Hamilton–Jacobi Treatment of Lagrangians with Linear Velocities

2003 ◽  
Vol 18 (23) ◽  
pp. 1591-1596 ◽  
Author(s):  
Eqab M. Rabei ◽  
Khaled I. Nawafleh ◽  
Yacoub S. Abdelrahman ◽  
H. Y. Rashed Omari
Keyword(s):  
Equations Of Motion ◽  
Lagrangian Systems ◽  
New Approach ◽  
Consistency Conditions ◽  
Action Integral ◽  
Jacobi Formulation

A new approach for solving mechanical problems of Linear Lagrangian systems using the Hamilton–Jacobi formulation is proposed. The equations of motion are recovered from the action integral. It has been proved that there is no need to follow the consistency conditions of the Dirac approach.

Quantization of fractional singular Lagrangian systems using WKB approximation

2018 ◽  
Vol 33 (36) ◽  
pp. 1850222 ◽  
Author(s):  
Eqab M. Rabei ◽  
Mohammed Al Horani
Keyword(s):  
Wave Function ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Singular Systems ◽  
Equations Of Motion ◽  
Singular System ◽  
Wkb Approximation ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Action Integral

In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

scholarly journals New Approach to the Planetary Theory

1979 ◽  
Vol 81 ◽  
pp. 69-72 ◽  
Author(s):  
Manabu Yuasa ◽  
Gen'ichiro Hori
Keyword(s):  
Perturbation Method ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Disturbing Function ◽  
Averaging Principle ◽  
The Sun ◽  
Planetary Theory ◽  
New Approach ◽  
Canonical Perturbation ◽  
Relative Coordinates

A new approach to the planetary theory is examined under the following procedure: 1) we use a canonical perturbation method based on the averaging principle; 2) we adopt Charlier's canonical relative coordinates fixed to the Sun, and the equations of motion of planets can be written in the canonical form; 3) we adopt some devices concerning the development of the disturbing function. Our development can be applied formally in the case of nearly intersecting orbits as the Neptune-Pluto system. Procedure 1) has been adopted by Message (1976).

A New Approach to the Rigid Finite Element Method in Modeling Spatial Slender Systems

2018 ◽  
Vol 18 (02) ◽  
pp. 1850017 ◽  
Author(s):  
Iwona Adamiec-Wójcik ◽  
Łukasz Drąg ◽  
Stanisław Wojciech
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Nonlinear Dynamic Analysis ◽  
New Approach ◽  
Static And Dynamic Analysis ◽  
Rigid Finite Element Method ◽  
Longitudinal Flexibility ◽  
Element Method

The static and dynamic analysis of slender systems, which in this paper comprise lines and flexible links of manipulators, requires large deformations to be taken into consideration. This paper presents a modification of the rigid finite element method which enables modeling of such systems to include bending, torsional and longitudinal flexibility. In the formulation used, the elements into which the link is divided have seven DOFs. These describe the position of a chosen point, the extension of the element, and its orientation by means of the Euler angles Z[Formula: see text]Y[Formula: see text]X[Formula: see text]. Elements are connected by means of geometrical constraint equations. A compact algorithm for formulating and integrating the equations of motion is given. Models and programs are verified by comparing the results to those obtained by analytical solution and those from the finite element method. Finally, they are used to solve a benchmark problem encountered in nonlinear dynamic analysis of multibody systems.

INTEGRABILITY AND ACTION FUNCTION IN MULTI-HAMILTONIAN SYSTEMS

2004 ◽  
Vol 19 (11) ◽  
pp. 863-870 ◽  
Author(s):  
S. I. MUSLIH
Keyword(s):  
Differential Equations ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Integral Function ◽  
Jacobi Method ◽  
Action Function ◽  
Action Integral ◽  
Method Integration ◽  
Total Differential

Multi-Hamiltonian systems are investigated by using the Hamilton–Jacobi method. Integration of a set of total differential equations which includes the equations of motion and the action integral function is discussed. It is shown that this set is integrable if and only if the total variations of the Hamiltonians vanish. Two examples are studied.

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.

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.

Nonlinear Dynamic Response of a Thin Plate in a Fractional Viscoelastic Medium under Internal Resonance 1:1:2

2014 ◽  
Vol 518 ◽  
pp. 60-65 ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Nonlinear Equations ◽  
Time Scales ◽  
Viscoelastic Medium ◽  
Equations Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Nonlinear Dynamic Response ◽  
New Approach ◽  
New Type

Dynamic behaviour of a nonlinear plate embedded in a fractional derivative viscoelastic medium and subjected to the conditions of the internal resonances two-to-one has been studied by Rossikhin and Shitikova in [1]. Nonlinear equations, the linear parts of which occur to be coupled, were solved by the method of multiple time scales. A new approach proposed in this paper allows one to uncouple the linear parts of equations of motion of the plate, while the same method, the method of multiple time scales, has been utilized for solving nonlinear equations. The new approach enables one to find a new type of the internal resonanse, i.e., one-to-one-to-two, as well as to solve the problems of vibrations of thin bodies more efficiently.

Simplification of manipulator dynamic formulations utilizing a dimensionless method

Robotica ◽  
1993 ◽  
Vol 11 (2) ◽  
pp. 139-147 ◽  
Author(s):  
Yueh-Jaw Lin ◽  
Hai-Yan Zhang
Keyword(s):  
Dynamic Model ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Dynamics Simulation ◽  
Dynamic Equations ◽  
New Approach ◽  
Magnitude Comparison ◽  
Dynamic Formulation ◽  
Order Of Magnitude ◽  
Physical Quantities

SUMMARYThis paper presents a new approach for simplifying dynamic equations of motion of robot manipulators by using a nondimensionalization scheme. With this approach the dynamic analysis is done in a nondimensional space. That is, it is required to establish a dimensionless coordinate system in which the dynamic equations of motion of manipulators are formulated. The characteristic parameters of the manipulators are then defined by choosing proper physical quantities as basic units for nondimensionalization. Within the nondimensional space the Lagrange method is applied to the manipulator to obtain a set of general dimensionless equations of motion. This dimensionless dynamic formulation of manipulators leads to an easier way to simplify the dynamic formulation by neglecting insignificant terms using the order of magnitude comparison. The dimensionless dynamic model and its simplified version of PUMA 560 robot are implemented using the proposed approach. It is found that the simplified dynamic model greatly reduces the computation burden of the inverse dynamics. Simulation results also show that the simplified model is extremely accurate. This implies that the proposed nondimensional simplification emethod is reliable.

scholarly journals ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS

1995 ◽  
Vol 10 (04) ◽  
pp. 579-610 ◽  
Author(s):  
V. MUKHANOV ◽  
A. WIPF
Keyword(s):  
String Theory ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Linear Constraints ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Diffeomorphism Invariance ◽  
Symmetry Transformations ◽  
Local Symmetries

In this paper we show how the well-known local symmetries of Lagrangian systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangian system. The non-linear constraints (which we have, for instance, in gravity, supergravity and string theory) generate the dynamics of the corresponding Lagrangian system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We show the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems, in particular those which are diffeomorphism-invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian and Lagrangian formalisms is found. The possible applications of our results are discussed.

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