The Method of Lie Series and of Lie Trans-forms

In the above paper,' Kormanik and Li have given a systematic procedure to obt.ain an analytic expresion for the stability boundary of a seeond-order nonlinear system, whose stable equilibrium point is located at the origin of t,he phase plane. In the present correspon- dence, we would like to make some comments concerning the useful- ness of the methods applied in that, paper. Lie series method has been used by Kormanik and Li1 to obtain a set of points which lie very close t.o exact stability boundary, by simultaneous solution of the system stat,e equations and the modified form of Zubov's partial differential equat.ion ((l) and (5)).* We have worked out. the example reported' on an IBM '7044 com- puter, using Lie series method and obtained the results as reported by the authors. We have also integrat.ed (1) and (5)' simultaneously, using a standard numerical integration technique-Fourth order Runge-Kutt.a method, and obtained the identical results (see Fig. (l)), with much less computational effort and reduced computing time. The total computer t.ime taken including execut.ion and com- pilation, using Runge-Kut,ta method was 20 s. We would accord- ingly like to know t.he claims the authors make about advantages of Lie series method over standard numerical integration techniques. Kormanik and Li1 have considered a simple example of Vander- pol-Oscillator where linlit cycle phenomenon is exhibited. It appears however that the techniques described' will be insufficient to obt.ain regions of attraction for systems having a stable equilibrium point but with phase-plane trajectories as shorn in Fig. 2. The basic problem in Fig. 2 is the computat.ion of t.he domain of at.traction of the equilibrium point.. This kind of problem is encountered while dealing with transient stabilit,y studies in power system, e.g., com- putation of domain of stability for a single machine connected to an infinit,e bus taking t.ransient saliency and damping into account. This same nature of phase plane t.rajectories is also exhibited in some phase-locked loop synchronization studies (I). One important problem encountered in 6he integration process is in the choice of t.he init.ia1 guess for t.he state variables. The authors suggest t.hat, in general, a set of initial values may be chosen as lying on sufficiently closed quadrdic surface around the st.able equilibrium point x = 0, which satisfies (2)' in the for the given +(x). If me t.ake a positive definite +(x), (2)' is always satisfied, as long as we stay within the st.ability domain, i.e., V(x) < 1. So we are un- able to appreciate what the authors wanted to convey by the term, small neighborhood. The necessity for a small neighborhood does not appear to be very stringent as long as above mentioned conditions are met. Furthermore, for the problem relatcd to Fig. 2, one would have

W. Grobner gave in his book “Lie-series and Their Applications” the definition of this series. With a linear differential operator D we can evaluate the terms of a convergent series. For testing, this method is applied to the two body problem. It could be shown that the evaluation of the first five terms are sufficient to get good results. Then, Lie series were used to integrate a restricted three body system, a three body system, and a restricted four body system. Because the terms are very complex, much computing time is required. The advantage of solving these problems with Lie series lies in the fact that the same procedure may be used for solving non-conservative systems; for example, a two body system with variable masses. For testing, the method of Lie series is also applied to solve non-autonomous systems.

: The report summarizes the recent work in the application of the LIE- series method to the solution of ordinary and partial differential equations. The power series method which is a special case of the Lie series method of chapter III is described in chapter II. Chapter III deals with the numerical evaluation of the Lie series perturbation formula. In chapter IV we prove Grobner's integral equation which leads to short proofs of the formulas of chapter III and to various generalizations of the method. A survey of these is presented at the end of this summary. Chapter V generalizes the concept of Runge-Kutta to methods with multiple nodes, which is possible with the use of the Lie differential operator D. Chapter VI deals with the step-size control and chapter VII shows the application of generalized Lie series to the calculation of switch-on transients occurring in the telegraphic equation.

The method of Lie series is used to construct a solution for the elliptic restricted three body problem. In a synodic pulsating coordinate system, the Lie operator for the motion of the third infinitesimal body is derived as function of coordinates, velocities and true anomaly of the primaries. The terms of the Lie series for the solution are then calculated with recurrence formulae which enable a rapid successive calculation of any desired number of terms. This procedure gives a very useful analytical form for the series and allows a quick calculation of the orbit.

Geodesics, in particular minimal geodesics, are of focal geodetic interest. The differential equations of a minimal geodesic can be written as a system of two second order ordinary differential equations, well known in the Lagrange portrait. Alternatively the system can be transformed by Legendre transformation into a system of four first order ordinary differential equations subject to the Hamilton portrait of a geodesic based on the generalized momenta. Assuming that the differntiable manifold }M, G µv } is partially covered by a set of orthogonal coordinates the cyclic coordinate P 1 generates the conservation of angular momentum, Clairaut constant, for any surface of revolution. The Lie series method is then used as one possible tool to describe analytically the geodesic flow on the sphere S 2R and on the ellipsoid of revolution E 2A,B . The solution of the initial value problem is given, whereas higher order terms of the Lie series expansion are calculated by recurrence relations. The solution of the boundary value problem is given by a bivariate series inversion via Riemann polar/normal coordinates. In order to overcome the problem of singularity a change of chart will be performed, whereas the minimal atlas of }M, G µv } is established by two charts.KeywordsAngular MomentumGeographical InformationRecurrence RelationHigh Order TermEntial EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper begins with a brief review of a form of the Lie series transformation, and then reports some new results in the study, using Lie series methods, of the orbit of Saturn’s satellite Hyperion. In particular, improved expressions are given for the long-period perturbations of the orbital elements which describe the motion in the orbit plane, and also first results for expressions for the short-period perturbations in the apse longitude, derived from the Lie series generating function.

This paper begins with a brief review of a form of the Lie series transformation, and then reports some new results in the study, using Lie series methods, of the orbit of Saturn’s satellite Hyperion. In particular, improved expressions are given for the long-period perturbations of the orbital elements which describe the motion in the orbit plane, and also first results for expressions for the short-period perturbations in the apse longitude, derived from the Lie series generating function.

Dynamics in the controlled center manifolds by Hamiltonian structure-preserving stabilization

The solution of the one-dimensional, multi-zone, multi-group diffusion equation using the lie series method

The theory of a one-dimensional bare reactor, treated by the lie series method

The Lie series recursive algorithm for Zubov's partial differential equation is used to generate two sets of points, where one represents the exact asymptotic stability boundary of an equilibrium state of the nonlinear system under consideration and the other is interior to it. Based on these two sets of data as training samples of two classes, a decision hypersurface can be determined such that it is a close approximation of the asymptotic stability boundary.

This paper concerns the estimation of stability domains for transient stability problems in power systems. Lieseries methods are used for generating a solution to Zubov&apos;s partial differential equation and pattern recognition algorithms are applied to obtain the boundary defining a region of stability. Critical clearing times for representative problems are obtained using the analytical expression for the stability boundary, and these values are verified by observing the nature of the swing curves obtained by numerical integration.

A Unified Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems

Normal forms in a cyclically graded Lie algebra

We discuss the classical dynamics of a CH-stretching and an OH-stretching vibration coupled to a hindered rotation around a CC (OC) bond of a CH3 group or an OH group. Our model is based on a two-dimensional system, in which zero angular momentum is assumed. The model is further simplified by considering only kinetic coupling between the CH (OH) stretching and the hindered rotation. Through numerical calculations, a new set of states is found, which originates from n:1 resonances between the internal rotation frequency and the stretching frequency, n being associated to the order of symmetry (n = 3 and 6 for the cases investigated). We also present a perturbative approach based on the Lie series method, which provides insight into these nonadiabatic states.