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 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.

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 A construction of the general relativistic Boltzmann equation

1984 ◽  
Vol 7 (3) ◽  
pp. 591-597 ◽  
Author(s):  
P. Dolan ◽  
A. C. Zenios
Keyword(s):  
Distribution Function ◽  
Boltzmann Equation ◽  
Kinetic Equation ◽  
Simple Model ◽  
State Space ◽  
Differential Forms ◽  
Relativistic Boltzmann Equation ◽  
Dimensional State Space ◽  
General Relativistic ◽  
Invariant Differential

Our work depends essentially on the notion of a one-particle seven-dimensional state-space. In constructing a general relativistic theory we assume that all measurable quantities arise from invariant differential forms. In this paper, we study only the case when instantaneous, binary, elastic collisions occur between the particles of the gas. With a simple model for colliding particles and their collisions, we derive the kinetic equation, which gives the change of the distribution function along flows in state-space.

scholarly journals An Efficient Contact Model for the Simulation of Cargo Airdrop Extraction Phase

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Leiming Ning ◽  
Jichang Chen ◽  
Mingbo Tong
Keyword(s):  
Degrees Of Freedom ◽  
Friction Model ◽  
Rigid Bodies ◽  
Contact Dynamics ◽  
Moving Frame ◽  
Contact Friction ◽  
Practical Implementation ◽  
Contact Formulation ◽  
Efficient Code ◽  
Extraction Phase

A high-fidelity cargo airdrop simulation requires the accurate modeling of the contact dynamics between an aircraft and its cargo. This paper presents a general and efficient contact-friction model for the simulation of aircraft-cargo coupling dynamics during an airdrop extraction phase. The proposed approach has the same essence as the finite element node-to-segment contact formulation, which leads to a flexible, straightforward, and efficient code implementation. The formulation is developed under an arbitrary moving frame with both aircraft and cargo treated as general six degrees-of-freedom rigid bodies, thus eliminating the restrictions of lateral symmetric assumptions in most existing methods. Moreover, the aircraft-cargo coupling algorithm is discussed in detail, and some practical implementation details are presented. The accuracy and capability of the present method are demonstrated through four numerical examples with increasing complexity and fidelity.

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.

Duality on the quantum space(3) with two parameters

Open Physics ◽  
2012 ◽  
Vol 10 (5) ◽  
Author(s):  
Muttalip Özavşar ◽  
Gürsel Yeşilot
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Differential Forms ◽  
Quantum Space ◽  
Dual Algebra ◽  
Commutation Relations ◽  
Dual Hopf Algebra ◽  
Invariant Differential ◽  
Leibniz Rules ◽  
Two Parameters

AbstractIn this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.

scholarly journals Model of the omni-channel Interaction in the Web 4.0 environment

2018 ◽  
Vol 44 ◽  
pp. 00023
Author(s):  
Anna Chernikova ◽  
Sergey Krasnov ◽  
Sergey Sergeyev
Keyword(s):  
Global Markets ◽  
Entire Group ◽  
Practical Implementation ◽  
Channel Interaction ◽  
Complex Topology ◽  
Scientific Approach ◽  
On Line ◽  
Cloud Servers ◽  
The Web

The practical implementation of the concept “Industry 4.0” entirely depends on the level of coordination between businesses, organizations and the entire group of consumers. The dominating comprehension of such interaction as a net of complex topology where nodes act as objects that have an interface a surface for contacts, results in a neuro-net-concept or the Web 4.0. Within the framework of the Russian national program “National Technological Initiative”, the development of this direction is considered as one of the main priorities. The quality of implementation of the system interaction in this paradigm is greatly influenced by algorithms used in the cloud servers of the nodes included into the structure of the Web 4.0. Besides, operation of businesses in the conditions of global markets will take place in the situation of strict competition and solutions forming the basis of the software have to be based on the scientific approach with the use of the methods of optimization and mathematical models. As the competition in the commercial sphere turned into the contest between the economic efficiency of on-line platforms, this work proposes a particular implementation of the algorithm of omni-channel interaction formed on the basis of a mathematical model.

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.

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