Variational Integrators for the Gravitational N-Body Problem

This paper describes a fourth-order integration algorithm for the gravitational N-body problem based on discrete Lagrangian mechanics. When used with shared timesteps, the algorithm is momentum conserving and symplectic. We generalize the algorithm to handle individual time steps; this introduces fifth-order errors in angular momentum conservation and symplecticity. We show that using adaptive block power of two timesteps does not increase the error in symplecticity. In contrast to other high-order, symplectic, individual timestep, momentum-preserving algorithms, the algorithm takes only forward timesteps. We compare a code integrating an N-body system using the algorithm with a direct-summation force calculation to standard stellar cluster simulation codes. We find that our algorithm has about 1.5 orders of magnitude better symplecticity and momentum conservation errors than standard algorithms for equivalent numbers of force evaluations and equivalent energy conservation errors.Comment: 31 pages, 8 figures. v2: Revised individual-timestepping description, expanded comparison with other methods, corrected error in predictor equation. ApJ, in pres

Chaotic Lagrangian Trajectories around an Elliptical Vortex Patch Embedded in a Constant and Uniform Background Shear Flow

The Lagrangian flow around a Kida vortex [J. Phys. Soc. Jpn. 5 0, 3517 (1981)], an elliptical two‐dimensional vortex patch embedded in a uniform and constant background shear, is described by a nonintegrable two‐degree‐of‐freedom Hamiltonian. For small values of shear, there exist large chaotic zones surrounding the vortex, often much larger than the vortex itself and extremely close to its boundary. Motion within the vortex is integrable. Implications for two‐dimensional turbulence are discussed

Chaotic Lagrangian Trajectories around an Elliptical Vortex Patch Embedded in a Constant and Uniform Background Shear Flow

The Lagrangian flow around a Kida vortex [J. Phys. Soc. Jpn. 5 0, 3517 (1981)], an elliptical two‐dimensional vortex patch embedded in a uniform and constant background shear, is described by a nonintegrable two‐degree‐of‐freedom Hamiltonian. For small values of shear, there exist large chaotic zones surrounding the vortex, often much larger than the vortex itself and extremely close to its boundary. Motion within the vortex is integrable. Implications for two‐dimensional turbulence are discussed

Pseudo-High-Order Symplectic Integrators

Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly with order in timestep, rendering higher-order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per timestep. This is achieved by only considering error terms of order epsilon, and neglecting those of order epsilon^2, epsilon^3 etc. Here we devise symplectic algorithms with 4 and 6 substeps per step which effectively behave as 4th and 6th-order integrators when epsilon is small. These algorithms are more efficient than the usual 2nd and 4th-order methods when applied to planetary systems.Comment: 14 pages, 5 figures. Accepted for publication in the Astronomical Journa

Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.Comment: Submitted to Phys. Rev. E, 29 Page

The role of chaotic resonances in the solar system

Our understanding of the Solar System has been revolutionized over the past decade by the finding that the orbits of the planets are inherently chaotic. In extreme cases, chaotic motions can change the relative positions of the planets around stars, and even eject a planet from a system. Moreover, the spin axis of a planet-Earth's spin axis regulates our seasons-may evolve chaotically, with adverse effects on the climates of otherwise biologically interesting planets. Some of the recently discovered extrasolar planetary systems contain multiple planets, and it is likely that some of these are chaotic as well.Comment: 28 pages, 9 figure

Life Stage Simulation Analysis: Estimating Vital-Rate Effects on Population Growth for Conservation

We developed a simulation method, known as life-stage simulation analysis (LSA) to measure potential effects of uncertainty and variation in vital rates on population growth (lambda) for purposes of species conservation planning. Under LSA, we specify plausible or hypothesized levels of uncertainty, variation, and covariation in vital rates fur a given population. We use these data under resampling simulations to establish random combinations of vital rates for a large number of matrix replicates and finally summarize results from the matrix replicates to estimate potential effects of each vital rate on lambda in a probability-based context. Estimates of potential effects are based on a variety of summary statistics, such as frequency of replicates having the same vital rate of highest elasticity, difference in elasticity values calculated under simulated conditions vs, elasticities calculated using mean invariant vital rates, percentage of replicates having positive population growth, and variation in lambda explained by variation in each vital rate. To illustrate, we applied LSA to viral rates for two vertebrates: desert tortoise (Gopherus agassizii) and Greater prairie Chicken (Tympanuchus cupido). Results fur the prairie chicken indicated that a single vital rate consistently had greatest effect on population growth. Results for desert tortoise, however, suggested that a variety of life stages could have strong effects on population growth. Additional simulations for the Greater Prairie Chicken under a hypothetical conservation plan also demonstrated that a variety of vital rates could be manipulated to achieve desired population growth. To improve the reliability of inference, we recommend that potential effects of vital rates on lambda be evaluated using a probability-based approach like LSA. LSA is an important complement to other methods that evaluate vital-rate effects on lambda, including classical elasticity analysis, retrospective methods of variance decomposition, and simulation of the effects of environmental stochasticity

Higher Order Force Gradient Symplectic Algorithms

We show that a recently discovered fourth order symplectic algorithm, which requires one evaluation of force gradient in addition to three evaluations of the force, when iterated to higher order, yielded algorithms that are far superior to similarly iterated higher order algorithms based on the standard Forest-Ruth algorithm. We gauge the accuracy of each algorithm by comparing the step-size independent error functions associated with energy conservation and the rotation of the Laplace-Runge-Lenz vector when solving a highly eccentric Kepler problem. For orders 6, 8, 10 and 12, the new algorithms are approximately a factor of $10^3$, $10^4$, $10^4$ and $10^5$ better.Comment: 23 pages, 10 figure