summary:Summary: Here, a relation between integrals and variational multipliers of a system of second-order ordinary differential equations is studied. A simple necessary and sufficient local condition for the existence of a multiplier is given

Klapka, Lubomír

Institute of Mathematics AS CR, v. v. i.

WSGP 20Lubomír KlapkaIntegrals and variational multipliers of second-order ordinary differential equationsIn: Jan Slovák and Martin Čadek (eds.): Proceedings of the 20th Winter School "Geometry andPhysics". Circolo Matematico di Palermo, Palermo, 2001. Rendiconti del Circolo Matematico diPalermo, Serie II, Supplemento No. 66. pp. 129--131.Persistent URL: http://dml.cz/dmlcz/701673Terms of use:© Circolo Matematico di Palermo, 2001Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czRENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Suppl. 66 (2001) pp. 129-131 INTEGRALS AND VARIATIONAL MULTIPLIERS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS LUBOMIR KLAPKA ABSTRACT. In this paper the relation between integrals and variational multipliers of a system of second-order ordinary differential equations is studied. A simple necessary and sufficient local condition for the existence of a multiplier is given. 1. INTRODUCTION There are a few different approaches to the inverse problem of the variational cal-culus for second-order differential equations (see, e.g. [1, 2, 6, 7]), but all of them use the notion of variational multipliers. The purpose of this paper is to study the local relation between variational multipliers and integrals of these differential equations. An appropriate mathematical apparatus was developed in [3, 4]. For simplicity, all mappings which are used in this paper are supposed to be infinitely differentiable and all expressions which are used here are local expressions only. 2. INTEGRALS AND MULTIPLIERS We say that a given system of differential equations of the form xk = f%xix)i (1) where k = 1,2,..., n, has an integral F if and only if F'(r, x, x) is a function such that dF/dt = 0 on any solution of (1). In this case, dF/dt = (dF/dxk)(xk - /*) . Integrals are said to be independent if they are independent as functions. We say that (1) has a variational multiplier if and only if there exists a non-singular matrix (o^(r,a:,i)), such that for some function L(t,x,x) - d (dL\ dL *&-ns*{W -e*- (2) 2000 Mathematics Subject Classification: 34A34, 34A55,, 37J05, 37J35, 70H03, 70H15, 93C15. Key words and phrases: Inverse problem of variational calculus, second-order ordinary differential equations, variational multipliers, integrals of differential equations, canonical transformations. The paper is in final form and no version of it will be submitted elsewhere. 1 3 0 L. KLAPKA Theorem. A system of differential equations (1) has a variational multiplier if and only if it has independent integrals Pi, P^ . . . , Pn, Q\ Q2 , . . . , Qn satisfying conditions 8PadQa 0PadQa дxk дxl дxl дxk = 0. (3) Proof. Suppose that PajQa are independent integrals of (1) satisfying the condi­tion (3). This condition implies the existence of a function S(t,x,x), such that dS _ dQa dx1 ~~ ~Pa Iw' ( 4 ) Then the function L = Pa(dQa/dt)-\-dS/dt does not depend on xk and so by straight­forward calculation we have (2), where = dPadQ^__dPadQ^ 9hl ~ dx1 dxk dxk dx1 ' [ j From (3) and (5) we get dPx A- • -AdPnAdQl • • - AdQn = (- det \gkl\)n dx1 A- • -AdxnA dx1 A - • • A dxn. Since the integrals are independent, the matrix (gkl) is non-singular. Hence, the system (1) has a variational multiplier. Conversely, suppose that a system of differential equations (1) has a variational multiplier. Then from (2) one gets d2L/dxkdxl = gk[. Since (gki) is a non-singular matrix, we can use the Hamiltonian formalism. Denote the Hamilton function by H. There exists a canonical transformation (pk,xk,t) -» (Pa,Qa,t) reducing H to zero (see, e.g. [4, 5]). In this case Pa,QQ are independent integrals of motion and pk dxk - Pa dQa — Hdt = dS, where 5 is a corresponding generating function. If we return to Lagrangean coordinates xk,xk,t we get, among others, the formula (4). The integrability condition of (4) with respect to S gives (3). This completes the proof. 3. REMARK The above Theorem is a direct consequence of Theorem 1. in [3], p. 3785, as well as Proposition 9.2.11 in [4], p. 180. However, they lead to another proof, based on the concept of a Lepagean 2-form. ACKNOWLEDGMENTS The author is indebted to Olga Krupkova for here attention to this work. The support by Grants CEZ:J10/98:192400002 and VS 96003 "Global Analysis" of the Czech Ministry of Education, Youth and Sports, and Grant No. 201/00/0724 of the Czech Grant Agency is gratefully acknowledged. REFERENCES [1] I. Anderson, G. Thompson, The inverse problem of the calculus of variations for ordinary dif­ferential equations, Memoirs Amer. Math. Soc. 98 (1992) No. 473. [2] J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc. 50 (1941) 71-128. [3] L. Klapka, Integrals of motion and semi-regular Lepagean forms in higher-order mechanics, J. Phys. A: Math. Gen. 16 (1983) 3783-3794. [4] O. Krupkova, The Geometry of Ordinary Variational Equations, Lecture Notes in Mathematics 1678 (Springer, Berlin, 1997). INTEGRALS AND VARIATIONAL MULTIPLIERS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS 1 3 1 [5] A. Lichnerowicz, Geometry of the canonical transformations, In : Dynamica l systems and micro-physics, control theory and mechanics, A. Blaquiere, G. Leitmann (eds.), Proc. 3rd Int. Semin. Udine, Italy, Sept. 4 - 9 , 1983 (Academic Press Inc., San Diego, California 1984), 287-327. [6] Z. Muzsnay, Sur le probleme inverse du calcul des variations, Thesis (Univ. Paul Sabatier, Toulouse, 1997). [7] W. Sarlet, Integrability aspects of the inverse problem of the calculus of variations. In: Differential Geometry and Applications, I. Kolaf, O. Kowalski, D. Krupka, J. Slovak (eds.), Proc. Conf. Brno, Czech Republic, Aug. 10 - 14, 1998 (Masaryk University, Brno, 1999) 563-574. SILESIAN UNIVERSITY AT OPAVA, MATHEMATICAL INSTITUTE BEZRUČOVO NÁM. 13, 746 01 OPAVA 1, CZECH REPUBLIC E-mail: Lubomír. KlapkaQmatih. s lu . cz 

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Integrals and variational multipliers of second-order ordinary differential equations

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Integrals and variational multipliers of second-order ordinary differential equations

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