Physics and Geometry

Introduction to Modern Dynamics ◽

10.1093/oso/9780198844624.003.0001 ◽

2019 ◽

pp. 3-52

Author(s):

David D. Nolte

Keyword(s):

Path Length ◽

Reference Frames ◽

Metric Spaces ◽

Complex Dynamics ◽

Geometric Approach ◽

Body Motion ◽

Visualization Tool ◽

Coordinate Transformations ◽

Flow Equations ◽

Rotating Frames

This chapter emphasizes the importance of a geometric approach to dynamics. The central objects of interest are trajectories of a dynamical system through multidimensional spaces composed of generalized coordinates. Trajectories through configuration space are parameterized by the path length element, which becomes an important feature in later chapters on relativity and metric spaces. Trajectories through state space are defined by mathematical flow equations whose flow fields and flow lines become the chief visualization tool for complex dynamics. Coordinate transformations and Jacobian matrices are used throughout this text, and the transformation to noninertial frames introduces fictitious forces like the Coriolis force that are experienced by observers in noninertial frames. Uniformly rotating frames provide the noninertial reference frames for the description of rigid-body motion.

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Measuring Electromagnetic Fields in Rotating Frames of Reference

Universe ◽

10.3390/universe6020031 ◽

2020 ◽

Vol 6 (2) ◽

pp. 31

Author(s):

Clive Speake ◽

Antonello Ortolan

Keyword(s):

Electromagnetic Fields ◽

Reference Frames ◽

Frame Of Reference ◽

Rotating Frame ◽

Coordinate Transformations ◽

Rotating System ◽

Spherical Shells ◽

Physics Community ◽

Rotating Frames ◽

Rotating Reference Frames

We review the problem of transforming electromagnetic fields between inertial and rotating reference frames. We compare the method of straightforward tensor coordinate transformations adopted by Schiff in his well-known paper of 1939 with the method of Orthogonal Tetrads (OT) that was applied to this problem in 1964 by Irvine. Although both methods are mathematically rigorous, the transformed fields have different forms depending on the method adopted. We emphasize that the OT method is expected to predict the fields that would actually be measured by an observer in a rotating frame of reference. We briefly discuss existing experimental evidence that supports the OT approach, but point out that there appears to be little awareness in the physics community of this problem or its resolution. We use both methods to transform the electrostatic and magnetic fields generated by rotating charged spherical shells from an inertial into a co-rotating system. We also briefly describe how such an arrangement of shells could be used to measure rotation relative to the fixed stars.

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A Fully Coupled Flow and 6-DOF Motion Solver in Multiple Reference Frames

Volume 1: Symposia ◽

10.1115/ajkfluids2015-3210 ◽

2015 ◽

Author(s):

Hua Shan ◽

Sung-Eun Kim ◽

Bong Rhee

Keyword(s):

Reference Frames ◽

Inertial Frame ◽

Equations Of Motion ◽

Body Motion ◽

Governing Equations ◽

Flow Equations ◽

Fixed Frame ◽

Actuator Disk ◽

Multiple Reference Frames ◽

Multiple Reference

In many computational fluid dynamics (CFD) applications involving a single rotating part, such as the flow through an open water propeller rotating at a constant rpm, it is convenient to formulate the governing equations in a non-inertial rotating frame. For flow problems consisting of both stationary and rotating parts, e.g. the stator and the rotor of a turbine, or the hull and propeller of a ship, the multiple reference frames (MRF) approach has been widely used. In most existing MRF models, the computation domain is divided into stationary and rotating zones. In the stationary zone, the flow equations are formulated in the inertial frame, while in the rotating zone, the equations are solved in the non-inertial rotating frame. Also, the flow is assumed to be steady in both zones and the flow solution in the rotating zone can be interpreted as the phase-locked time average result. Compared with other approaches, such as the actuator disk (body-force) model, the MRF approach is superior because it accounts for the actual geometry of the rotating part, e.g. propeller blades. A more complicated situation occurs when the flow solver is coupled to the six degrees of freedom (6-DOF) equations of rigid-body motion in predicting the maneuver of a self-propelled surface or underwater vehicle. In many applications, the propeller is replaced by the actuator disk model. The current work attempts to extend the MRF approach to the 6-DOF maneuvering problems. The governing equations for unsteady incompressible flow in a non-inertial frame have been extended to the flow equations in multiple reference frames: a hull-fixed frame that undergoes translation and rotation predicted by the 6-DOF equations of motion and a propeller-fixed frame in relative rotation with respect to the hull. Because of the large disparity between time scales in the 6-DOF rigid body motion of the hull and the relative rotational motion of the propeller, the phase-locked solution in the propeller MRF zone is considered a reasonable approximation for the actual flow around the propeller. The flow equations are coupled to the 6-DOF equations of motion using an iterative coupling algorithm. The coupled solver has been developed as part of NavyFOAM. The theoretical framework and the numerical implementation of the coupled solver are outlined in this paper. Some numerical test results are also presented.

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A geometric approach to establishment of datum reference frames

Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C) ◽

10.1109/robot.1999.772562 ◽

2003 ◽

Cited By ~ 1

Author(s):

J.B. Gou ◽

Y.X. Chu ◽

Z.X. Li

Keyword(s):

Reference Frames ◽

Geometric Approach

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Parallel updating and weighting of multiple spatial maps for visual stability during whole body motion

Journal of Neurophysiology ◽

10.1152/jn.00576.2015 ◽

2015 ◽

Vol 114 (6) ◽

pp. 3211-3219 ◽

Cited By ~ 10

Author(s):

J. J. Tramper ◽

W. P. Medendorp

Keyword(s):

Reference Frame ◽

Reference Frames ◽

Spatial Information ◽

Target Location ◽

Visual Space ◽

Body Motion ◽

Whole Body ◽

Spatial Representations ◽

Integration Model ◽

The Brain

It is known that the brain uses multiple reference frames to code spatial information, including eye-centered and body-centered frames. When we move our body in space, these internal representations are no longer in register with external space, unless they are actively updated. Whether the brain updates multiple spatial representations in parallel, or whether it restricts its updating mechanisms to a single reference frame from which other representations are constructed, remains an open question. We developed an optimal integration model to simulate the updating of visual space across body motion in multiple or single reference frames. To test this model, we designed an experiment in which participants had to remember the location of a briefly presented target while being translated sideways. The behavioral responses were in agreement with a model that uses a combination of eye- and body-centered representations, weighted according to the reliability in which the target location is stored and updated in each reference frame. Our findings suggest that the brain simultaneously updates multiple spatial representations across body motion. Because both representations are kept in sync, they can be optimally combined to provide a more precise estimate of visual locations in space than based on single-frame updating mechanisms.

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Flexible Multibody Dynamics Based on a Fully Cartesian System of Support Coordinates

Journal of Mechanical Design ◽

10.1115/1.2919191 ◽

1993 ◽

Vol 115 (2) ◽

pp. 294-299 ◽

Cited By ~ 11

Author(s):

N. Vukasovic ◽

J. T. Celigu¨eta ◽

J. Garci´a de Jalo´n ◽

E. Bayo

Keyword(s):

Multibody Dynamics ◽

Reference Frames ◽

Deformable Body ◽

Equations Of Motion ◽

Flexible Multibody Dynamics ◽

Body Motion ◽

Cartesian System ◽

Flexible Multibody ◽

Local Reference ◽

Rigid Multibody

In this paper we present an extension to flexible multibody systems of a system of fully cartesian coordinates previously used in rigid multibody dynamics. This method is fully compatible with the previous one, keeping most of its advantages in kinematics and dynamics. The deformation in each deformable body is expressed as a linear combination of Ritz vectors with respect to a local frame whose motion is defined by a series of points and vectors that move according to the rigid body motion. Joint constraint equations are formulated through the points and vectors that define each link. These are chosen so that a minimum use of local reference frames is done. The resulting equations of motion are integrated using the trapezoidal rule combined with fixed point iteration. An illustrative example that corresponds to a satellite deployment is presented.

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NONLINEAR AND COMPLEX DYNAMICS IN ECONOMICS

Macroeconomic Dynamics ◽

10.1017/s1365100514000091 ◽

2014 ◽

Vol 19 (8) ◽

pp. 1749-1779 ◽

Cited By ~ 16

Author(s):

William A. Barnett ◽

Apostolos Serletis ◽

Demitre Serletis

Keyword(s):

Dynamical Systems ◽

Complex Dynamics ◽

State Of The Art ◽

Geometric Approach ◽

Stable Region ◽

Physical Sciences ◽

Dynamical Complexity ◽

Policy Relevance ◽

Potential Applications ◽

Transcritical Bifurcations

This paper is an up-to-date survey of the state of the art in dynamical systems theory relevant to high levels of dynamical complexity, characterizing chaos and near-chaos, as commonly found in the physical sciences. The paper also surveys applications in economics and finance. This survey does not include bifurcation analyses at lower levels of dynamical complexity, such as Hopf and transcritical bifurcations, which arise closer to the stable region of the parameter space. We discuss the geometric approach (based on the theory of differential/difference equations) to dynamical systems and make the basic notions of complexity, chaos, and other related concepts precise, having in mind their (actual or potential) applications to economically motivated questions. We also introduce specific applications in microeconomics, macroeconomics, and finance and discuss the policy relevance of chaos.

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Covariant hydrodynamics of Hamiltonian systems

Dal'nevostochnyi Matematicheskii Zhurnal ◽

10.47910/femj202114 ◽

2021 ◽

Vol 21 (2) ◽

pp. 166-179

Author(s):

A.I. Gudimenko ◽

Keyword(s):

Hamiltonian Systems ◽

Reference Frames ◽

Mechanical Systems ◽

Velocity Fields ◽

Time Dependent ◽

Hamiltonian Mechanics ◽

Coordinate Transformations ◽

Time Axis ◽

Hydrodynamic Reduction

The theory of hydrodynamic reduction of non-autonomous Hamiltonian mechanics (V. Kozlov, 1983) is presented in the geometric formalism of bundles over the time axis R. In this formalism, time is one of the coordinates, not a parameter; the connections describe reference frames and velocity fields of mechanical systems. The equations of the theory are presented in a form that is invariant with respect to time-dependent coordinate transformations and the choice of reference frames.

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Analysis of Pointing Errors Reveals Properties of Data Representations and Coordinate Transformations Within the Central Nervous System

Neural Computation ◽

10.1162/089976600300014746 ◽

2000 ◽

Vol 12 (12) ◽

pp. 2823-2855 ◽

Cited By ~ 36

Author(s):

J. McIntyre ◽

F. Stratta ◽

J. Droulez ◽

F. Lacquaniti

Keyword(s):

Central Nervous System ◽

Nervous System ◽

Reference Frames ◽

Target Location ◽

Motor Command ◽

Coordinate Transformations ◽

Internal Reference ◽

Internal Representations ◽

Pointing Task ◽

The Central Nervous System

The execution of a simple pointing task invokes a chain of processing that includes visual acquisition of the target, coordination of multimodal proprioceptive signals, and ultimately the generation of a motor command that will drive the finger to the desired target location. These processes in the sensorimotor chain can be described in terms of internal representations of the target or limb positions and coordinate transformations between different internal reference frames. In this article we first describe how different types of error analysis can be used to identify properties of the internal representations and coordinate transformations within the central nervous system. We then describe a series of experiments in which subjects pointed to remembered 3D visual targets under two lighting conditions (dim light and total darkness) and after two different memory delays (0.5 and 5.0 s) and report results in terms of variable error, constant error, and local distortion. Finally, we present a set of simulations to help explain the patterns of errors produced in this pointing task. These analyses and experiments provide insight into the structure of the underlying sensorimotor processes employed by the central nervous system.

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