No. 5 Discrete variational principle the first integrals of the In view of the face that only the momentum integrals can be obtained by the abo

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1 Vol 14 No 5, May 005 cfl 005 Chin. Phys. Soc /005/14(05)/ Chinese Physics IOP Publishing Ltd Discrete variational principle the first integrals of the conservative holonomic systems in event space * Zhang Hong-Bin(Φ ) a)b)y, Chen Li-Qun( ΛΠ) a)c), Liu Rong-Wan(Ξ± ) a) a) Shanghai Institute of Mathematics Mechanics, Shanghai University, Shanghai 0007, China b) Department of Physics, Anhui Chaohu College, Chaohu 38000, China c) Department of Mechanics, Shanghai University, Shanghai 00436, China (Received 7 October 004; revised manuscript received November 004) It is shown in this paper that first integrals of discrete equation of motion f the conservative holonomic systems can be determined explicitly by investigating the invariance properties of the discrete Lagrangian in event space. The result obtained is a discrete analogue of Noether's theem in the calculus of variations. Two examples are given to illustrate the applications of the result. Keywds: event space, discrete mechanics, conservative holonomic system, Noether's theem, first integral PACC: 030, Introduction Due to its applications to optimization engineering problems, because of recent emphases on numerical methods, the discrete calculus of variations becomes an imptant mathematical discipline [1;] in analyses of discrete systems, the systems that are governed by difference equations. Briefly, the central problem of the discrete calculus of variations is to determine a finite sequence q M 1 ;q M ; ;q N of real numbers f which the sum J fq k g = k=m F (k; q k ;q k1 ) (1) is extremal, where the given function F (x; y; z) is assumed to be continuously differentiable. A necessary condition f J to have an extremum f a given sequence fq k g(k = M 1;M; ;N) is that fq k g satisfy the second-der difference equation F y (k; q k ;q k1 )+F z (k + 1;q k+1 ;q k ) = 0; where we have denoted (k = M;M + 1; ;N) y; z) F y (k; q k ;q k1 ) = k ;z=q ; (3) y; z) F z (k +1;q k+1 ;q k ) = k+1 ;z=q k : (4) Equation (), which is in general nonlinear, is called the discrete Euler equation because of its similarity to the classical Euler equation in the calculus of variations. Its derivation has been carried out (see Ref.[1]) by treating J as an dinary function of the N M + variables q M1 ;q M ; ;q N applying the usual rules of the k = 0: (k = M 1;M; ;N): (5) In 1973 Logan [3] extended Noether's theem to a discrete case showed that first integrals of Eq.() could be obtained by studying the invariance properties of F (x; y; z), further studied multi-freedom higher-der problems. In 1981 Maeda [4] constructed the constants of motion f the discrete time system in the case of continuous symmetries. In the literature [5], we studied the first integrals of the discrete nonconservative nonholonomic systems. Λ Project suppted by the National Natural Science Foundation of China (Grant No ) the Science Research of the Education Bureau of Anhui Province, China (Grant No 004kj94). y Cresponding auth. hbzhang00@eyou.com

2 No. 5 Discrete variational principle the first integrals of the In view of the face that only the momentum integrals can be obtained by the above approaches, in this paper we address the problem of constructing first integrals of discrete equations of motion f conservative holonomic systems by investigating the invariance properties of the discrete Lagrangian in event space. We can obtain not only the momentum integrals, but also the energy integral. In Section, by perfming an analogue of the method the process in continuous mechanics, a discrete LagrangeD'Alembert principle a discrete EulerLagrange equation in event space are obtained. In Section 3, by investigating the invariance properties of the discrete Lagrangian in event space, we give the discrete first integrals of systems. In Section 4, two simple examples are presented to illustrate the applications of the results. Finally, in Section 5, a summary is given..the discrete variational principle of conservative holonomic systems in event space We study a mechanical system whose configuration is determined by the n generalized codinates q s (s = 1; ; ;n) construct a space R n+1 (event space) in which the codinates of a point are q s t. We introduce the notion x 1 = t; x s+1 = q s ; (s = 1; ; ;n) (6) then all the variables x ff (ff = 1; ; ;n+1)may be given as a function of some parameter fi. A choice of it has no special meaning, e.g. see Ref.[6]. Let x ff = x ff ; (ff = 1; ; ;n+1) (7) be such curves of class C such that (x ff ) 0 = dxff dfi ; (8) where the (x ff ) 0 are not all zero simultaneously. We set _x ff = dxff dt = (xff ) 0 (x 1 ) 0 ; (9) where x_ ff denotes the derivative with respect to t (x ff ) 0 denotes the derivative with respect to fi. It is straightfward to show that f any function F = F (t; q; _q), [7;8] we have ~F [x ff ; (x ff ) 0 ] = (x 1 ) 0 F»x ff ; (x ) 0 (x 1 ) ; ; (xn+1 ) 0 : (10) 0 (x 1 ) 0 Obviously, the function F ~ is zero der homogenous f (x ff ) 0. F the Lagrangian L(t; q; _q) of a given system in configuration space, the Lagrangian Λ[x ff ; (x ff ) 0 ] in event space are defined by the following equations: Λ[x ff ; (x ff ) 0 ] = (x 1 ) 0 L»x ff ; (x ) 0 (x 1 ) ; ; (xn+1 ) 0 ; (11) 0 (x 1 ) 0 we have n+1 ff ) 0 (xff ) 0 = Λ[x ff ; (x ff ) 0 ]: (1) Hamilton's principle has the fm Z fi1 fi 0 Λ[x ff ; (x ff ) 0 ]dfi = 0; x ff jfi =fi 0 = x ff jfi =fi 1 = 0: (13) The D'AlembertLagrange principle can be written in the fm n+1 ff ) ff x ff = 0; (14) ff=1 the parameter equations of motion f the conservative holonomic system are @(x ff ) 0 = 0: We now define the discrete action sum in event space as ~J d (x ff ) = Λ d [x ff (fi 1);x ff ]; (16) where Λ d [x ff (fi 1);x ff ] is the discrete Lagrangian in event space, which can be obtained by substituting x ff x ff (fi 1) f (x ff ) 0 x ff (fi 1) f x ff respectively in given Lagrangian. Then the discrete LagrangeD'Alembert principle is given by ~ Jd (x ff ) = Λ d [x ff (fi 1);x ff ] = 0: (17) Equation (17) is the discrete equivalent of Eq.(13). Proposition In event space, f the conservative holonomic systems with the discrete Lagrangian Λ d [x ff (fi 1);x ff ], the discrete EulerLagrange equations can be derived from the principle (17) as d [x fi (fi 1);x fi ff d[x fi ;x fi (fi + ff = 0: (ff; fi = 1; ; ;n+1): (18)

3 890 Zhang Hong-Bin et al Vol. 14 Proof: From Eq.(16), one [x fi (fi 1);x fi ff x ff (fi 1) (fi 1) d[x fi (fi 1);x fi ff x ff N [x fi (fi 1);x fi ] ff d[x fi ;x fi (fi ff x ff = 0; (ff; fi = 1; ; ;n+1) (19) where we have used a discrete integration by parts (rearranging the summation) the fact that x ff (M 1) = x ff (N) = 0. If we now require that Eq.(19) be zero f any choice of x ff, then we obtain the discrete EulerLagrange equation (18) in event space. F convenience, Eq.(18) may be rewritten as follows (In the following we will assume the summation convention of summing over repeated indices on different levels): where D Λ d;ff [x fi (fi 1);x fi ] + D 1 Λ d;ff [x fi ;x fi (fi + 1)] = 0; (ff; fi = 1; ; ;n+1) (0) D Λ d;ff [x fi (fi 1);x fi ] d[x fi (fi 1);x fi ff ; (1) D 1 Λ d;ff [x fi ;x fi (fi d[x fi ;x fi (fi + ff : () 3. First integrals of the discrete conservative holonomic systems As in the continuous case, a systematic procedure f establishing first integrals of the discrete Euler Lagrange equation can be developed from a direct study of the invariance properties, the discrete Lagrangian Λ d [Rx ff ;x ff ], where R is the lag operat [9] defined by R r x ff = x ff (fi r): (ff = 1; ; ;n+1): (3) Let us introduce the following infinitesimal transfmation: (x ff ) Λ = x ff +"u ff ; (ff = 1; ; ;n+1) (4) where " is a parameter u ff = u ff [x fi ;fi] is a sequence depending upon fi x fi ; fi = M; ;N 1. Definition The discrete Lagrangian Λ d [Rx ff, x ff ] is generalized-difference-invariant with respect to the infinitesimal transfmation (4), if there exists such a sequence v[x fi (fi 1);x fi ;fi]; fi = M; ;N that ffiλ d f"u ff [x fi ;fi]g ="f v[x fi (fi 1);x fi ;fi]g; (ff; fi = 1; ; ;n+1) (5) f each fi, where ffiλ d is given by following fmula: [3] ffiλ d [ffix ff ] =fd Λ d;ff [x fi (fi 1);x fi ;fi] + D 1 Λ d;ff [x fi ;x fi (fi +1);fi +1]gffix ff + fffix ff (fi 1)D 1 Λ d;ff [x fi (fi 1);x fi ;fi]g; (ff; fi = 1; ; ;n+1) (6) where is the difference operat, x ff = x ff (fi + 1) x ff. Based on the discrete LagrangeD'Alembert principle, the following theem concerning first integral can be proved. Theem If discrete Lagrangian Λ d [Rx ff, x ff ;fi] is generalized-difference-invariant with respect to the infinitesimal transfmation (4), if Eq.(0) holds, then u ff [x fi (fi 1);fi 1]D 1 Λ d;ff [x fi (fi 1);x fi ;fi] + v[x fi (fi 1);x fi ;fi] = const: (ff; fi = 1; ; ;n+1) (7) Proof: From Eqs.(5) (6), one has fd Λ d;ff [x fi (fi 1);x fi ;fi] + D 1 Λ d;ff [x fi ;x fi (fi +1);fi +1]g"u ff [x fi ;fi] + f"u ff [x fi (fi 1);fi 1] D 1 Λ d;ff [x fi (fi 1);x fi ;fi]g =" v[x fi (fi 1);x fi ;fi]: Simplifying using Eq.(0), one obtains fu ff [x fi (fi 1);fi 1]D 1 Λ d;ff [x fi (fi 1);x fi ;fi] + v[x fi (fi 1);x fi ;fi]g = 0; (8) whence Eq.(7) holds. This completes the proof.

4 No. 5 Discrete variational principle the first integrals of the Equation (7), which is a first-der difference relation, represents a first integral of the second-der discrete EulerLagrange equation given by Eq.(0). 4. Illustrative examples To illustrate relevant aspects of the they developed in the preceding sections, two simple examples are presented. Example 1 Consider a conservative holonomic system with its Lagrangian as follows: Let L = 1 m _q : (9) x 1 = t; x = q: (30) Hence, in event space the Lagrangian can be written as Λ = 1 m[(x ) 0 ] (x 1 ) 0 : (31) Then the discrete Lagrangian is Λ d = 1 m[x x (fi 1)] x 1 x 1 (fi 1) : (3) From Eq.(0), the discrete EulerLagrange equations can be separated into the configuration time components as follows: m x x (fi 1) x 1 x 1 (fi 1) m x (fi +1) x = 0; (33) x 1 (fi +1) x 1 1 m[x x (fi 1)] [x 1 x 1 (fi 1)] + 1 (fi +1) x ] m[x = 0: (34) [x 1 (fi +1) x 1 ] Direct verification shows that Λ d [Rx ff ;x ff ;fi] = 1 m[x Rx ] x 1 Rx 1 (35) is difference-invariant (with v = 0) with respect to the infinitesimal transfmation (x 1 ) Λ = x 1 ; (36) (x ) Λ = x +"; (37) (x 1 ) Λ = x 1 +"; (38) (x ) Λ = x ; (39) respectively. Therefe, by use of Theem, we immediately obtain the first integrals of Eqs.(33) (34) respectively, D 1 Λ d; [x ff (fi 1);x ff ] = const; (40) D 1 Λ d;1 [x ff (fi 1);x ff ] = const; (41) m x x (fi 1) = const; (4) x 1 x 1 (fi 1) 1 x (fi 1)] m[x = const: (43) [x 1 x 1 (fi 1)] Obviously, Eq.(4) is the momentum integral of the discrete system (3), Eq.(43) is the energy integral. Example Lagrangian is Consider the Kepler problem. Its L = 1 (_q 1 + _q )+μ(q 1 + q ) 1 : (q 1 + q ) 1 6= 0: (44) Let x 1 = t; x = q 1 ; x 3 = q : (45) Hence, in event space the Lagrangian can be written as follows: Λ = 1 1 (x 1 ) 0 f[(x ) 0 ] +[(x 3 ) 0 ] g+μ(x 1 ) 0 [(x ) +(x 3 ) ]: (46) Then discrete Lagrangian is Λ d = 1 [x x (fi 1)] +[x 3 x 3 (fi 1)] x 1 x 1 (fi 1) + μ[x 1 x 1 (fi 1)]f[x (fi 1)] +[x 3 (fi 1)] g: (47) From Eq.(0), the discrete EulerLagrange equations can be obtained in the following fm: 1 [x x (fi 1)] +[x 3 x 3 (fi 1)] [x 1 x 1 (fi 1)] + 1 [x (fi +1) x ] +[x 3 (fi +1) x 3 ] [x 1 (fi +1) x 1 ] + μ[x (fi 1)] + μ[x 3 (fi 1)] μ[x ] μ[x 3 ] = 0; (48) x x (fi 1) x 1 x 1 (fi 1) x (fi +1) x x 1 (fi +1) x 1 +μ[x 1 (fi +1) x 1 ]x = 0; (49) x 3 x 3 (fi 1) x 1 x 1 (fi 1) x3 (fi +1) x 3 x 1 (fi +1) x 1 +μ[x 1 (fi +1) x 1 ]x 3 = 0: (50)

5 89 Zhang Hong-Bin et al Vol. 14 It is easily to verify that Eq.(47) is difference-invariant (with v = 0) with respect to the infinitesimal transfmation (x 1 ) Λ = x 1 +"; (51) (x ) Λ = x ; (5) (x 3 ) Λ = x 3 : (53) In terms of Theem, we may immediately obtain the first integral of Eq.(48) D 1 Λ d;1 [x ff (fi 1);x ff ;fi] = const; (54) 1 [x x (fi 1)] +[x 3 x 3 (fi 1)] [x 1 x 1 (fi 1)] + μ[x (fi 1)] + μ[x 3 (fi 1)] = const: (55) The Eq.(55) is the energy integral of the discrete system (47). In addition, we introduce the following infinitesimal transfmation: (x 1 ) Λ = x 1 ; (56) (x ) Λ = x "x 3 ; (57) (x 3 ) Λ = x 3 +"x : (58) After some simple calculations, one finds that Eq.(47) is also difference-invariant (with v = 0) with respect to the above transfmation. By use of Theem, the first integral of Eqs.(49) (50) can be derived as follows: u [x ff (fi 1);fi 1]D 1 Λ d; [x ff (fi 1);x ff ;fi] + u 3 [x ff (fi 1);fi 1] D 1 Λ d;3 [x ff (fi 1);x ff ;fi] = const; (59) x x 3 (fi 1) x (fi 1)x 3 x 1 x 1 (fi 1) = const: (60) Equation (60) is the angle-momentum integral of the discrete system (47). 5. Conclusion In this paper, we define the discrete variational principle of conservative holonomic systems in event space. By investigating the invariance properties of the discrete Lagrangian, we deduce a discrete Noether theem. This theem may give not only the momentum integral of the equation of motion, but also the energy integral. References [1] Cadzow J A 1970 Int. J. Control [] Marsden J E West M 001 Acta Numer [3] Logan J D 1973 Aequat. Math [4] Maeda S 1981 Math. Japonica 6 85 [5] Zhang H B, Chen L Q Liu R W 005 Chin. Phys [6] Synge J L 1960 Classical Dynamics (Berlin: Springer) [7] Mei F X 1990 Acta Mech. Sin [8] Mei F X 000 Appl. Mech. Rev [9] Miller K S 1968 Linear Difference Equations (New Yk: Benjamin)