Variational Principle of Instability of Atmospheric Motions

Problems of instability of rotating atmospheric motions are investigated by using nonlinear governing equations and the variational principle. The method suggested in this paper is universal for obtaining criteria of instability in all models with all possible basic flows. For example, the model can be barotropic or baroclinic, layer or continuous, quasi-geostrophic or primitive equations; the basic flow can be zonal or nonzonal, steady or unsteady.Although the basic flows possess a great deal of variety, they all are the stationary points in the functional space determined by an appropriate invariant functional. The basic flow is an unsteady one if the conservation of angular momentum is included in the associated functional.The second variation, linear or nonlinear, gives the criteria of instability. Especially, the general criteria of instability for unsteady basic flow, orographically disturbed flow as well as nongeostrophic flow are first obtained by the method described in this paper.It is also shown that the difference between the criteria of instability obtained by the linear theory and our variational principle clearly indicates the importance of using nonlinear governing equations.In the appendix the theory is extended to cases such as in a β-plane where the fluid does not possess finite total energy, hence the variational principle can not be directly applied. However, a generalized Liapbunoff norm can still be obtained on the basis of variational consideration.

RELATIVISTIC VARIATION PRINCIPLES AND EQUATION OF MOTION FOR VARIABLE MASS CONTROLLABLE MECHANICAL SYSTEM

With classical variable mass and relativistic variable mass cases being considered.the relativistic D' Alembert principles of Lagrange form Nielsen form and Appell. form for variable mass controllable mechanical system are given the relativistic Chaplygin equation. Nielsen equation and Appell equation .for variable mass controllable mechanical system in quasi-coordinates and generalized- coordinates are obtained, and the equations of motion of relativistic controllable mechanical system for holonomic system and constant mass system are diseussed

Comments on Zeng's Paper “Variational Principle of Instability of Atmospheric Motions”

After my paper (Zeng, 1986b) was published and another (Zeng, 1989) was submitted to the journal, I found two papers written by Arnold (1966) and McIntyre et al. (1987) and received some reprints of Ripa’s papers (1983; 1984; 1987; 1988) in the same field. I thank Drs. Mu Mu and Pedro Ripa very much for showing and sending me these interesting papers.

AN UNCERTAINTY PRINCIPLE ON THE GROUP OF RIGID MOTIONS OF THE PLANE

Let M(2)be the group of rigid motions of the plane.The Fourier transform and the Piancherel formula on M(2)can be explicitly given by the general group representation theory.Using this fact.we establish a kind of uncertainty principle on M(2).The result can easily be generalized to higher dimensional cases.An application of the result yields an uncertainty principle on the Euclidean spaces obtained by R.S.Strichartz.

Progress in high-performance stick-slip piezoelectric actuators: a review

The ultra-precision field is popular for its micro-nanometer positioning accuracy and large working stroke.Piezoelectric actuators based on the stick-slip operational principle exhibit superior performance characteristics,making them stand out with unique advantages in this field.This paper provides a comprehensive review of the developments in stick-slip piezoelectric actuators over recent years.Starting with a detailed explanation of their operating principles,the article proceeds with a brief introduction to the more commonly used driving waveforms and their applications.Subsequently,various design and optimization technologies for existing com-pliant mechanisms are presented.Furthermore,stick-slip piezoelectric actuators are categorized based on different motion forms,including linear,rotary,and multi-degree of freedom types.Each category is thoroughly examined in terms of structural design and performance features.Following this,the discussion shifts toward controller method research and friction modeling analysis,featuring a particular emphasis on the advancements related to displacement back-lash suppression studies.This systematic summary aims to provide a reference for researchers within related fields,thereby facilitating the further development and application of stick-slip piezoelectric actuators.