Exact solutions for a class of integrable Henon-Heiles-type systems

We study the exact solutions of a class of integrable Henon-Heiles-type systems (according to the analysis of Bountis et al. (1982)). These solutions are expressed in terms of two-dimensional Kleinian functions. Special periodic solutions are expressed in terms of the well-known Weierstrass function. We extend some of our results to a generalized Henon-Heiles-type system with n+1 degrees of freedom.Comment: RevTeX4-1, 13 pages, Submitted to J. Math. Phy

Mechanical similarity as a generalization of scale symmetry

In this paper we study the symmetry known as mechanical similarity (LMS) and present for any monomial potential. We analyze it in the framework of the Koopman-von Neumann formulation of classical mechanics and prove that in this framework the LMS can be given a canonical implementation. We also show that the LMS is a generalization of the scale symmetry which is present only for the inverse square potential. Finally we study the main obstructions which one encounters in implementing the LMS at the quantum mechanical level.Comment: 9 pages, Latex, a new section adde

Breakdown of Lindstedt Expansion for Chaotic Maps

In a previous paper of one of us [Europhys. Lett. 59 (2002), 330--336] the validity of Greene's method for determining the critical constant of the standard map (SM) was questioned on the basis of some numerical findings. Here we come back to that analysis and we provide an interpretation of the numerical results by showing that no contradiction is found with respect to Greene's method. We show that the previous results based on the expansion in Lindstedt series do correspond to the transition value but for a different map: the semi-standard map (SSM). Moreover, we study the expansion obtained from the SM and SSM by suppressing the small divisors. The first case turns out to be related to Kepler's equation after a proper transformation of variables. In both cases we give an analytical solution for the radius of convergence, that represents the singularity in the complex plane closest to the origin. Also here, the radius of convergence of the SM's analogue turns out to be lower than the one of the SSM. However, despite the absence of small denominators these two radii are lower than the ones of the true maps for golden mean winding numbers. Finally, the analyticity domain and, in particular, the critical constant for the two maps without small divisors are studied analytically and numerically. The analyticity domain appears to be an perfect circle for the SSM analogue, while it is stretched along the real axis for the SM analogue yielding a critical constant that is larger than its radius of convergence.Comment: 12 pages, 3 figure

Symbolic dynamics for the NN-centre problem at negative energies

We consider the planar $N$-centre problem, with homogeneous potentials of degree -\a<0, \a \in [1,2). We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets

Emergent Semiclassical Time in Quantum Gravity. I. Mechanical Models

Strategies intended to resolve the problem of time in quantum gravity by means of emergent or hidden timefunctions are considered in the arena of relational particle toy models. In situations with `heavy' and `light' degrees of freedom, two notions of emergent semiclassical WKB time emerge; these are furthermore equivalent to two notions of emergent classical `Leibniz--Mach--Barbour' time. I futhermore study the semiclassical approach, in a geometric phase formalism, extended to include linear constraints, and with particular care to make explicit those approximations and assumptions used. I propose a new iterative scheme for this in the cosmologically-motivated case with one heavy degree of freedom. I find that the usual semiclassical quantum cosmology emergence of time comes hand in hand with the emergence of other qualitatively significant terms, including back-reactions on the heavy subsystem and second time derivatives. I illustrate my analysis by taking it further for relational particle models with linearly-coupled harmonic oscillator potentials. As these examples are exactly soluble by means outside the semiclassical approach, they are additionally useful for testing the justifiability of some of the approximations and assumptions habitually made in the semiclassical approach to quantum cosmology. Finally, I contrast the emergent semiclassical timefunction with its hidden dilational Euler time counterpart.Comment: References Update

Spatio-temporal genetic tagging of a cosmopolitan planktivorous shark provides insight to gene flow, temporal variation and site-specific re-encounters

Migratory movements in response to seasonal resources often influence population structure and dynamics. Yet in mobile marine predators, population genetic consequences of such repetitious behaviour remain inaccessible without comprehensive sampling strategies. Temporal genetic sampling of seasonally recurring aggregations of planktivorous basking sharks, Cetorhinus maximus, in the Northeast Atlantic (NEA) affords an opportunity to resolve individual re-encounters at key sites with population connectivity and patterns of relatedness. Genetic tagging (19 microsatellites) revealed 18% of re-sampled individuals in the NEA demonstrated inter/multi-annual site-specific re-encounters. High genetic connectivity and migration between aggregation sites indicate the Irish Sea as an important movement corridor, with a contemporary effective population estimate (Ne) of 382 (CI = 241–830). We contrast the prevailing view of high gene flow across oceanic regions with evidence of population structure within the NEA, with early-season sharks off southwest Ireland possibly representing genetically distinct migrants. Finally, we found basking sharks surfacing together in the NEA are on average more related than expected by chance, suggesting a genetic consequence of, or a potential mechanism maintaining, site-specific re-encounters. Long-term temporal genetic monitoring is paramount in determining future viability of cosmopolitan marine species, identifying genetic units for conservation management, and for understanding aggregation structure and dynamics

Critical Perspective: Named Reactions Discovered and Developed by Women

Named organic reactions. As chemists, we’re all familiar with them: who can forget the Diels−Alder reaction? But how much do we know about the people behind the names? For example, can you identify a reaction named for a woman? How about a reaction discovered or developed by a woman but named for her male adviser? Our attempts to answer these simple questions started us on the journey that led to this Account. We introduce you to four reactions named for women and nine reactions discovered or developed by women. Using information obtained from the literature and, whenever possible, through interviews with the chemists themselves, their associates, and their advisers, we paint a more detailed picture of these remarkable women and their outstanding accomplishments. Some of the women you meet in this Account include Irma Goldberg, the only woman unambiguously recognized with her own named reaction. Gertrude Maud Robinson, the wife of Robert Robinson, who collaborated with him on several projects including the Piloty−Robinson pyrrole synthesis. Elizabeth Hardy, the Bryn Mawr graduate student who discovered the Cope rearrangement. Dorothee Felix, a critical member of Albert Eschenmoser’s research lab for over forty years who helped develop both the Eschenmoser−Claisen rearrangement and the Eschenmoser−Tanabe fragmentation. Jennifer Loebach, the University of Illinois undergraduate who was part of the team in Eric Jacobsen’s lab that discovered the Jacobsen−Katsuki epoxidation. Keiko Noda, a graduate student in Tsutomu Katsuki’s lab who also played a key role in the development of the Jacobsen−Katsuki epoxidation. Lydia McKinstry, a postdoc in Andrew Myers’s lab who helped develop the Myers asymmetric alkylation. Rosa Lockwood, a graduate student at Boston College whose sole publication is the discovery of the Nicholas reaction. Kaori Ando, a successful professor in Japan who helped develop the Roush asymmetric alkylation as a postdoc at MIT. Bianka Tchoubar, a critically important member of the organic chemistry community in France who developed the Tiffeneau−Demjanov rearrangement. The accomplishments of the women in this Account illustrate the key roles women have played in the discovery and development of reactions used daily by organic chemists around the world. These pioneering chemists represent the vanguard of women in the field, and we are confident that many more of the growing number of current and future female organic chemists will be recognized with their own named reactions

Four-body co-circular central configurations

We classify the set of central configurations lying on a common circle in the Newtonian four-body problem. Using mutual distances as coordinates, we show that the set of four-body co-circular central configurations with positive masses is a two-dimensional surface, a graph over two of the exterior side-lengths. Two symmetric families, the kite and isosceles trapezoid, are investigated extensively. We also prove that a co-circular central configuration requires a specific ordering of the masses and find explicit bounds on the mutual distances. In contrast to the general four-body case, we show that if any two masses of a four-body co-circular central configuration are equal, then the configuration has a line of symmetry.Peer ReviewedPostprint (author’s final draft