Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

Alberto Castro Ortega

doi:10.1007/s10509-015-2612-0

Home/Archive/arXiv

arXiv · 1601.05015

Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

Alberto Castro Ortega

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work the existence of periodic solutions is studied for the Hamiltonian functions (Formula presented.) where the first term consist of a harmonic oscillator and the second term are homogeneous polynomials of degree 5 defined by two real parameters (Formula presented.) . Using the averaging method of second order we provide the sufficient conditions on the parameters to guarantee the existence of periodic solutions for positive energy and we study the stability of these periodic solutions.

Explore related subjects

astro-ph.GA

Keep this discovery

BibTeX RIS

Alberto Castro Ortega. 2016-01-19. Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five. https://doi.org/10.1007/s10509-015-2612-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fast Dynamical Modelling of Milky Way Globular Clusters -- II. Impacts of Black Hole Prescriptions

The populations of stellar-mass black holes (BHs) in globular clusters (GCs) play a key role in their dynamical evolution, however the mechanisms surrounding their formation and retention are uncertain. In this work, we extend the analysis of Paper I by fitting coupled rapid cluster evolution and multimass equilibrium models to a large sample of Milky Way GCs, under a variety of prescriptions for stellar evolution, BH formation and supernovae (SN) natal kicks. We explore the impacts of adopting SSE or PARSEC (through SEVN) prescriptions for BH initial-final mass relations, the rapid or delayed SN fallback mechanisms, and an ad hoc grid of kick strengths ejecting between 40 and 80 per cent of all BHs formed. All models reproduce the same present-day conditions despite starting from notably different initial BH populations, due to the correlation found between the initial cluster densities and initial BH mass fractions. A linear relationship is found between the (log) initial half-mass density and the initial BH mass fraction, with the SEVN models resulting in median densities ($\rho_{h,0} \sim 10^{7.2\pm1.1}\,{M_\odot pc^{-3}}$) nearly an order of magnitude higher than those of SSE ($\rho_{h,0} \sim 10^{6.4\pm0.9}\,{M_\odot pc^{-3}}$). We also find that both the bottom-light initial mass functions and the present-day BH mass fractions previously inferred are relatively robust against the stellar evolution models and natal kick prescriptions assumed. Finally, we discuss the implications of these results on the expected numbers and properties of dynamical binary-BH mergers, and the growth of intermediate-mass BHs.

astro-ph.GA↗

arXiv2026

SPURS: An Ultra-deep View Inside the Compact, Nitrogen-Enriched Nuclei of Little Red Dots

We present the first ultra-deep rest-UV spectroscopy of four UV-bright Little Red Dots (LRDs), obtained from the SPURS Cycle 4 Large Program. The spectra reveal broad CIV (FWHM $\approx2700-2800$ km s$^{-1}$) in two LRDs, alongside narrow-line densities elevated above star-forming galaxies ($n_e\sim10^4-10^5$ cm$^{-3}$, reaching $10^6$ cm$^{-3}$ in the most extreme source) and nitrogen-enhancements in all four LRDs. We detect broad HeII emission (FWHM $\approx930$ km s$^{-1}$) in one LRD, and two others with fast P-Cygni absorption ($\gtrsim2200$ km s$^{-1}$). Strong interstellar absorption lines and Ly$\alpha$ damping wings reveal the UV continuum is deeply embedded in neutral gas ($N_{\rm HI}\gtrsim10^{22}$ cm$^{-2}$) in all four LRDs. Detections of fluorescent FeII and OI emission and fine-structure absorption indicate this gas lies close to the UV-emitting region. In archival $z>4$ samples, we find nitrogen and strong CIII] emission are significantly more common in LRDs than in the galaxy population. The transmission of broad CIV, tracing the broad-line region or cocoon, depends on rest-optical color within our sample, consistent with an orientation-dependent picture in which bluer, less obscured sightlines offer a more direct, polar view of the central engine and its outflows. We find several potential signatures of very massive stars, whose winds may contribute to nitrogen enhancement. We investigate other abundance patterns expected from supermassive stars but our results are inconclusive. Our results place the UV-emitting region within $\lesssim8$ pc of the LRD nucleus, consistent with an actively assembling nuclear star cluster. Dynamical interactions in this extremely dense environment, including tidal disruption of stars, may explain the high incidence of nitrogen enhancements in LRDs.

astro-ph.GA↗

arXiv2026

Nitrogen-Loud Quasars from the Dark Energy Spectroscopic Instrument. I. Sample Selection and Basic Properties

We present the largest sample to date of nitrogen-loud (N-loud) quasars with strong broad N IV] $\lambda1486$ and/or N III] $\lambda1750$ emission lines over the redshift range $1.6 < z < 4.3$, selected from the Dark Energy Spectroscopic Instrument (DESI) Data Release 1. The final sample contains 1,993 N-loud quasars, corresponding to about 1.2% of the parent quasar sample. The $L_{1450}$ distribution of the N-loud quasars is broadly similar to that of the DESI parent sample, but their redshift distribution is distinct, with a stronger concentration around $z \sim 2.5$--3. Their composite spectrum displays a broadly similar UV continuum shape to that of the parent quasars, while showing significantly enhanced broad nitrogen emission features, including N V, N IV], and N III]. Other metal emission features also show a moderate enhancement. Relative to a control sample matched in redshift and UV continuum luminosity, the N-loud quasars show systematically narrower broad C IV and Mg II emission lines, lower single-epoch virial black hole masses, and higher Eddington ratios, suggesting that N-loud quasars may preferentially appear during a relatively rapid black hole accretion phase. The radio-loud fraction is 10.1%, with the highest fraction among objects exhibiting both N III] and N IV] emission. The catalog provides a statistical baseline for future studies of nitrogen enhancement and its physical origin.

astro-ph.GA↗