A comparison of symplectic and Hamilton's principle algorithms for autonomous and non-autonomous systems of ordinary differential equations

2001 ◽  
Vol 39 (3-4) ◽  
pp. 289-306 ◽  
Author(s):  
Begoña Cano ◽  
H.Ralph Lewis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Hamilton’S Principle ◽  
Hamilton's Principle

scholarly journals On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme

2021 ◽  
Vol 29 (1) ◽  
pp. 63-72
Author(s):  
Yu Ying ◽  
Mikhail D. Malykh
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Projective Geometry ◽  
Autonomous Systems ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Duality Principle ◽  
Integrals Of Motion ◽  
Quadratic Integral

The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as t0.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.

scholarly journals Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. P. Markakis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Closed Form Solution ◽  
First Integrals ◽  
Form Solution ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Abel Equations

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

PDE Modeling of a Flexible Two-Link Manipulator

2001 ◽  
Author(s):  
X. Zhang ◽  
S. S. Nair ◽  
V. Chellaboina
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Experimental Studies ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Partial Differential

Abstract The partial differential equations of a flexible two-link manipulator are derived using the Hamilton’s Principle. The model is validated by simulation as well as experimental studies using a two-link setup in the laboratory.

Influence of Cross-Section Distortion on the Bending Vibration of Thin-Walled Beams of H Section

1964 ◽  
Vol 6 (3) ◽  
pp. 211-218 ◽  
Author(s):  
A. D. S. Barr ◽  
T. Duthie
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Reasonable Agreement ◽  
Experimental Results ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Bending Vibration ◽  
Cross Sectional ◽  
Thin Walled ◽  
The Cross

Approximate differential equations describing the bending vibration of beams of thin-walled H section, in which the distortion of the cross-section in its own plane is taken into account, are derived from Hamilton's principle using an assumed form for the cross-section deformation. Only the simplest of the cross-sectional deformation configurations which will couple with ordinary bending is considered. The variation with wavelength of the two spectra of frequencies which result from this coupling of the bending and cross-sectional motions is shown for several section geometries. Theoretical curves show reasonable agreement with experimental results from free beams.

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