scholarly journals Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Linyu Peng ◽  
Huafei Sun ◽  
Xiao Sun
Keyword(s):  
Hamiltonian System ◽  
Geometric Structure ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Hamiltonian Dynamics ◽  
Jacobi Field ◽  
Linear Hamiltonian System ◽  
Conformal Metric ◽  
Jacobi Equations ◽  
Special Case

We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, anNdegrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

DIFFERENT TYPES OF ATTRACTORS IN THE THREE BODY PROBLEM PERTURBED BY DISSIPATIVE TERMS

2011 ◽  
Vol 21 (08) ◽  
pp. 2195-2209 ◽  
Author(s):  
JOHN D. HADJIDEMETRIOU ◽  
GEORGE VOYATZIS
Keyword(s):  
Hamiltonian System ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Large Mass ◽  
Three Body Problem ◽  
Resonant Orbits ◽  
Different Types ◽  
Families Of Periodic Orbits ◽  
Three Body ◽  
Special Case

We study the evolution of a conservative dynamical system with three degrees of freedom, where small nonconservative terms are added. The conservative part is a Hamiltonian system, describing the motion of a planetary system consisting of a star, with a large mass, and of two planets, with small but not negligible masses, that interact gravitationally. This is a special case of the three body problem, which is nonintegrable. We show that the evolution of the system follows the topology of the conservative part. This topology is critically determined by the families of periodic orbits and their stability. The evolution of the complete system follows the families of the conservative part and is finally trapped in the resonant orbits of the Hamiltonian system, in different types of attractors: chaotic attractors, limit cycles or fixed points.

PERIODIC AND HOMOCLINIC ORBITS IN A TWO-PARAMETER UNFOLDING OF A HAMILTONIAN SYSTEM WITH A HOMOCLINIC ORBIT TO A SADDLE-CENTER

1995 ◽  
Vol 05 (02) ◽  
pp. 397-408 ◽  
Author(s):  
O. Yu. KOLTSOVA ◽  
L.M. LERMAN
Keyword(s):  
Hamiltonian System ◽  
Homoclinic Orbit ◽  
Degrees Of Freedom ◽  
Original System ◽  
Homoclinic Orbits ◽  
Linear Hamiltonian System ◽  
Twist Map ◽  
Invariant Formulation ◽  
Point Of Intersection ◽  
Two Parameter

For a two-degrees-of-freedom Hamiltonian system with a homoclinic orbit (loop) to a saddle-center we prove that the Poincaré map on a section to the loop within the Hamiltonian level containing a saddle-center is a twist map with discontinuity at the point of intersection with the loop. It explains the reason for existence of a countable set of periodic orbits near the loop. The types of these orbits are determined. We discover homoclinic doubling and tripling in a generic two-parameter Hamiltonian unfolding of such a Hamiltonian system. Besides this, we also study the nonautonomous linear Hamiltonian system obtained by means of the linearization of the original system at the loop. We derive the invariant formulation of the genericity condition that is needed for studying the original system.

Coupled systems of two-dimensional turbulence

2013 ◽  
Vol 732 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Hamiltonian System ◽  
Energy Conservation ◽  
Strong Correlation ◽  
Conservation Law ◽  
Hamiltonian Dynamics ◽  
Coupled Systems ◽  
The Other ◽  
Two Dimensional ◽  
Coherent Vortices ◽  
Fluid Particles

AbstractOrdinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

scholarly journals Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 518 ◽  
Author(s):  
Alessandro Sergi ◽  
Gabriel Hanna ◽  
Roberto Grimaudo ◽  
Antonino Messina
Keyword(s):  
Constant Temperature ◽  
Degrees Of Freedom ◽  
Open Quantum Systems ◽  
Hamiltonian Dynamics ◽  
Quantum Systems ◽  
Classical Dynamics ◽  
Open Quantum ◽  
Time Translation ◽  
Translation Symmetry ◽  
Lie Brackets

Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein.

scholarly journals Use of Jacobians for inverse kinematics of articulated robots : a study on approximate solutions

2020 ◽  
Author(s):  
◽  
Uriel Jacket Tresor Demby's
Keyword(s):  
Numerical Solution ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Fuzzy Inference ◽  
Numerical Approach ◽  
Data Driven ◽  
Approximate Methods ◽  
Inference System ◽  
First Case

In the context of articulated robotic manipulators, the Forward Kinematics (FK) is a highly non-linear function that maps joint configurations of the robot to poses of its endeffector. Furthermore, while in the most useful cases these functions are neither injective (one-to-one) nor surjective (onto), depending on the robot configuration -- i.e. the sequence of prismatic versus revolute joints, and the number of Degrees of Freedom (DoF) -- the associated Inverse Kinematics (IK) problem may be practically or even theoretically impossible to be solved analytically. Therefore, in the past decades, several approximate methods have been developed for many instances of IK problems. The approximate methods can be divided into two distinct categories: data-driven and numerical approaches. In the first case, data-driven approaches have been successfully used for small workspace domains (e.g., task-driven applications), but not fully explored for large ones, i.e. in task-independent applications where a more general IK is required. Similarly, and despite many successful implementations over the years, numerical solutions may fail if an improper matrix inverse is employed (e.g., Moore-Penrose generalized inverse). In this research, we propose a systematic, robust and accurate numerical solution for the IK problem using the Unit-Consistent (UC) and the Mixed (MX) Inverse methods to invert the Jacobians derived from the Denavit-Hartenberg (D-H) representation of the FK for any robot. As we demonstrate, this approach is robust to whether the system is underdetermined (less than 6 DoF) or overdetermined (more than 6 DoF). We compare the proposed numerical solution to data driven solutions using different robots -- with DoF varying from 3 to 7. We conclude that numerical solutions are easier to implement, faster, and more accurate than most data-driven approaches in the literature, specially for large workspaces as in task-independent applications. We particularly compared the proposed numerical approach against two data-driven approaches: Multi-Layer Perceptron (MLP) and Adaptive Neuro-Fuzzy Inference System (ANFIS), while exploring various architectures of these Neural Networks (NN): i.e. number of inputs, number of outputs, depth, and number of nodes in the hidden layers.

scholarly journals A novel numerical approach for fracture analysis in orthotropic media

2017 ◽  
Vol 20 (K2) ◽  
pp. 5-13
Author(s):  
Minh Ngoc Nguyen ◽  
Nha Thanh Nguyen ◽  
Tinh Quoc Bui ◽  
Thien Tich Truong
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Numerical Approach ◽  
Fracture Analysis ◽  
Orthotropic Materials ◽  
Displacement Fields ◽  
Novel Approach ◽  
Interpolation Procedure ◽  
Orthotropic Media ◽  
Standard Finite Element Method

This paper presents a novel approach for fracture analysis in two-dimensional orthotropic domain. The proposed method is based on consecutive-interpolation procedure (CIP) and enrichment functions. The CIP were recently introduced as an improvement for standard Finite Element method, such that higher-accurate and higher-continuous solution can be obtained without smoothing operation and without increasing the number of degrees of freedom. To avoid re-meshing, the enrichment functions are employed to mathematically describe the jump in displacement fields and the singularity of stress near crack tip. The accuracy of the method for analysis of cracked body made of orthotropic materials is studied. For that purpose, various examples with different geometries and boundary conditions are considered. The results of stress intensity factors, a key quantity in fracture analysis, are validated by comparing with analytical solutions and numerical solutions available in literatures.

Finite Element Analysis of 2-D Structures by New Strain Based Triangular Element

2018 ◽  
Vol 35 (3) ◽  
pp. 305-313 ◽  
Author(s):  
C. Rebiai
Keyword(s):  
Dynamic Analysis ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Time Integration ◽  
Triangular Element ◽  
Membrane Element ◽  
Lumped Mass ◽  
Element Analysis ◽  
Benchmark Tests ◽  
Three Degrees Of Freedom

ABSTRACTIn this investigation, a new simple triangular strain based membrane element with drilling rotation for 2-D structures analysis is proposed. This new numerical model can be used for linear and dynamic analysis. The triangular element is named SBTE and it has three nodes with three degrees of freedom at each node. The displacements field of this element is based on the assumed functions for the various strains satisfying the compatibility equations. This developed element passed both patch and benchmark tests in the case of bending and shear problems. For the dynamic analysis, lumped mass with implicit/explicit time integration are employed. The obtained numerical results using the developed element converge toward the analytical and numerical solutions in both analyses.

An Improved Method for Calculating Free Vibrations in Systems of Several Degrees of Freedom

1938 ◽  
Vol 5 (2) ◽  
pp. A61-A66
Author(s):  
Winston M. Dudley
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Matrix Theory ◽  
Free Vibrations ◽  
Initial Disturbance ◽  
Improved Method ◽  
Matrix Methods ◽  
Computing Machine ◽  
The Matrix ◽  
Time Required

Abstract In 1934 two English investigators (1) published a method for calculating the various modes and frequencies of vibration of a system having several degrees of freedom. Their method, which is based on matrices, greatly shortens the time spent in obtaining numerical solutions in many important problems, notably those with immovable foundations. In this paper is presented a new theorem which (a) makes possible a further reduction of nearly one half in the time required, so that solutions up to 20 deg or more of freedom are now practical and (b) makes it then possible to determine the motion of the system after any initial disturbance in a few minutes, instead of several hours as required by older methods. It is useful in the latter respect whether the modes have been determined by matrix methods, or not. Although the paper gives simpler proofs than any previously published, knowledge of the matrix theory is not required in using the method. Problems are analyzed by a tabular process, in which an ordinary computing machine helps greatly. Comments based on computing experience are given. A simple numerical example has been given elsewhere (1).

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