Construction of a quasiconserved quantity in the Henon-Heiles problem using a single set of variables

The problem of finding the coefficients of a simple series expansion for a quasiconserved quantity K for the Henon-Heiles Hamiltonian H using a single set of variables is solved. In the past, this type of approach has been problematic because the solution to the equations determining the coefficients in the expansion is not unique. As a result, the existence of a consistent expression for K to all orders had not previously been established. We show how to deal with this arbitrariness in the expansion coefficients for K in a consistent way. Due to this arbitrariness, we find a class of expansions for K, in contrast to the single unique expansion for K generated by the normal-form approach of Gustavson [Astron. J. 71, 670 (1966)]. It may be possible to devise a criterion for deciding which one of our expansions is optimally convergent, although we do not deal with this question here. We proceed by introducing a single set of dynamic variables that have simple symmetry properties and that also diagonalize the problem of finding the coefficients of K. No canonical transformations are required. A straightforward constructive procedure is given for generating the power series to any order for quantities having the symmetry of the Hamiltonian that -are formally conserved. This leads to a very practical method for calculating a quasiconserved quantity in the Henon-Heiles problem. A comparison is made through several orders of the terms generated by this approach and those generated in the original Gustavson expansion in normal form

Some Do\u27s and Don\u27t\u27s for Using Computers in Science Instruction

As science researchers and professors, we have had a fairly broad collection of experiences in using computers, from batch-processing and number crunching in large computer systems using FORTRAN and symbolic languages to teaching problem-solving courses for nonscience students using personal computers. While we claim no special computer expertise, we both own computers and are frequently called upon to offer advice to our colleagues who are greater novices than we are. Hence, on the basis of reflections on our experiences and observations of what we have seen other people and institutions do, we have compiled a list of “Do’s” and “Don’ts” about computer-assisted instruction, software, and hardware

Possible conserved quantity for the Henon-Heiles problem

We study a power-series expansion for a conserved quantity K in the case of the two-dimensional Henon-Heiles potential. An alternative technique to that of Gustavson [Astron. J. 71, 670 (1966)] is applied to find the coefficients in the expansion for K. The technique is used to determine twelve orders for the conserved quantity K, more than twice as many as that calculated by Gustavson. We investigate the degree of constancy of our truncated K in regions where the motion is known to be chaotic and also where it is nonchaotic

Numerical study of a high-order quasiconserved quantity in the Henon-Heiles problem

Recent efforts to derive and study a quasiconserved quantity K in the Henon-Heiles problem in terms of a single set of variables are discussed. Numerical results are given, showing how the value of such a quantity varies with time and order in a power-series expansion for K in terms of monomials of the coordinates and velocities. The lowest order in the power series for K corresponds to n =4 and the highest order to n =27, so that 24 orders are included in the series. The results are compared with an earlier study by the authors [Phys. Rev. A 42, 1931 (1990)] that included an expansion for K for orders n =4 to n =15. In general, even in regions where the earlier study suggested that the series for K might be converging, our more recent results [Phys. Rev. A 44, 925 (1991)], involving twice as many orders, suggest that the series diverges

Frontal and striatal alterations associated with psychopathic traits in adolescents

Neuroimaging research has demonstrated a range of structural deficits in adults with psychopathy, but little is known about structural correlates of psychopathic tendencies in adolescents. Here we examined structural magnetic resonance imaging (sMRI) data obtained from 14-year-old adolescents (n=108) using tensor-based morphometry (TBM) to isolate global and localized differences in brain tissue volumes associated with psychopathic traits in this otherwise healthy developmental population. We found that greater levels of psychopathic traits were correlated with increased brain tissue volumes in the left putamen, left ansa peduncularis, right superiomedial prefrontal cortex, left inferior frontal cortex, right orbitofrontal cortex, and right medial temporal regions and reduced brain tissues volumes in the right middle frontal cortex, left superior parietal lobule, and left inferior parietal lobule. Post hoc analyses of parcellated regional volumes also showed putamen enlargements to correlate with increased psychopathic traits. Consistent with earlier studies, findings suggest poor decision-making and emotional dysregulation associated with psychopathy may be due, in part, to structural anomalies in frontal and temporal regions whereas striatal structural variations may contribute to sensation-seeking and reward-driven behavior in psychopathic individuals. Future studies will help clarify how disturbances in brain maturational processes might lead to the developmental trajectory from psychopathic tendencies in adolescents to adult psychopathy