Pendulum simple, mathematical

There is a considerable literature [10-13] devoted to finding approximate formulas for the frequency of the simple pendulum for non-zero amplitudes, usually based on mathematical arguments designed to approximate elliptic functions. 

There is actually a considerable literature on the approximate amplitude dependence of the simple pendulum [9-11], although this is the only one we know of which is based on approximating the physics rather than the mathematics. The formula is remarkably accurate even for initial angular displacements of 90° from the downward vertical. The corresponding equations for the spherical pendulum in generalised coordinates are altogether more complicated, very... 

How can the result of unique steady state be consistent with the observed oscillation in Figure 5.9 The answer is that the steady state, which mathematically exists, is physically impossible since it is unstable. By unstable, we mean that no matter how close the system comes to the unstable steady state, the dynamics leads the system away from the steady state rather than to it. This is analogous to the situation of a simple pendulum, which has an unstable steady state when the weight is suspended at exactly at 180° from its resting position. (Stability analysis, which is an important topic in model analysis and in differential equations in general, is discussed in detail in a number of texts, including [146].)... 

In order to understand the mathematical importance of the chemostat, one must look at the broader picture of the subject of nonlinear differential equations. Linear differential equations have been studied for more than two hundred years their solutions have a rich structure that has been well worked out and exploited in physics, chemistry, and biology. Avast and challenging new world opens up when one turns to nonlinear differential equations. There is an almost incomprehensible variety of non-linearities to be studied, and there is little common structure among them. Models of the physical and biological world provide classes of nonlinearities that are worthy of study. Some of the classic and most studied nonlinear differential equations are those associated with the simple pendulum. Other famous equations include those associated with the names of... 

Due to the exploratory nature of these investigations at this stage, rather than pretending to study realistic systems, we will limit ourselves to the consideration of simple models, which we believe contain an indication of methods to be further pursued and a useful phenomenology in nuce. Accordingly, in this Introductory Section, we will show that the semiclassical approach naturally leads to a search for a quasiseparable variable, and to adiabatic and diabatic representations (see the Appendix). A general analysis of transition between modes will then illustrate the role of local breakdown of adiabaticity and a semiclassical study of the pendulum motion will provide an introduction to the mathematical techniques involved. 

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