been largely lacking. Here we investigate this fundamental problem and we show that the finite-scale Hamiltonian dynamics is governed by effective dynamical invariants, which are significantly different from the dynamical invariants that describe the asymptotic Hamiltonian dynamics. The effective invariants depend both on the scale of resolution and the region of the phase space under consideration, and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of th…
We provide a geometric method to stabilize asymptotically with phase an arbitrary fixed periodic orbit of a locally generic three-dimensional Hamiltonian dynamical system. The main advantage of this method is that one needs not know a parameterization of the orbit to be stabilized, but only the values of the Hamiltonian and a fixed Casimir (of the Poisson configuration manifold) at that orbit. The stabilization procedure is illustrated in the case of the Rikitake model of geomagnetic reversal.
Explicit methods in extended phase space for inseparable Hamiltonian problems
本文提出了一种在扩展相空间中显式Leapfrog积分不可分离哈密顿系统的 方法，通过新定义的哈密顿量实现数值积分，并探讨了其在非哈密顿问题中的应用及投影优化。
We study the spectral location of strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the …
Hamiltonian Dynamics of Bayesian Inference Formalised by Arc Hamiltonian Systems
该研究引入定义域为度量空间的无穷维哈密顿弧场系统，证明其能量守恒并推导流存在条件，同时将贝叶斯推理的先验后验差异表征为最小自由能的一阶变分，揭示了自由能的潜在不变性。
In this article we show that the braid type of a set of $1$-periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric $d_{\rm Hofer}$. We call this new phenomenon braid stability for the Hofer metric. We apply braid stability to study the stability of the topological entropy $h_{\rm top}$ of Hamiltonian diffeomorphisms on surfaces with respect to small perturbations with respect to $d_{\rm Hofer}$. We show that $h_{\rm top}$ is lower semicontinuous on the space of Hamiltonian diffeomor…
A note on the RAGE Theorem and phase-averaged dispersion for the Fibonacci Hamiltonian
本文提出更弱的谱测度条件以确保量子非局域化，并改进斐波那契准晶体的相平均非局域化界。
In this paper, from the viewpoint of completeness of Marsden-Weinstein reduction, we illustrate how to give the definitions of a controlled Hamiltonian (CH) system and a reducible controlled Hamiltonian system with symmetry; and how to describe the dynamics of a CH system and the controlled Hamiltonian equivalence; as well as how to give the regular point reduction and the regular orbit reduction for a CH system with symmetry, by analyzing carefully the geometrical and topological structures of the phase space and the reduced phase space of the corresponding Hamiltonian system. We also introdu…
We establish kinetic Hamiltonian flows in density space embedded with the $L^2$-Wasserstein metric tensor. We derive the Euler-Lagrange equation in density space, which introduces the associated Hamiltonian flows. We demonstrate that many classical equations, such as Vlasov equation, Schr{ö}dinger equation and Schr{ö}dinger bridge problem, can be rewritten as the formalism of Hamiltonian flows in density space.
It is very well known that periodic orbits of autonomous Hamiltonian systems are generically organized into smooth one-parameter families (the parameter being just the energy value). We present a simple example of an integrable Hamiltonian system (with an arbitrary number of degrees of freedom greater than one) with a unique periodic orbit in the phase space (which is not compact). Similar examples are given for Hamiltonian systems with a unique invariant torus (of any prescribed dimension) carrying conditionally periodic motions. Parallel examples for Hamiltonian systems with a compact phase …
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