Variational principle for limit cycles of the Rayleigh van der Pol equation
|
|
- Wilfrid Lambert Carter
- 5 years ago
- Views:
Transcription
1 PHYICAL REVIEW E VOLUME 59, NUMBER 5 MAY 999 Variational principle for limit cycles of the Rayleigh van er Pol equation R. D. Benguria an M. C. Depassier Faculta e Física, Pontificia Universia Católica e Chile, Casilla 306, antiago 22, Chile Receive 2 October 998 We show that the amplitue of the limit cycle of Rayleigh s equation can be obtaine from a variational principle. We use this principle to reobtain the asymptotic values for the perio an amplitue of the Rayleigh an van er Pol equations. Limit cycles of general Liénar systems can also be erive from a variational principle X PAC numbers: x, Ky, Hq, W I. INTRODUCTION Limit cycles or self-excite oscillations appear in a wie class of problems in electronics, biology, astrophysics, to name a few. The first an most stuie examples of equations which exhibit limit cycles are the Rayleigh equation ÿ 3 ẏ ẏ 3 y0, which was introuce to show the appearance of sustaine vibrations in acoustics, an the van er Pol equation 2 ẍx 2 ẋx0, erive for a certain electrical circuit. These systems exhibit a single perioic solution of perio an amplitue epening on. It was soon notice that these two equations are closely relate. Inee, taking the erivative of Rayleigh s equation an calling ẏx, we obtain van er Pol s equation for x. Therefore, the limit cycles for both equations have the same perio an the amplitue x max of the limit cycle of the van er Pol equation is the maximum value of ẏ. More general systems of the form ẍ f xẋx0 were stuie by Liénar 3, who gave conitions on f for the existence of a unique limit cycle. The conitions for the existence an uniqueness of limit cycles for systems of the form ẍ f (x)ẋg(x)0, or generalize Liénar systems, have also been establishe 4. For functions f (x) which o not satisfy Liénar s conitions, the existence, number, an location of limit cycles is an open problem. For small epartures of the Hamiltonian case ẍx0 these questions can be answere by Melnikov s theory 5,6. A new, nonperturbative, approach to solving this problem has been propose recently 7,8. In this paper we will be concerne with the Rayleigh van er Pol equation. There is a large amount of work on this equation, both rigorous an numerical. The perio an amplitue have been etermine mainly by perturbation theory for the small an large limits. Accurate methos 9 have been evelope to obtain regular series for intermeiate values of. The leaing orer results in the limit 0 are the perio T2 an the amplitue of the limit cycle A2. In the limit of the perio is, in leaing orer, T ln(2) an the amplitue for the van er Pol equation is again A2. Higher orer corrections for both the small an large 0 limits have been obtaine. Detaile results an aitional references on these an other points can be foun in 0 3, among other books. The purpose of this work is to present an alternative approach by noticing that a variational principle can be formulate for Rayleigh s equation an also for general Liénar equations 3. We erive this variational principle an show how to recover from it the perio an amplitue in the limiting regions of the parameter. By using this variational principle together with appropriate trial functions one can obtain the amplitue an the perio of the limit cycle for arbitrary values of. II. VARIATIONAL PRINCIPLE We shall work on Rayleigh s equation an erive the amplitue an perio for Rayleigh s equation. Given the connection between the two equations mentione in the Introuction, one can also obtain the amplitue of the limit cycle for the van er Pol equation. We shall use a metho employe previously to obtain variational principles for other nonlinear eigenvalue problems 5 7 for a review see 4. Due to the symmetry of Rayleigh s equation, the limit cycle extens between a minimum x min A an a maximum x max A. Moreover, in phase space, if the point (ẏ,y) belongs to the limit cycle, then the point (ẏ,y) also belongs to it. Therefore we may consier the positive upper half ẏ0 of the phase plane, where half a perio will elapse when going from the points (ẏ0,x min )to(ẏ0,x max ). Then the equation for the limit cycle in phase space can be written as the nonlinear eigenvalue problem, p p y p3 3 p y0 with pa0 an p0, where we have calle p(y)ẏ(y). The nonlinear eigenvalue is the amplitue A which appears in the bounary conitions. It is convenient to efine a new variable uy/a in terms of which the equation for p is given by with p()0 an p0. p p u p3 3 p Ru0, X/99/595/48895/$5.00 PRE The American Physical ociety
2 4890 R. D. BENGURIA AND M. C. DEPAIER PRE 59 Two parameters appear naturally in this problem, RA/ an A. In terms of the function p(u), the perio of the limit cycle is given by u T2A p. Having written the problem in phase space, we may aapt a metho to obtain variational principles evelope for other problems. The metho is simple enough that the full etail can be given here. Let g(u) be a positive continuous, with continuous erivative but otherwise arbitrary function. Multiplying Eq. 4 by g(u) an integrating in u between 0 an, we obtain, after integrating by parts an making use of the bounary conitions on p, R uguu gu,pu u, 5 ĝuexp u vtt exp u p p t, which is a nonsingular positive function. Notice that the minimizing g is uniquely efine moulo a multiplicative constant. Our main result is then Rmin g 2 8 guf vu u, 9 uguu where the minimization is over all positive functions g(u) which are continuous an have continuous erivative in 0,. Alternatively, one may express g an v in terms of pˆ, an consier the more intuitive function pˆ as the trial function. In terms of pˆ the above results rea where gu,pu gp 2 p2 g 3 gp3 Rmin pˆ 6 u e [pˆ /pˆ ]t pˆ 33pˆu u ue [pˆ/pˆ ]t u. gp 2 p2 v 3 p3. Here we have efine v(u)g(u)/ g(u). At each value of u we may regar as a function of p; since g is positive, has a single maximum at a positive value pˆ given by pˆ 2 v v To avoi confusion between the true solution of the equation p(u) an the trial functions pˆ (u), we will continue with the use of g(u) an also v), the original trial function. Notice also that we shall be intereste in the maximum value of v(u), which gives the maximum of pˆ, which in turn is the amplitue of the limit cycle of the van er Pol equation. For each trial function g we obtain a value of pˆ an the corresponing approximate perio is given by Therefore (g,p)(g,pˆ ). Replacing this in the ientity Eq. 5, we obtain u T4A vv R 2 guf vu u, 7 uguu where F(v)2v(4v 2 )v4v 2. Therefore, given a trial function g(u) an a value of, through Eq. 7 we obtain a boun on R; the original parameters A an are given in terms of an R by AR /2 an R /2. The equality in Eq. 7 will hol whenever the trial function g(u) is such that pˆ is exactly p, the true solution for the limit cycle. From Eq. 6 we see that this will occur for gĝ satisfying from which it follows that g g vp p, If we wish to obtain approximate values for the amplitue an perio, we may choose ifferent trial functions an optimize the boun numerically. Instea, we shall stuy analytically the large an small limits to reobtain the known results in these limits. III. AYMPTOTIC REULT In this section we reobtain the known results in the limits of small an large. Notice that the variational principle 9 is of the form Rmin Ig,g/Jg,g with I g(u)f v(u) u/2 an J ug(u)u. In orer to minimize the above quotient, it is better to introuce a Lagrange multiplier, an extremize instea the quantity Ig,gJg,g Lw,w,uu, where is a Lagrange multiplier an where, rather than using variables g an g, we have introuce the variables u wu vss an wuvu.
3 PRE 59 VARIATIONAL PRINCIPLE FOR LIMIT CYCLE OF The Euler-Lagrange equation tw L L w 0, for this problem, is FvvvFvFvu0 or, more explicitly, 3 u v2 vv 2 42v 3 2v 2 2v 2 4u0. 2 This equation for v(u) written in terms of p is, as expecte, the original Eq. 4 in phase space with R/2. A. mall limit Let us consier first the limit of small. In orer to o this, we first obtain the approximate solution v(u) of Eq. 2 for small, which will epen on. For small, Eq. 2 for v becomes 3 u v2 vv 2 4u0 which can be integrate to obtain v 2 vv u2 C, where C is a constant. Noticing that the above equation can be written as 2pˆ 22u 2 /6C an that the bounary conitions on pˆ are pˆ ()0, evaluating at u or ) we obtain the value for the constant C2/6. Defining /2 we have that for small an finally v 2 vv 2 42u 2 20, vuu 2 u 2. Thus, for small in leaing orer we have I 2 exp u vtt F vu u 2 F vu u an J exp u vtt uu u u vtt. Using the expression 3 for v(u) we obtain 3 I 3 4 4, so that from Eq. 7 FIG.. Graph of the function u(v). R 3 J 4 34, The right sie of Eq. 4 is minimize at 4, leaing to the approximation R4/ for small values of. Therefore, AR /2 2, which is the correct result in the small limit. The perio, evaluate from Eq. 0, with v(u) given by Eq. 3, ist 2. B. Large limit Let us examine now the limit of large. For large, we obtain from Eq. 2 u2v 3 22v 2 v To fix the value of we coul procee as above. However, the same results can be obtaine in a more intuitive manner which we evelop here. The function u(v) is plotte in Fig.. Its maximum occurs at v0 an its value is 8. ince the maximum value of u is recall that with our choice of variables u ranges from towe must have 8. With this choice for, the maximum value for v which occurs at u is 3/2. We have then our first conclusion, namely that the maximum of v at large is 3/2 an the corresponing 2 maximum for p is (v max v max 4)/22. This maximum for p of the Rayleigh equation correspons to the limiting amplitue of the van er Pol equation. The amplitue for the Rayleigh equation can be rea from the variational principle. For large values of the function ĝ(u) given by Eq. 8 is highly peake at the maximum of u v(t)t, i.e., at u since in this case v0). Therefore, for large, from Eq. 9 we obtain R 2 Fv 2 F0 2 3,
4 4892 R. D. BENGURIA AND M. C. DEPAIER PRE 59 FIG. 2. Numerical results for the amplitue of the Rayleigh equation as a function of. The asymptotic behavior A 2/3 can be notice here. an the asymptotic behavior for the amplitue is A In Fig. 2 we show the results of the numerical integration of Rayleigh s equation where we verify this asymptotic behavior. The perio can be calculate from Eq. 0. In fact, by changing the inepenent variable from u to v, an by using Eq. 5, we obtain 0 T4A 3/2 vv 2 4 u v v4a arcsinh3/4. Using Eq. 6 in Eq. 7 we finally obtain T32 arcsinh3/432 ln2 7 which is the correct leaing orer for the perio for large. The two functions v(u) which we have obtaine as limiting values may also be use as convenient trial functions to obtain numerical bouns on R. In Fig. 3 we show the boun obtaine using as a trial function the small approximation for v(u). With this, the simplest trial function, we obtain at 4 an error of less than.7%. Improvements of the maximum value of p, or equivalently on the amplitue of the van er Pol equation, can be obtaine only by improving the trial function. IV. CONCLUION We have shown that the equation satisfie by the limit cycle of Rayleigh s equation erives from the variational FIG. 3. Bouns on the amplitue of the limit cycle of the Rayleigh equation obtaine with a simple trial function. principle Eq. 9 which enables one to obtain the amplitue, in principle, with any esire accuracy. Once sufficient accuracy is obtaine, the perio an amplitue for the van er Pol equation follow from it. We have reobtaine the limiting values for small an large eviations from the linear problem analytically. Our purpose in this work has been to show the existence of the variational principle, an the metho of erivation. It is clear that a variational principle can also be formulate for Liénar systems of the form given in Eq. 3. The conition for the existence of a unique limit cycle are that f is even, F(x) 0 x f (s)s satisfies F0 for 0x x 0, F0 for xx 0, F is monotone increasing for xx 0, an F as x. These properties guarantee that the limit cycle of the associate equation ÿf(ẏ)y0 obtaine calling xẏ an integrating can be erive from a variational principle as well. everal questions remain to be answere. First, for other types of functions f the equation may possess more than one limit cycle. It is irect to show that the phase space equation for the limit cycles of ÿf(ẏ)y0, for any polynomial F, can be erive from a variational principle. This means that all the limit cycles correspon to an extremum of a certain functional. It is an open question whether it is possible to count or estimate the positions of such extrema. Finally, the possibility of extening these results to generalize Liénar systems of the form ẍ f (x)ẋg(x)0 is another problem that remains open. ACKNOWLEDGMENT This work was supporte by Fonecyt Project Nos an Chile. The work of R.B. has been supporte by a Cátera Presiencial en Ciencias Chile an by the John imon Guggenheim Memorial Founation. Lor Rayleigh, Philos. Mag. 5, B. van er Pol, Philos. Mag. er. 7 2, A. Liénar, Rev. Gen. Electr., 23, N. Levinson an O. K. mith, Duke Math. J. 9, L. Perko, Differential Equations an Dynamical ystems pringer-verlag, New York, M. A. F. anjuán, Phys. Rev. E 57, H. Giacomini an. Neukirch, Phys. Rev. E 56, H. Giacomini an. Neukirch, Phys. Rev. E 57, C. M. Anersen an J. F. Geer, IAM oc. In. Appl. Math.
5 PRE 59 VARIATIONAL PRINCIPLE FOR LIMIT CYCLE OF J. Appl. Math. 42, J. Grasman, Asymptotic Methos for Relaxation Oscillations an Applications pringer-verlag, New York, 987. M. Farkas, Perioic Motions pringer, Berlin, Y.Y. Qian et al., Theory of Limit Cycles, Translations of Mathematical Monographs Vol. 66 American Mathematical ociety, Provience, N. N. Bogoliubov an I. A. Mitropolsky, Asymptotic Methos in the Theory of Nonlinear Oscillations Goron an Breach, New York, R. D. Benguria an M. C. Depassier, Contemp. Math. 27, R. D. Benguria an M. C. Depassier, Commun. Math. Phys. 75, R. D. Benguria an M. C. Depassier, Phys. Rev. Lett. 77, R. D. Benguria an M. C. Depassier, Phys. Rev. Lett. 77,
Linear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationMath 300 Winter 2011 Advanced Boundary Value Problems I. Bessel s Equation and Bessel Functions
Math 3 Winter 2 Avance Bounary Value Problems I Bessel s Equation an Bessel Functions Department of Mathematical an Statistical Sciences University of Alberta Bessel s Equation an Bessel Functions We use
More informationA Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix
More informationA nonlinear inverse problem of the Korteweg-de Vries equation
Bull. Math. Sci. https://oi.org/0.007/s3373-08-025- A nonlinear inverse problem of the Korteweg-e Vries equation Shengqi Lu Miaochao Chen 2 Qilin Liu 3 Receive: 0 March 207 / Revise: 30 April 208 / Accepte:
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationNOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationThe effect of dissipation on solutions of the complex KdV equation
Mathematics an Computers in Simulation 69 (25) 589 599 The effect of issipation on solutions of the complex KV equation Jiahong Wu a,, Juan-Ming Yuan a,b a Department of Mathematics, Oklahoma State University,
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationFirst Order Linear Differential Equations
LECTURE 6 First Orer Linear Differential Equations A linear first orer orinary ifferential equation is a ifferential equation of the form ( a(xy + b(xy = c(x. Here y represents the unknown function, y
More informationDifferentiability, Computing Derivatives, Trig Review. Goals:
Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationOn the number of isolated eigenvalues of a pair of particles in a quantum wire
On the number of isolate eigenvalues of a pair of particles in a quantum wire arxiv:1812.11804v1 [math-ph] 31 Dec 2018 Joachim Kerner 1 Department of Mathematics an Computer Science FernUniversität in
More informationNumber of limit cycles of the Liénard equation
PHYSICAL REVIEW E VOLUME 56, NUMBER 4 OCTOBER 1997 Number of limit cycles of the Liénard equation Hector Giacomini and Sébastien Neukirch Laboratoire de Mathématiques et Physique Théorique, CNRS UPRES
More informationImplicit Differentiation. Lecture 16.
Implicit Differentiation. Lecture 16. We are use to working only with functions that are efine explicitly. That is, ones like f(x) = 5x 3 + 7x x 2 + 1 or s(t) = e t5 3, in which the function is escribe
More informationarxiv: v1 [math-ph] 5 May 2014
DIFFERENTIAL-ALGEBRAIC SOLUTIONS OF THE HEAT EQUATION VICTOR M. BUCHSTABER, ELENA YU. NETAY arxiv:1405.0926v1 [math-ph] 5 May 2014 Abstract. In this work we introuce the notion of ifferential-algebraic
More information'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21
Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting
More informationQubit channels that achieve capacity with two states
Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationHarmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method
1 Harmonic Moelling of Thyristor Briges using a Simplifie Time Domain Metho P. W. Lehn, Senior Member IEEE, an G. Ebner Abstract The paper presents time omain methos for harmonic analysis of a 6-pulse
More informationDifferentiability, Computing Derivatives, Trig Review
Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationA new proof of the sharpness of the phase transition for Bernoulli percolation on Z d
A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationMath 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More informationGLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
Electronic Journal of Differential Equations, Vol. 015 015), No. 99, pp. 1 14. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu ftp eje.math.txstate.eu GLOBAL SOLUTIONS FOR D COUPLED
More informationON THE OPTIMALITY SYSTEM FOR A 1 D EULER FLOW PROBLEM
ON THE OPTIMALITY SYSTEM FOR A D EULER FLOW PROBLEM Eugene M. Cliff Matthias Heinkenschloss y Ajit R. Shenoy z Interisciplinary Center for Applie Mathematics Virginia Tech Blacksburg, Virginia 46 Abstract
More informationSemiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom
PHYSICAL REVIEW A 69, 063409 (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics,
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationFinal Exam Study Guide and Practice Problems Solutions
Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making
More informationCOUPLING REQUIREMENTS FOR WELL POSED AND STABLE MULTI-PHYSICS PROBLEMS
VI International Conference on Computational Methos for Couple Problems in Science an Engineering COUPLED PROBLEMS 15 B. Schrefler, E. Oñate an M. Paparakakis(Es) COUPLING REQUIREMENTS FOR WELL POSED AND
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationG j dq i + G j. q i. = a jt. and
Lagrange Multipliers Wenesay, 8 September 011 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationFormulation of statistical mechanics for chaotic systems
PRAMANA c Inian Acaemy of Sciences Vol. 72, No. 2 journal of February 29 physics pp. 315 323 Formulation of statistical mechanics for chaotic systems VISHNU M BANNUR 1, an RAMESH BABU THAYYULLATHIL 2 1
More informationG4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.
G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether
More informationState-Space Model for a Multi-Machine System
State-Space Moel for a Multi-Machine System These notes parallel section.4 in the text. We are ealing with classically moele machines (IEEE Type.), constant impeance loas, an a network reuce to its internal
More informationExperimental Robustness Study of a Second-Order Sliding Mode Controller
Experimental Robustness Stuy of a Secon-Orer Sliing Moe Controller Anré Blom, Bram e Jager Einhoven University of Technology Department of Mechanical Engineering P.O. Box 513, 5600 MB Einhoven, The Netherlans
More informationand from it produce the action integral whose variation we set to zero:
Lagrange Multipliers Monay, 6 September 01 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More information1.4.3 Elementary solutions to Laplace s equation in the spherical coordinates (Axially symmetric cases) (Griffiths 3.3.2)
1.4.3 Elementary solutions to Laplace s equation in the spherical coorinates (Axially symmetric cases) (Griffiths 3.3.) In the spherical coorinates (r, θ, φ), the Laplace s equation takes the following
More information. ISSN (print), (online) International Journal of Nonlinear Science Vol.6(2008) No.3,pp
. ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.6(8) No.3,pp.195-1 A Bouneness Criterion for Fourth Orer Nonlinear Orinary Differential Equations with Delay
More informationAgmon Kolmogorov Inequalities on l 2 (Z d )
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,
More informationA Sketch of Menshikov s Theorem
A Sketch of Menshikov s Theorem Thomas Bao March 14, 2010 Abstract Let Λ be an infinite, locally finite oriente multi-graph with C Λ finite an strongly connecte, an let p
More informationarxiv: v1 [physics.class-ph] 20 Dec 2017
arxiv:1712.07328v1 [physics.class-ph] 20 Dec 2017 Demystifying the constancy of the Ermakov-Lewis invariant for a time epenent oscillator T. Pamanabhan IUCAA, Post Bag 4, Ganeshkhin, Pune - 411 007, Inia.
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationAN INTRODUCTION TO NUMERICAL METHODS USING MATHCAD. Mathcad Release 14. Khyruddin Akbar Ansari, Ph.D., P.E.
AN INTRODUCTION TO NUMERICAL METHODS USING MATHCAD Mathca Release 14 Khyruin Akbar Ansari, Ph.D., P.E. Professor of Mechanical Engineering School of Engineering an Applie Science Gonzaga University SDC
More information4.2 First Differentiation Rules; Leibniz Notation
.. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 307. First Differentiation Rules; Leibniz Notation In this section we erive rules which let us quickly compute the erivative function f (x) for any polynomial
More informationApplication of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate
Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite
More informationMath 2163, Practice Exam II, Solution
Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so
More informationII. First variation of functionals
II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent
More informationALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS
ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS MARK SCHACHNER Abstract. When consiere as an algebraic space, the set of arithmetic functions equippe with the operations of pointwise aition an
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationarxiv: v1 [cond-mat.stat-mech] 9 Jan 2012
arxiv:1201.1836v1 [con-mat.stat-mech] 9 Jan 2012 Externally riven one-imensional Ising moel Amir Aghamohammai a 1, Cina Aghamohammai b 2, & Mohamma Khorrami a 3 a Department of Physics, Alzahra University,
More informationThermal Modulation of Rayleigh-Benard Convection
Thermal Moulation of Rayleigh-Benar Convection B. S. Bhaauria Department of Mathematics an Statistics, Jai Narain Vyas University, Johpur, Inia-3400 Reprint requests to Dr. B. S.; E-mail: bsbhaauria@reiffmail.com
More informationCalculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10
Calculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10 This is a set of practice test problems for Chapter 4. This is in no way an inclusive set of problems there can be other types of problems
More informationLESSON 23: EXTREMA OF FUNCTIONS OF 2 VARIABLES OCTOBER 25, 2017
LESSON : EXTREMA OF FUNCTIONS OF VARIABLES OCTOBER 5, 017 Just like with functions of a single variable, we want to find the minima (plural of minimum) and maxima (plural of maximum) of functions of several
More informationSolutions to Math 41 Second Exam November 4, 2010
Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationStable and compact finite difference schemes
Center for Turbulence Research Annual Research Briefs 2006 2 Stable an compact finite ifference schemes By K. Mattsson, M. Svär AND M. Shoeybi. Motivation an objectives Compact secon erivatives have long
More informationSturm-Liouville Theory
LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory
More informationSINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
Communications on Stochastic Analysis Vol. 2, No. 2 (28) 289-36 Serials Publications www.serialspublications.com SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationOn the uniform tiling with electrical resistors
On the uniform tiling with electrical resistors M. Q. Owaiat Department of Physics, Al-Hussein Bin Talal University, Ma an,7, Joran E-mail: Owaiat@ahu.eu.jo Abstract We calculate the effective resistance
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationFinal Exam: Sat 12 Dec 2009, 09:00-12:00
MATH 1013 SECTIONS A: Professor Szeptycki APPLIED CALCULUS I, FALL 009 B: Professor Toms C: Professor Szeto NAME: STUDENT #: SECTION: No ai (e.g. calculator, written notes) is allowe. Final Exam: Sat 1
More informationComputing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions
Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5
More informationTMA4195 Mathematical modelling Autumn 2012
Norwegian University of Science an Technology Department of Mathematical Sciences TMA495 Mathematical moelling Autumn 202 Solutions to exam December, 202 Dimensional matrix A: τ µ u r m - - s 0-2 - - 0
More informationLINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form
LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1 We consier ifferential equations of the form y + a()y = b(), (1) y( 0 ) = y 0, where a() an b() are functions. Observe that this class of equations inclues equations
More informationApplications of the Wronskian to ordinary linear differential equations
Physics 116C Fall 2011 Applications of the Wronskian to orinary linear ifferential equations Consier a of n continuous functions y i (x) [i = 1,2,3,...,n], each of which is ifferentiable at least n times.
More informationSpectral Flow, the Magnus Force, and the. Josephson-Anderson Relation
Spectral Flow, the Magnus Force, an the arxiv:con-mat/9602094v1 16 Feb 1996 Josephson-Anerson Relation P. Ao Department of Theoretical Physics Umeå University, S-901 87, Umeå, SWEDEN October 18, 2018 Abstract
More informationA Weak First Digit Law for a Class of Sequences
International Mathematical Forum, Vol. 11, 2016, no. 15, 67-702 HIKARI Lt, www.m-hikari.com http://x.oi.org/10.1288/imf.2016.6562 A Weak First Digit Law for a Class of Sequences M. A. Nyblom School of
More information2 GUANGYU LI AND FABIO A. MILNER The coefficient a will be assume to be positive, boune, boune away from zero, an inepenent of t; c will be assume con
A MIXED FINITE ELEMENT METHOD FOR A THIRD ORDER PARTIAL DIFFERENTIAL EQUATION G. Li 1 an F. A. Milner 2 A mixe finite element metho is escribe for a thir orer partial ifferential equation. The metho can
More informationDynamic Equations and Nonlinear Dynamics of Cascade Two-Photon Laser
Commun. Theor. Phys. (Beiing, China) 45 (6) pp. 4 48 c International Acaemic Publishers Vol. 45, No. 6, June 5, 6 Dynamic Equations an Nonlinear Dynamics of Cascae Two-Photon Laser XIE Xia,,, HUANG Hong-Bin,
More informationSharp Thresholds. Zachary Hamaker. March 15, 2010
Sharp Threshols Zachary Hamaker March 15, 2010 Abstract The Kolmogorov Zero-One law states that for tail events on infinite-imensional probability spaces, the probability must be either zero or one. Behavior
More informationEntanglement is not very useful for estimating multiple phases
PHYSICAL REVIEW A 70, 032310 (2004) Entanglement is not very useful for estimating multiple phases Manuel A. Ballester* Department of Mathematics, University of Utrecht, Box 80010, 3508 TA Utrecht, The
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationSimilar Operators and a Functional Calculus for the First-Order Linear Differential Operator
Avances in Applie Mathematics, 9 47 999 Article ID aama.998.067, available online at http: www.iealibrary.com on Similar Operators an a Functional Calculus for the First-Orer Linear Differential Operator
More informationLagrangian and Hamiltonian Dynamics
Lagrangian an Hamiltonian Dynamics Volker Perlick (Lancaster University) Lecture 1 The Passage from Newtonian to Lagrangian Dynamics (Cockcroft Institute, 22 February 2010) Subjects covere Lecture 2: Discussion
More informationLaplace s Equation in Cylindrical Coordinates and Bessel s Equation (II)
Laplace s Equation in Cylinrical Coorinates an Bessel s Equation (II Qualitative properties of Bessel functions of first an secon kin In the last lecture we foun the expression for the general solution
More informationConservation laws a simple application to the telegraph equation
J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness
More informationUnit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule
Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin
More informationLinear and quadratic approximation
Linear an quaratic approximation November 11, 2013 Definition: Suppose f is a function that is ifferentiable on an interval I containing the point a. The linear approximation to f at a is the linear function
More informationAverage value of position for the anharmonic oscillator: Classical versus quantum results
verage value of position for the anharmonic oscillator: Classical versus quantum results R. W. Robinett Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 682 Receive
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationAbstract A nonlinear partial differential equation of the following form is considered:
M P E J Mathematical Physics Electronic Journal ISSN 86-6655 Volume 2, 26 Paper 5 Receive: May 3, 25, Revise: Sep, 26, Accepte: Oct 6, 26 Eitor: C.E. Wayne A Nonlinear Heat Equation with Temperature-Depenent
More informationNested Saturation with Guaranteed Real Poles 1
Neste Saturation with Guarantee Real Poles Eric N Johnson 2 an Suresh K Kannan 3 School of Aerospace Engineering Georgia Institute of Technology, Atlanta, GA 3332 Abstract The global stabilization of asymptotically
More informationASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVE FOR VISCOUS CONSERVATION LAW IN TWO SPACE DIMENSIONS
TANSACTIONS OF THE AMEICAN MATHEMATICAL SOCIETY Volume 35, Number 3, Pages 13 115 S -9947(999-4 Article electronically publishe on September, 1999 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVE FO VISCOUS
More informationMath 115 Section 018 Course Note
Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of
More informationDistribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 1, 9733 1996 ARTICLE NO 56 Distribution Theory for Discontinuous Test Functions an Differential Operators with Generalize Coefficients P Kurasov* Department
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationfv = ikφ n (11.1) + fu n = y v n iσ iku n + gh n. (11.3) n
Chapter 11 Rossby waves Supplemental reaing: Pelosky 1 (1979), sections 3.1 3 11.1 Shallow water equations When consiering the general problem of linearize oscillations in a static, arbitrarily stratifie
More informationHyperbolic Moment Equations Using Quadrature-Based Projection Methods
Hyperbolic Moment Equations Using Quarature-Base Projection Methos J. Koellermeier an M. Torrilhon Department of Mathematics, RWTH Aachen University, Aachen, Germany Abstract. Kinetic equations like the
More information