XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

Adaptive Diversification and Speciation as Pattern Formation in Partial Differential Equation Models

2011 ◽  
Author(s):  
Michael Doebeli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Pattern Formation ◽  
Differential Equations ◽  
Adaptive Dynamics ◽  
Partial Differential ◽  
Adaptive Diversification ◽  
Differential Equation Models ◽  
Phenotype Distribution

This chapter discusses partial differential equation models. Partial differential equations can describe the dynamics of phenotype distributions of polymorphic populations, and they allow for a mathematically concise formulation from which some analytical insights can be obtained. It has been argued that because partial differential equations can describe polymorphic populations, results from such models are fundamentally different from those obtained using adaptive dynamics. In partial differential equation models, diversification manifests itself as pattern formation in phenotype distribution. More precisely, diversification occurs when phenotype distributions become multimodal, with the different modes corresponding to phenotypic clusters, or to species in sexual models. Such pattern formation occurs in partial differential equation models for competitive as well as for predator–prey interactions.

XIII.—Partial Differential Equations and the Calculus of Variations

1927 ◽  
Vol 46 ◽  
pp. 126-135 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Calculus Of Variations ◽  
Order Equation ◽  
Second Order ◽  
Physical Problem ◽  
Partial Differential ◽  
Hamiltonian Principle

A partial differential equation of physics may be defined as a linear second-order equation which is derivable from a Hamiltonian Principle by means of the methods of the Calculus of Variations. This principle states that the actual course of events in a physical problem is such that it gives to a certain integral a stationary value.

scholarly journals Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1853
Author(s):  
Christiane Quesne
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Supersymmetric Quantum Mechanics ◽  
Invariance Condition ◽  
Partial Differential ◽  
Shape Invariance ◽  
Shape Invariant ◽  
Infinite Set

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

Sign in / Sign up

Export Citation Format

Share Document