scholarly journals The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

2017 ◽  
Vol 17 (1) ◽  
Author(s):  
Alberto Abbondandolo ◽  
Luca Asselle ◽  
Gabriele Benedetti ◽  
Marco Mazzucchelli ◽  
Iskander A. Taimanov
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Cotangent Bundle ◽  
Energy Levels ◽  
High Energy ◽  
Intermediate Range ◽  
Magnetic Flows ◽  
Almost All

AbstractWe consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range

scholarly journals The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

2016 ◽  
Vol 08 (03) ◽  
pp. 545-570 ◽  
Author(s):  
Luca Asselle ◽  
Gabriele Benedetti
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Critical Value ◽  
Closed Manifold ◽  
Hamiltonian Flow ◽  
Cotangent Bundles ◽  
Almost All

Let [Formula: see text] be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian [Formula: see text] and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if [Formula: see text] is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of [Formula: see text].

scholarly journals HOFER–ZEHNDER CAPACITY AND HAMILTONIAN CIRCLE ACTIONS

2004 ◽  
Vol 06 (06) ◽  
pp. 913-945 ◽  
Author(s):  
LEONARDO MACARINI
Keyword(s):  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Energy Levels ◽  
Decomposition Theorem ◽  
Hyperbolic Surface ◽  
Circle Action ◽  
Zero Section ◽  
Open Dense Subset ◽  
Circle Actions ◽  
Magnetic Flows

We introduce the Hofer–Zehnder Γ-semicapacity [Formula: see text] of a symplectic manifold (M,ω) (or Γ-sensitive Hofer–Zehnder capacity) with respect to a subset Γ⊂π1(M)[Formula: see text] and prove that given a geometrically bounded symplectic manifold (M,ω) and an open subset N⊂M admitting a Hamiltonian free circle action with order greater than two then N has bounded Hofer–Zehnder Γ-semicapacity, where Γ⊂π1(N) is the subgroup generated by the orbits of the action. We give several applications of this result. Using Biran's decomposition theorem, we prove the following: let (M2n,Ω) be a closed Kähler manifold (n>2) with [Ω]∈H2(M,ℤ) and Σ a complex hypersurface representing the Poincaré dual of k[Ω], for some k∈ℕ. Suppose either that Ω vanishes on π2(M) or that k>2. Then there exists a decomposition of M into an open dense subset E such that E\Σ has finite Hofer–Zehnder Γ-semicapacity and an isotropic CW-complex, where Γ⊂π1(E\Σ) is the subgroup generated by the obvious circle action on the normal bundle of Σ. Moreover, we prove that if (M,Σ) is subcritical then M\Σ has finite Hofer–Zehnder Γ-semicapacity. We also show that given a hyperbolic surface M and TM endowed with the twisted symplectic form ω0+π*Ω, where Ω is the Kähler form on M, then the Hofer–Zehnder Γ-semicapacity of the domain Uk bounded by the hypersurface of kinetic energy k minus the zero section M0 is finite if k<1/2, where Γ⊂π1(Uk) is the subgroup generated by the fibers of SM. Finally, we consider the problem of the existence of periodic orbits on prescribed energy levels for magnetic flows. We prove that given any weakly exact magnetic field Ω on any compact Riemannian manifold M there exists a sequence of contractible periodic orbits of arbitrarily small energy, extending a previous result of Polterovich.

KNOTS AND LINKS IN INTEGRABLE HAMILTONIAN SYSTEMS

1998 ◽  
Vol 07 (02) ◽  
pp. 123-153 ◽  
Author(s):  
J. CASASAYAS ◽  
J. MARTINEZ ALFARO ◽  
A. NUNES
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Energy Levels ◽  
Solid Torus ◽  
Integrable Hamiltonian Systems ◽  
Hamiltonian Flows ◽  
Smale Flows ◽  
Knots And Links

The main purpose of this paper is to prove that Bott integrable Hamiltonian flows and non-singular Morse-Smale flows are closely related. As a consequence, we obtain a classification of the knots and links formed by periodic orbits of Bott integrable Hamiltonians on the 3-sphere and on the solid torus. We also show that most of Fomenko's theory on the topology of the energy levels of Bott integrable Hamiltonians can be derived from Morgan's results on 3-manifolds that admit non-singular Morse-Smale flows.

Digestibility of nutrients and milk productivity of cows when using the concentrate mixture with the inclusion of rapeseed presscake

2020 ◽  
pp. 48-53
Author(s):  
L. Yarmots ◽  
G. Yarmots ◽  
A. Belenkaya
Keyword(s):  
Protein Digestibility ◽  
High Energy ◽  
Negative Influence ◽  
The Body ◽  
Milk Productivity ◽  
Lactating Cows ◽  
Low Protein ◽  
Concentrate Mixture ◽  
Almost All ◽  
Experimental Group

For ruminants, especially high-yielding animals in addition to the complete supply of animals with protein, its digestibility in the rumen is important. With low protein digestibility in the rumen, the released ammonia will be more effectively used by the rumen microflora, and the undigestible protein in the subsequent sections of the digestive tract can serve as a source of amino acids for the body. The use of concentrate mixtures with the inclusion of local, affordable and cheaper grain feeds, in particular a high-energy and protein ingredient- rapeseed presscake makes it possible to increase the milk productivity of cows throughout lactation. These presscakes are well balanced in their amino acid composition and belong to feeds whose protein has a low degree of digestibility in the rumen. The purpose of the researches was to study the digestibility of nutrients and milk productivity of cows when using the concentrate mixture with the inclusion of rapeseed presscake. In the scientific and economic experiment has been carried out on lactating cows, where the cows of the experimental group in the concentrate mixture of peas has been replaced with rapeseed presscake the digestibility of nutrients in the ration, energy metabolism and milk productivity have been studied. Studies have shown that almost all the nutrients were significantly better digested by the animals of the experimental group. Energy in milk was more allocated by cows of the experimental group by 6,29 MJ. From cows of the experimental group for 100 and 305 days of lactation has been obtained more milk by 6,27 and 7,06 %, respectively, than from control herdmates. The biochemical parameters of blood were within the limits of the physiological norm in animals of both groups. Thus, the replacement of peas with rapeseed presscake in the concentrate mixture did not have a negative influence on the metabolic processes and helped to increase the milk productivity of cows.

scholarly journals The Use of Lasers in Dental Materials: A Review

Materials ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3370
Author(s):  
Emmanouil-George C. Tzanakakis ◽  
Evangelos Skoulas ◽  
Eudoxie Pepelassi ◽  
Petros Koidis ◽  
Ioannis G. Tzoutzas
Keyword(s):  
Energy Levels ◽  
Dental Materials ◽  
High Strength ◽  
Surface Characteristics ◽  
High Energy ◽  
Zirconia Ceramics ◽  
Material Management ◽  
Laboratory Procedures ◽  
Laser Devices ◽  
Exact Point

Lasers have been well integrated in clinical dentistry for the last two decades, providing clinical alternatives in the management of both soft and hard tissues with an expanding use in the field of dental materials. One of their main advantages is that they can deliver very low to very high concentrated power at an exact point on any substrate by all possible means. The aim of this review is to thoroughly analyze the use of lasers in the processing of dental materials and to enlighten the new trends in laser technology focused on dental material management. New approaches for the elaboration of dental materials that require high energy levels and delicate processing, such as metals, ceramics, and resins are provided, while time consuming laboratory procedures, such as cutting restorative materials, welding, and sintering are facilitated. In addition, surface characteristics of titanium alloys and high strength ceramics can be altered. Finally, the potential of lasers to increase the adhesion of zirconia ceramics to different substrates has been tested for all laser devices, including a new ultrafast generation of lasers.

scholarly journals Normal Form Near Orbit Segments of Convex Hamiltonian Systems

2021 ◽  
Author(s):  
Shahriar Aslani ◽  
Patrick Bernard
Keyword(s):  
Normal Form ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Vector Fields ◽  
Energy Surface ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
The Other ◽  
Finsler Metrics ◽  
The One

Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [ 8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [ 5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [ 11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [ 1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [ 9, 10]. It is proved in [ 10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [ 10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [ 10,Lemma 3.1] itself taken from [ 3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [ 3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [ 3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [ 7] on Finsler metrics, as [ 3] did. Although our normal form is weaker than the one claimed in [ 10], it is actually sufficient to prove the main results of [ 6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [ 3, Lemma C.1].

Effects of dietary energy levels on ovarian and uterine expression of IGF-1R and EGFR mRNA in prepubertal gilts

2008 ◽  
Vol 5 (2) ◽  
pp. 159-164
Author(s):  
Li Bo ◽  
Zhang He ◽  
Zhang Jing ◽  
Sun Bo-Xing ◽  
Chen Lu ◽  
...  
Keyword(s):  
Growth Factor ◽  
Energy Intake ◽  
Energy Levels ◽  
Growth Factor Receptor ◽  
High Energy ◽  
Diet Group ◽  
Free Access ◽  
Access To Water ◽  
Moderate Energy ◽  
Epidermal Growth

AbstractNine prepubertal gilts (JunMu No. 1) were randomly allocated into three groups (n=3) and fed with a high-energy diet (Group H), a low-energy diet (Group L), or a moderate-energy diet (Group M) for 14 days. Free access to water was provided throughout the research period. Ovaries and uteri were collected after the energy treatments, and processed for determination of the absolute quantities of insulin-like growth factor receptor (IGF-1R) and epidermal growth factor receptor (EGFR) mRNA, using real-time polymerase chain reaction (PCR). The expression of IGF-1R and EGFR mRNA in ovaries and uteri was significantly ranked as: Group H>Group M>Group L (P<0.05). This result suggests that high energy intake markedly enhanced the ovarian and uterine expression of IGF-1R and EGFR in prepubertal gilts, whereas insufficient energy intake markedly inhibited such expression. IGF-1R and EGFR may be involved in mediating the effects of energy intake on the development of the reproductive system in prepubertal gilts.

Synchronized Periodic Solutions of a Class of Periodically Driven Nonlinear Oscillators

1988 ◽  
Vol 55 (3) ◽  
pp. 721-728 ◽  
Author(s):  
Gamal M. Mahmoud ◽  
Tassos Bountis
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Nonlinear Oscillators ◽  
High Energy ◽  
Second Order ◽  
Multiple Time ◽  
Particle Beams ◽  
Periodically Driven ◽  
Time Scaling ◽  
A New Technique

We consider a class of parametrically driven nonlinear oscillators: x¨ + k1x + k2f(x,x˙)P(Ωt) = 0, P(Ωt + 2π) = P(Ωt)(*) which can be used to describe, e.g., a pendulum with vibrating length, or the displacements of colliding particle beams in high energy accelerators. Here we study numerically and analytically the subharmonic periodic solutions of (*), with frequency 1/m ≅ √k1, m = 1, 2, 3,…. In the cases of f(x,x˙) = x3 and f(x,x˙) = x4, with P(Ωt) = cost, all of these so called synchronized periodic orbits are obtained numerically, by a new technique, which we refer to here as the indicatrix method. The theory of generalized averaging is then applied to derive highly accurate expressions for these orbits, valid to the second order in k2. Finally, these analytical results are used, together with the perturbation methods of multiple time scaling, to obtain second order expressions for regions of instability of synchronized periodic orbits in the k1, k2 plane, which agree very well with the results of numerical experiments.

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