The differential equations for the feynman amplitude of a single-loop graph with four vertices

1978 ◽  
Vol 23 (1) ◽  
pp. 63-66 ◽  
Author(s):  
V. A. Golubeva ◽  
V. Z. �nol'skii
Keyword(s):  
Differential Equations ◽  
Feynman Amplitude ◽  
Single Loop ◽  
Loop Graph

scholarly journals Inelastic Collisions for Small Momentum Transfers

1964 ◽  
Vol 17 (4) ◽  
pp. 553 ◽  
Author(s):  
J Cunningham
Keyword(s):  
Scalar Product ◽  
Inelastic Collisions ◽  
Small Momentum ◽  
Single Loop ◽  
Loop Graph

Let us denote by Pi> i = 1, 2, ... , 5, the four-momenta of the internal lines of the five-point single-loop graph (p~ = fLi). Each vertex may then be labeled by two suffices and we shall denote the four-momenta of the external particles (measured formally as incoming) by P12 = PI' P23 = P2, ... ,P51 = P5 (P~ = M~). Scalar product variables may be constructed as follows

ON THE SIMPLE STATIONARY CONFIGURATIONS OF SINGLE-LOOP SPATIAL N-REVOLUTE OVERCONSTRAINED LINKAGES

1996 ◽  
Vol 20 (1) ◽  
pp. 17-39 ◽  
Author(s):  
Chung-Ching Lee
Keyword(s):  
Computer Graphics ◽  
Differential Equations ◽  
Range Of Motion ◽  
Closed Form ◽  
Single Loop ◽  
Closed Form Solutions ◽  
Systematic Formulation ◽  
Overconstrained Linkages

Based on the derived matrix and its differential equations, a systematic formulation is presented to either identify the simple stationary configurations of movable spatial 4R, 5R and 6R overconstrained linkages or prove none of them occurs at all. Some examples are given to confirm the correctness and validity of the derived mathematical criterion. In addition, the closed-form solutions of linkage joint variables are well-established and with the help of computer graphics, geometrical meanings of linkage configurations are described. This approach can be used to provide a foundation for understanding the range of motion in overconstrained linkage application.

Solution of the Forward Dynamics of a Single-loop Linkage Using Power Series

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Kinematic Chain ◽  
Linear Constraint ◽  
Rate Coefficients ◽  
Forward Dynamics ◽  
Single Loop ◽  
Single Matrix ◽  
Instantaneous Screw ◽  
Kinematic Solution

The kinematic differential equations express the paths taken by points, lines, and coordinate frames attached to a rigid body in terms of the instantaneous screw for the motion of that body. Such differential equations are linear but with a time-varying coefficient and hence solvable by power series. A single-loop kinematic chain may be expressed by a system of such differential equations subject to a linear constraint. A single matrix factorization followed by a sequence of substitutions of linear-system right-hand-side terms determines successive orders of the joint rate coefficients in the kinematic solution for this mechanism. The present work extends this procedure to the forward dynamics problem, applying it to a Clemens constant-velocity coupling expressed as a spatial 9R closed kinematic chain.

Series Solution for Finite Displacement of Single-Loop Spatial Linkages

2012 ◽  
Vol 4 (2) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Coordinate Frame ◽  
Recursive Procedure ◽  
Single Loop ◽  
Analytical Technique ◽  
Finite Displacement ◽  
Instantaneous Screw

The kinematic differential equation for a spatial point trajectory accepts the time-varying instantaneous screw of a rigid body as input, the time-zero coordinates of a point on that rigid body as the initial condition and generates the space curve traced by that point over time as the solution. Applying this equation to multiple points on a rigid body derives the kinematic differential equations for a displacement matrix and for a joint screw. The solution of these differential equations in turn expresses the trajectory over the course of a finite displacement taken by a coordinate frame in the case of the displacement matrix, by a joint axis line in the case of a screw. All of the kinematic differential equations are amenable to solution by power series owing to the expression for the product of two power series. The kinematic solution for finite displacement of a single-loop spatial linkage may, hence, be expressed either in terms of displacement matrices or in terms of screws. Each method determines coefficients for joint rates by a recursive procedure that solves a sequence of linear systems of equations, but that procedure requires only a single factorization of a 6 by 6 matrix for a given initial posture of the linkage. The inverse kinematics of an 8R nonseparable redundant-joint robot, represented by one of the multiple degrees of freedom of a 9R loop, provides a numerical example of the new analytical technique.

Three Dimensional Vibration of a Ballooning String

1995 ◽  
Author(s):  
Kevin J. Hall ◽  
Fang Zhu ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Matrix Operator ◽  
String Length ◽  
Partial Differential ◽  
Single Loop ◽  
Varying Coefficients ◽  
Coupled Equations

Abstract In many textile manufacturing processes, yarn is rotated at high speed forming a balloon. In this paper, Hamilton’s principle is used to derive the nonlinear partial differential equations of a ballooning string. Jacobian elliptical sine functions satisfy the nonlinear steady state equations. The steady state eyelet tension is related to the string length for a constant balloon height. For high tension and low string length cases, single loop balloons occur. As the string length increases, tension decreases and multiple loop solutions are obtained. The nonlinear partial differential equations are linearized about the steady state solutions, resulting in three coupled equations with spatially varying coefficients. The equations involve a positive definite mass matrix operator, skew symmetric gyroscopic matrix operator, and symmetric stiffness matrix operator. It is shown using a Galerkin approach that only single loop balloons are stable for practical yarn elasticity. The natural frequencies of the single loop balloon increase with decreasing balloon size and increasing yam stiffness. The effect of yarn elasticity on the first three vibration modes of a single loop balloon is analyzed. The steady state and stability analyses are experimentally verified.

Partial Differential Equations

2020 ◽  
Author(s):  
A. K. Nandakumaran ◽  
P. S. Datti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential

p-adic Differential Equations

2009 ◽  
Author(s):  
Kiran S. Kedlaya
Keyword(s):  
Differential Equations

Partial Differential Equations in Fluid Dynamics

2008 ◽  
Author(s):  
Isom H. Herron ◽  
Michael R. Foster
Keyword(s):  
Fluid Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential
Sign in / Sign up

Export Citation Format

Share Document