Academic literature on the topic 'Variables action-Angles'

AbstractWe show how to capture the behaviour of the phase-space distribution function (DF) of a Galactic disc stellar population at a resonance. This is done by averaging the Hamiltonian over fast angle variables and re-expressing the DF in terms of a new set of canonical actions and angles variables valid in the resonant region. We then assign to the resonant DF the time average along the orbits of the axisymmetric DF expressed in the new set of actions and angles. This boils down to phase-mixing the DF in terms of the new angles, such that the DF for trapped orbits only depends on the new set of actions. This opens the way to quantitatively fitting the effects of the bar and spirals to Gaia data in terms of distribution functions in action space.

The chaotic motion of a ray path in a deep water acoustic waveguide with internal-wave-induced fluctuations of the sound speed is investigated. A statistical approach for the description of chaotic rays is discussed. The behavior of ray trajectories is studied using Hamiltonian formalism expressed in terms of action-angle variables. It is shown that the range dependence of the action variable of chaotic ray can be approximated by a random Wiener process. On the basis of this result, analytical expressions for probability density functions of ray parameters are derived. Distributions of coordinates, momenta (grazing angles), and actions of sound rays are evaluated. Numerical simulation shows that statistical characteristics of ray parameters weakly depend on a particular realization of random perturbation giving rise to ray chaos.

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

AbstractThe present-day response of a Galactic disc stellar population to a non-axisymmetric perturbation of the potential, in the form of a bar or spiral arms, can be treated, away from the main resonances, through perturbation theory within the action-angle coordinates of the unperturbed axisymmetric system. The first order moments of such a perturbed distribution function (DF) in the presence of spiral arms give rise to non-zero radial and vertical mean stellar velocities, called breathing modes. Such an Eulerian linearized treatment however diverges at resonances. The Lagrangian approach to the impact of non-axisymmetries at resonances avoids this problem. It is based on the construction of new orbital tori in the resonant trapping region, which come complete with a new system of angle-action variables. These new tori can be populated by phase-averaging the unperturbed DF over the new tori. This boils down to phase-mixing the DF in terms of the new angles, such that the DF for trapped orbits only depends on the new set of actions. This opens the way to quantitatively fitting the effects of the bar and spirals to Gaia data with an action-based DF.

The Hannay angle of classical mechanics is generalized so that it is invariant under gauge transformations, which are a restricted class of canonical transformations. A distinction between the Hamiltonian and the energy is essential to make in time-dependent problems. A time-dependent generalized harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. The energy, the time derivative of which is the power, is obtained from the equation of motion and related to the action variable. Hamilton’s equations give the time rate of change of the angle and action variables. The generalized Hannay angle is shown to be zero, and remains invariant under gauge transformations. On the other hand, if the original Hamiltonian is chosen as the energy, a nonzero generalized Hannay angle is obtained, but the power is given incorrectly. Nevertheless in the adiabatic limit, the total angle, which is the sum of the dynamical and Hannay angles, is equal to the one calculated from the correct energy.

We identified biomechanical variables indicative of lower extremity dysfunction, distinct from age-related gait adaptations, and examined interrelationships among these variables to better understand the neuromuscular adaptations in gait. Sagittal plane ankle, knee, and hip peak angles, moments, and powers and spatiotemporal parameters were acquired during preferred-speed gait in 120 subjects: 45 healthy young, 37 healthy elders, and 38 elders with functional limitations due to lower extremity musculoskeletal pathology, primarily arthritis. Multiple analysis of covariance with discriminate analysis, adjusted for gait speed, was used to identify the variables discriminating groups. Correlation analysis was used to explore interrelationships among these variables within each group. Healthy elders were discriminated (sensitivity 76%, specificity 82%) from young adults via decreased late-stance ankle plantar flexion angle, increased late-stance knee power absorption, and early-stance hip extensor power generation. Disabled elders were discriminated (sensitivity 74%, specificity 73%) from healthy elders via decreased late-stance ankle plantar flexor moment and power generation, increased early-stance ankle dorsiflexor moment, and late-stance hip flexor moment and power absorption. Relationships among variables showed a higher degree of coupling for the disabled elders compared with the healthy groups, suggesting a reduced ability to alter motor strategies. Our data suggest that, beyond age-related changes, elders with lower extremity dysfunction rely excessively on passive action of hip flexors to provide propulsion in late stance and contralateral ankle dorsiflexors to enhance stability. These findings support a growing body of evidence that gait changes with age and disablement have a neuromuscular basis, which may be informative in a motor control framework for physical therapy interventions.

It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ⩽ n – 1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.

Adequate mathematical models and computational algorithms are developed in this study to investigate specific features of the deformation processes of elastic rotational shells at large displacements and arbitrary rotation angles of the normal line. A finite difference method (FDM) is used to discretize the original continuum problem in spatial variables, replacing the differential operators with a second-order finite difference approximation. The computational algorithm for solving the nonlinear boundary value problem is based on a quasi-dynamic form of the ascertainment method with the construction of an explicit two-layer time-difference scheme of second-order accuracy. The influence of physical and mechanical characteristics of isotropic and composite materials on the deformation features of elastic spherical shells under the action of surface loading of “tracking” type is investigated. The results of the studies conducted have shown that the physical and mechanical characteristics of isotropic and composite materials significantly affect the nature of the deformation of the clamped spherical shell in both the subcritical and post-critical domains. The developed mathematical models and computational algorithms can be applied in the future to study shells of rotation made of hyperelastic (non-linearly elastic) materials and soft shells.

To solve the maneuvering decision problem in air combat of unmanned combat aircraft vehicles (UCAVs), in this paper, an autonomous maneuver decision method is proposed for a UCAV based on deep reinforcement learning. Firstly, the UCAV flight maneuver model and maneuver library of both opposing sides are established. Then, considering the different state transition effects of various actions when the pitch angles of the UCAVs are different, the 10 state variables including the pitch angle, are taken as the state space. Combined with the air combat situation threat assessment index model, a two-layer reward mechanism combining internal reward and sparse reward is designed as the evaluation basis of reinforcement learning. Then, the neural network model of the full connection layer is built according to an Asynchronous Advantage Actor–Critic (A3C) algorithm. In the way of multi-threading, our UCAV keeps interactively learning with the environment to train the model and gradually learns the optimal air combat maneuver countermeasure strategy, and guides our UCAV to conduct action selection. The algorithm reduces the correlation between samples through multi-threading asynchronous learning. Finally, the effectiveness and feasibility of the method are verified in three different air combat scenarios.

To study the movement characteristics and separation mechanism of safflower petals and their impurities under the action of airflow and lower the impurity rate in the cleaning operation process, integration of computational fluid dynamics (CFD) and discrete element method (DEM) codes was performed to study the motion and sorting behavior of impurity particles and safflower petals under different airflow inclination angles, dust removal angles and inlet airflow velocities by establishing a true particle model. In this model, the discrete particle phase was applied by the DEM software, and the continuum gas phase was described by the ANSYS Fluent software. The Box-Behnken experimental design with three factors and three levels was performed, and parameters such as inlet airflow velocity, airflow inclined angle, and dust remover angle were selected as independent variables that would influence the cleaning impurity rate and the cleaning loss rate. A mathematical model was established, and then the effects of various parameters and their interactions were analyzed. The test results show that the cleaning effect is best when the inlet airflow velocity is 7 m/s, the airflow inclined angle is 0°, and the dust remover angle is 25°. Confirmatory tests showed that the average cleaning impurity rate and cleaning loss rate were 0.69% and 2.75%, respectively, which dropped significantly compared with those from previous optimization. An experimental device was designed and set up; the experimental results were consistent with the simulation results, indicating that studying the physical behavior of safflower petals-impurity separation in the airflow field by using the DEM-CFD coupling method is reliable. This result provides a basis for follow-up studies of separation and cleaning devices for lightweight materials such as safflower petals.

Les mouvements humains sont toujours complexes. Même une tâche simple comme prendre un verre d'eau implique de nombreux degrés de liberté i.e., différents groupes de muscles, plusieurs articulations et un nombre infini de trajectoires possible pour le bras. Néanmoins, les mouvements sont facilement disponibles aux sujets sains et semble être naturellement optimisé par le système nerveux central. Cela est souvent modélisé par la minimisation d'un paramètre donné du système, tel que l'énergie ou l'à-coup, qui semblent être des candidats naturels. Malheureusement, ces approches sont souvent limitées dans leur portée et ne peuvent pas décrire les mouvements périodiques dans des environnements changeant dans le temps. Dans de tels systèmes, les invariants adiabatiques sont des observables pertinentes issues de la mécanique hamiltonienne. L'objectif de cette thèse de doctorat est d'étudier le rôle et l'utilisation des invariants adiabatiques dans le contrôle moteur humain. Pour ce faire, nous avons réalisé une série d'expériences. Tout d'abord, nous les avons étudiées en tant que contrainte pour la stabilité globale de la marche, même lorsqu'elle est exposée à une tâche altérant la variabilité, telle que le maintien d'un rythme dicté par un métronome. Ensuite, nous avons utilisé des résultats récents en physique pour évaluer la variabilité inhérente à la marche à longue distance en tant que phénomène de diffusion de la distribution des invariants adiabatiques. Enfin, nous les avons explorés dans des environnements temporels changeants, notamment en modifiant la "gravité" à la fois dans une centrifugeuse et dans un contexte de vol parabolique, où ils semblent être des quantités pertinentes pour montrer les changements dans les stratégies motrices. Les différents résultats de cette thèse indiquent que les invariants adiabatiques révèlent des contraintes génériques cachées affectant les mouvements humains périodiques
Human motion is inherently complex. Even an ordinary task like lifting a glass of water involves many degrees of freedom i.e., different muscle groups, multiple joints and an infinite number of trajectories for the arm. Nevertheless, motion is readily available to healthy subjects, and seems to be naturally optimized by the central nervous system. This is often modelized as the minimization of a given parameter of the system e.g., energy or jerk, which appear as natural candidates. Unfortunately, these approaches are often limited in their scopes, and cannot describe periodic motion in time-changing environments. In such systems, adiabatic invariants are relevant observables originating from Hamiltonian mechanics. The aim of this doctoral dissertation is to investigate the role and use of adiabatic invariants in human motor control. This was done in a series of experiments. First, we studied them as a constraint for the global stability of gait, even when exposed to a variability-altering task, such as metronome keeping. Then, we used recent results in physics to assess the inherent variability of long-range walking as a diffusion phenomenon of the distribution of adiabatic invariants. Finally, we explored them in time-changing environments, specifically by altering “gravity” both in a centrifuge and a parabolic flight context, where they seem to be relevant quantities to show changes in motor strategies. The different findings in this dissertation point to adiabatic invariants revealing generic hidden constraints affecting periodic human motion

Safe driving behaviour during lane change is the function of selecting and processing task relevant cues from the ongoing driving environment; enabling the goal-directed preparatory process. Such preparatory information typically facilitates reaction time to the anticipated event. However, it is unclear how the additional information other than task specific cues from the driving environment act on preparatory processes while driving a car. We implemented a pre-cue paradigm in a simulated lane change task (LCT) to answer this question. In contrast to the standard paradigm, additional information was presented either just before the preparatory stimulus (pre-cue) or the target stimulus (pre-target) that was either congruent, incongruent, or neutral to the lane change direction. Reaction time and amplitudes of steering out and in angles (A1 and A2) were measured as dependent variables. Results showed that reaction time and steering in amplitude A2 were increased when the additional information was presented before the final target for intended action and similarly when the additional information presented in the same lane changed direction (congruent). Later one accounts for contingent attentional capture. To accommodate the entire pattern of results observed in the study, we tentatively suggest that any information which is not relevant for the intended action have considerable influence on attention and action preparation on the basis of the temporal and visuo-spatial positioning. A strong effect is found especially at the time of the final determination of the upcoming driving manoeuvre.

An objective of this study is to simulate the backward walking motion of a full-body digital human model. The model consists of 55 degree of freedom – 6 degrees of freedom for global translation and rotation and 49 degrees of freedom representing the kinematics of the entire body. The resultant action of all the muscles at a joint is represented by the torque for each degree of freedom. The torques and angles at a joint are treated as unknowns in the optimization problem. The B-spline interpolation is used to represent the time histories of the joint angles and the well-established robotics formulation of the Denavit-Hartenberg method is used for kinematics analysis of the mechanical system. The recursive Lagrangian formulation is used to develop the equations of motion, and was chosen because of its known computational efficiency. The backwards walking problem is formulated as a nonlinear optimization problem. The control points of the B-splines for the joint angle profiles are treated as the design variables. For the performance measure, total dynamic effort that is represented as the integral of the sum of the squares of all the joint torques is minimized using a sequential quadratic programming algorithm. The solution is simulated in the Santos™ environment. Results of the optimization problem are the torque and joint angle profiles. The torques at the key joints and the ground reaction forces are compared to those for the forward walk in order to study the differences between the two walking patterns. Simulation results are approximately validated with the experimental data which is motion captured in the VSR Lab at the University of Iowa.