Differential Invariants of the (2+1)-Dimensional Breaking Soliton Equation

2016 ◽  
Vol 71 (9) ◽  
pp. 855-862
Author(s):  
Zhong Han ◽  
Yong Chen
Keyword(s):  
Recurrence Relations ◽  
Differential Invariant ◽  
Lie Symmetry ◽  
Differential Invariants ◽  
Moving Frame ◽  
Moving Frames ◽  
Soliton Equation ◽  
Invariant Algebra ◽  
Equivariant Moving Frames

AbstractWe construct the differential invariants of Lie symmetry pseudogroups of the (2+1)-dimensional breaking soliton equation and analyze the structure of the induced differential invariant algebra. Their syzygies and recurrence relations are classified. In addition, a moving frame and the invariantization of the breaking soliton equation are also presented. The algorithms are based on the method of equivariant moving frames.

scholarly journals Moving Frames for Lie Pseudo–Groups

2008 ◽  
Vol 60 (6) ◽  
pp. 1336-1386 ◽  
Author(s):  
Peter J. Olver ◽  
Juha Pohjanpelto
Keyword(s):  
Recurrence Relations ◽  
Differential Forms ◽  
Group Actions ◽  
Differential Invariants ◽  
Moving Frames ◽  
Constructive Theory ◽  
Invariant Differential

AbstractWe propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. Themoving frame provides an effectivemeans for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.

Two Algorithms for a Moving Frame Construction

2003 ◽  
Vol 55 (2) ◽  
pp. 266-291 ◽  
Author(s):  
Irina A. Kogan
Keyword(s):  
Differential Forms ◽  
Differential Invariants ◽  
Moving Frame ◽  
Entire Group ◽  
Practical Implementation ◽  
Moving Frames ◽  
Significant Development ◽  
Jet Bundles ◽  
Invariant Differential ◽  
Frame Construction

AbstractThe method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.

scholarly journals A Moving Frame Trajectory Tracking Method of a Flying-Wing UAV Using the Differential Geometry

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yi Zhu ◽  
Xin Chen ◽  
Chuntao Li
Keyword(s):  
Flight Control ◽  
Trajectory Tracking ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Elliptic Cylinder ◽  
Dynamic Equations ◽  
Moving Frame ◽  
Moving Frames ◽  
Tracking Method ◽  
Tracking Process

The problem of UAV trajectory tracking is a difficult issue for scholars and engineers, especially when the target curve is a complex curve in the three-dimensional space. In this paper, the coordinate frames during the tracking process are transformed to improve the tracking result. Firstly, the basic concepts of the moving frame are given. Secondly the transfer principles of various moving frames are formulated and the Bishop frame is selected as a final choice for its flexibility. Thirdly, the detailed dynamic equations of the moving frame tracking method are formulated. In simulation, a moving frame of an elliptic cylinder helix is formulated precisely. Then, the devised tracking method on the basis of the dynamic equations is tested in a complete flight control system with 6 DOF nonlinear equations of the UAV. The simulation result shows a satisfactory trajectory tracking performance so that the effectiveness and efficiency of the devised tracking method is proved.

scholarly journals Stabilizing Ship Motion With a Dual System Inertial Disk

2019 ◽  
Author(s):  
Torstein R. Storaas ◽  
Kasper Virkesdal ◽  
Gitle S. Brekke ◽  
Thorstein Rykkje ◽  
Thomas Impelluso
Keyword(s):  
Oil And Gas ◽  
New Technologies ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Moving Frames ◽  
New Approach ◽  
Lie Group Theory ◽  
Moving Frame Method ◽  
Frame Method ◽  
Modern Mathematics

Abstract Norwegian industries are constantly assessing new technologies and methods for more efficient and safer maintenance in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze ship stability moderated by a dual gyroscopic inertial device. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of two inertial disk devices, it accounts for the prescribed spin of the disks. It separates out the prescribed variables. This work displays the results in 3D on cell phones. It represents a prelude to testing in a wave tank.

scholarly journals Modeling a Knuckle-Boom Crane Control to Reduce Pendulum Motion Using the Moving Frame Method

2019 ◽  
Author(s):  
Maren Eriksen Eia ◽  
Elise Mari Vigre ◽  
Thorstein Ravneberg Rykkje
Keyword(s):  
Equations Of Motion ◽  
Moving Frame ◽  
Web Pages ◽  
Moving Frames ◽  
Pendulum Motion ◽  
Moving Frame Method ◽  
Frame Method ◽  
The Masses ◽  
And Control ◽  
Boom Crane

Abstract A Knuckle Boom Crane is a pedestal-mounted, slew-bearing crane with a joint in the middle of the distal arm; i.e. boom. This distal boom articulates at the ‘knuckle (i.e.: joint)’ and that allows it to fold back like a finger. This is an ideal configuration for a crane on a ship where storage space is a premium. This project researches the motion and control of a ship mounted knuckle boom crane to minimize the pendulum motion of a hanging load. To do this, the project leverages the Moving Frame Method (MFM). The MFM draws upon Lie group theory — SO(3) and SE(3) — and Cartan’s Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. The work reported here accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass on the ship. The equations of motion are solved numerically using a 4th order Runge Kutta (RK4), while solving for the rotation matrix for the ship using the Cayley-Hamilton theorem and Rodriguez’s formula for each timestep. This work displays the motion on 3D web pages, viewable on mobile devices.

Modeling Stablization of Crane and Ship by Gyroscopic Control Using the Moving Frame Method

2018 ◽  
Author(s):  
Josef Flatlandsmo ◽  
Torbjørn Smith ◽  
Ørjan O. Halvorsen ◽  
Johnny Vinje ◽  
Thomas J. Impelluso
Keyword(s):  
Oil And Gas ◽  
New Technologies ◽  
Equations Of Motion ◽  
Long Term Results ◽  
Moving Frame ◽  
Moving Frames ◽  
Learning Modules ◽  
Moving Frame Method ◽  
Frame Method

Norwegian industries are constantly assessing new technologies and methods for more efficient and safer production in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze the motion induced by a crane and controlled by a gyroscopic inertial device mounted on a ship. The crane is a simple two-link system that transfers produce and equipment to and from barges. An inertial flywheel — a gyroscope — is used to stabilize the barge during transfer. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of both the crane and the stabilizing inertial device. Furthermore, this work allows for buoyancy and motor induced torques. Furthermore, this work displays the results in 3D on cell phones. The long-term results of this work leads to a robust 3D active compensation method for loading/unloading operations offshore. Finally, the interactivity between the crane and the stabilizing gyro anticipates the impending time of artificial intelligence when machines, equipped with on-board CPU’s and IP addresses, are empowered with learning modules to conduct their operations.

Modelling Buoy Motion at Sea

2019 ◽  
Author(s):  
Thorstein R. Rykkje ◽  
Tord Tørressen ◽  
Håvard Løkkebø
Keyword(s):  
Equations Of Motion ◽  
Moving Frame ◽  
Web Pages ◽  
Moving Frames ◽  
Lie Group Theory ◽  
Moving Frame Method ◽  
Frame Method ◽  
And Mathematics ◽  
The Masses ◽  
Theoretical Results

Abstract This project creates a model to assess the motion induced on a buoy at sea, under wave conditions. We use the Moving Frame Method (MFM) to conduct the analysis. The MFM draws upon concepts and mathematics from Lie group theory — SO(3) and SE(3) — and Cartan’s notion of Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. This work accounts for the masses and geometry of all components and for buoyancy forces and added mass. The resulting movement will be displayed on 3D web pages using WebGL. Finally, the theoretical results will be compared with experimental data obtained from a previous project done in the wave tank at HVL.

Production and Analytics of a Multi-Linked Robotic System

2019 ◽  
Author(s):  
Thorstein R. Rykkje ◽  
Eystein Gulbrandsen ◽  
Andreas Fosså Hettervik ◽  
Morten Kvalvik ◽  
Daniel Gangstad ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Robotic System ◽  
Moving Frame ◽  
Moving Frames ◽  
New Approach ◽  
Lie Group Theory ◽  
Senior Design ◽  
Moving Frame Method ◽  
Frame Method

Abstract This paper extends research into flexible robotics through a collaborative, interdisciplinary senior design project. This paper deploys the Moving Frame Method (MFM) to analyze the motion of a relatively high multi-link system, driven by internal servo engines. The MFM describes the dynamics of the system and enables the construction of a general algorithm for the equations of motion. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. The result is a dynamic 3D analytical model for the motion of a snake-like robotic system, that can take the physical sizes of the system and return the dynamic behavior. Furthermore, this project builds a snake-like robot driven by internal servo engines. The multi-linked robot will have a servo in each joint, enabling a three-dimensional movement. Finally, a test is performed to compare if the theory and the measurable real-time results match.

scholarly journals Charmonia in moving frames

2018 ◽  
Vol 175 ◽  
pp. 14006 ◽  
Author(s):  
S. Prelovsek ◽  
G. Bali ◽  
S. Collins ◽  
D. Mohler ◽  
M. Padmanath ◽  
...  
Keyword(s):  
Irreducible Representation ◽  
Energy Region ◽  
Rotational Symmetry ◽  
Moving Frame ◽  
Moving Frames ◽  
Lattice Simulation ◽  
Preliminary Results ◽  
Additional Information ◽  
Scattering Matrices ◽  
Zero Momentum

Lattice simulation of charmonium resonances with non-zero momentum provides additional information on the two-meson scattering matrices. However, the reduced rotational symmetry in a moving frame renders a number of states with different JP in the same lattice irreducible representation. The identification of JP for these states is particularly important, since quarkonium spectra contain a number of states with different JP in a relatively narrow energy region. Preliminary results concerning spin-identification are presented in relation to our study of charmonium resonances in flight on the Nf = 2 + 1 CLS ensembles.

Differential invariants of ℝ*-structures

1996 ◽  
Vol 119 (2) ◽  
pp. 341-356 ◽  
Author(s):  
Antonio Valdés
Keyword(s):  
Differential Geometry ◽  
Differential Invariant ◽  
Equivalence Problem ◽  
Differential Invariants ◽  
Natural Action ◽  
Base Manifold ◽  
Analytic Case ◽  
Algebraic Invariants

A differential invariant of a G-structure is a function which depends on the r-jet of the G-structure and such that it is invariant under the natural action of the pseudogroup of diffeomorphisms of the base manifold. The importance of these objects is clear, since they seem to be the natural obstructions for the equivalence of G-structures. Hopefully, if all the differential invariants coincide over two r–jets of G-structure then they are equivalent under the action of the pseudogroup. If all the differential invariants coincide for every r it is hoped that the G-structures are formally equivalent, and so equivalent in the analytic case. This is the equivalence problem of E. Cartan. In this paper we deal with the problem of finding differential invariants on the bundles of ℝ*-structures, following the program pointed out in [3]. There are several reasons that justify the study of this type of G-structures. The first one is that it is a non-complicated example that helps to understand the G-structures with the property for the group G of having a vanishing first prolongation (i.e. of type 1). The simplicity comes from the fact that the algebraic invariants of ℝ* are very simple. The differential geometry of this type of structure, however, has much in common with general G-structures of type 1. Also, ℝ*-structures are objects of geometrical interest. They can be interpreted as ‘projective parallelisms’ of the base manifold and they can also be interpreted as a generalization of Blaschke's notion of a web.

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