Lie-point symmetries of the Lagrangian system on time scales

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1 ie-point symmetries of the agrangian system on time scales Cai Ping-Ping Song-Duan Fu Jing-i Fang-Yu Hong Institute of Mathematical Physics Zhejiang Sci-Tech University Hangzhou 008 China Eastern iaodong University Shenyang 8000 China Abstract This letter investigates the ie point symmetries and conserved uantities of the agrangian systems on time scales which unify the ie symmetries of the two cases for the continuous and the discrete agrangian systems. By defining the infinitesimal transformations generators and using the invariance of differential euations under infinitesimal transformations the determining euations of the ie symmetries on time scales are established. Then the structure euations and the form of conserved uantities with delta derivatives are obtained. The letter also gives brief discussion on the ie symmetries for the discrete systems. Finally several examples are designed to illustrate these results. PACS: 0.0.-a; 45.0.Jj Keywords: time scale ie symmetry agrangian system infinitesimal transformation conserved uantity delta derivative. Introduction The theory of time scales is a relatively new field introduced by Stefan Hilger in 988 [] in order to unify and generalize difference and differential euations. Time scale calculus theory is applicable to any field in which dynamic processes can be described with discrete or continuous models. The study of the calculus of variations in the context of time scales has its beginning in 004 with the paper of Martin Bohner. Since the pioneer paper [] the classical results of the calculus of variations on continuous-time ( ) and discrete-time( ) have been unified and generalized to a time scale : Euler-agrange euations [4]; necessary optimality conditions for variational problems subject to isoperimetric constraints [56]; high-order delta derivatives [7-9]; wea maximum principle for variable endpoints optimal control problems [0]; integration on time scales []; the boundary value problems []; applications of time scale to economics [4]. In recent years D. F. M. Torres and Corresponding author. sfujngli@6.com Supported by the National Natural Science Foundations of China (No. 078) and the National Natural Science Foundations of Zhejiang Province of China (Grant No. Y604)

2 others made use of the Euler-agrange euations on time scales to generalize one of the most beautiful results of the calculus of variations-the celebrated Noether's theorem [45]. It is generally nown that the principle of symmetry and the laws of physics have close relations. And the development of modern physics such as uantum mechanics uantum field theory and nuclear physics shows that the principle of symmetry has become the most important principles of exploring the laws of motion of microparticles. The analysis of the symmetries and conserved uantities of physical systems is very important to study the dynamical behavior of the systems and their ualitative properties. There are two modern methods to find the conserved laws that is Noether symmetry method and ie symmetry method. The Noether method is maing good progress [6-9]. The ie group theory has been approved to be a powerful tool to solve differential euations to study constrained mechanical systems to discuss controllable dynamical systems to investigate mechanico-electrical systems and to establish properties of their solution space. These aspects of ie group theory have been described in many literatures [0-7]. ie group theory has also been applied to discrete euations such as differential-difference euations discrete dynamical systems and discrete mechanico-electrical systems [89]. In this paper the ie method is introduced to explore the symmetries of the agrangian systems on an arbitrary time scale. The determining euations the structure euations and forms of conserved uantities with delta derivatives of the ie symmetries are given. The ie-form invariant (Theorem ) unifies and extends the previous formulations of ie s method in the discrete-time and continuous domains [0-9]. The ie symmetries of the discrete agrangian systems are also discussed and several examples to illustrate the application of the results are given.. Basics on time scale calculus A time scale is a nonempty closed subset of real numbers and we usually denote it by the symbol. The two most popular examples are and. We define the forward and bacward jump operator σ ρ: by

3 σ(t)=inf {s :s>t} and t sups : s t for all t (supplemented by inf Ø=sup and supø=inf ). The graininess function μ: [0 ) is defined by t t t. () Hence the graininess function is constant 0 if while it is constant for. However a time scale could have nonconstant graininess. A point t is called right-scattered right-dense left-scattered and left-dense if t t t t t t and t t holds respectively. Throughout we let ab with a<b. For an interval [ab] we simply write [ab] when this is not ambiguous. We also define a b : a b\ b b and a b : a b\ b b. We say that a function f : is delta differentiable at t provided there exists a real number f t such that for all ε>0 there is a neighborhood t t U of t with f t f s f t t s t s for all s U. For differentiable f the formula f f f () is very useful and easy to prove. If f and g are both differentiable then so is fg with fg f g f g () where we abbreviate Next a function f : f by f. is called rd-continuous if it is continuous in right-dense points and if its left-sided limits exist in left-dense points. ByC all rd-continuous functions whilec rd rd we denote the set of denotes the set of all differentiable functions with rd-continuous derivative. It is nown that rd-continuous functions possess an antiderivative i.e. there exists a function F : with F t f t and in this case

4 an integral of f from a to b (ab ) is defined by b t t Fb Fa f. (4) a. Variational relationships on time scales. Exchange relationship between the isochronous variation and the delta derivatives Consider two infinitely closed orbits α and α+dα. We denote the generalized coordinates by =(tα) =(tα+dα) corresponding to the two infinitely closed orbits respectively in giving time on time scale. We define the isochronous variation as t d t. (5) Extending =(tα+dα) to the linear terms of dα we obtain t t d t d. (6) Substituting E. (6) into E. (5) we have t d. (7) Similarly we have t d. (8) t t According to E. (6) we get t t d t d. (9) t Comparing Es. (9) and (8) we obtain. (0) Similarly we have. () We call Es. (0) and () the exchanging relationships with respect to the delta derivatives and isochronous variation.. The isochronous variation on time scales Now we study the infinitely closed orbits α and α+dα. The generalized coordinates are given by =(tα) and * = * (tα) for any t where t=t(α) so we have 4

5 t. Taing total variation for we obtain t t d t d. () t Since Δt is the variational of time with respect to α therefore t t d. () Substituting Es. (7) and () into E. () we get t. (4) We call E. (4) the relationship between the isochronous variation and the total variation on time scale. Using Es. (0) and (4) we have t. (5) Differentiating both sides of E. (4) with respect to t we obtain t t. (6) According to Es. (5) and (6) we get t t t t t t. (7) Similarly we have t (8) t t t t From Es. (8) and (9) we have. (9) t t t t t. (0) Differentiating both sides of E. (7) with respect to t we obtain t t t t t t. By virtue of Es. (0) and () we can obtain () t t t. () 4. ie symmetries of agrangian systems on time scales 5

6 4. Euations of motion of the systems on time scales We consider the fundamental problem of the calculus of variations on time scales as defined by Bohner []: I t min b t t t a a b under given boundary conditions a A b B () where σ is the forward jump operator and is the delta derivative of with respect to and the agrangian : n n is a C function with respect to its arguments. By i we will denote the partial derivative of with respect to the ith variable i=. Admissible function ( ) are assumed to bec. rd The following results nown as the Euler-agrange euation and the DuBois-Reymond euation are necessary optimality conditions for optimal trajectories of delta variational problems. Theorem (Euler-agrange Euation []). If ( ) is a minimizer of problem () then ( ) satisfies the euation t t t t t t t. (4) Theorem (DuBois-Reymond Euation for Delta Problems [4]). If C rd is a local minimizer of problem () then ( ) satisfies the euation t t t t t t t t. We observe that the mechanical system with agrangian t euation of motion t t t t t 0 (5) has the t. (6) t In general it is supposed that the system (6) is nonsingular i.e. 0. (7) Expanding E. (6) we can determine the generalized acceleration as 6

7 h t. (8) 4. Infinitesimal transformations and determining euations on time scales Introduce the infinitesimal transformations in terms of time and coordinates t t t or their expanded form t t t t t t t t (9) (0) where ε is an infinitesimal parameter and τ(t) ξ(t) are the generators of infinitesimal transformations. E. (0) is a one-parameter ie-point group of transformations. Introducing the vector s generator under the infinitesimal transformations 0 t () which can be prolonged to the two- and three-point schemes t () () where t t t t t t t t t and τ(t) ξ(t) are infinitesimal generators. Then based on the invariance of the differential Es. (8) under the infinitesimal transformations (0) if and only if we can have ht 0 (4) h h. (5) Es. (5) are called the determining euations on time scales which the generators τ(t) and ξ(t) should satisfy. We hereinafter give the definition of ie symmetries of the agrangian systems on time scales. 7

8 Definition. If generators τ(t) ξ(t) satisfy the determining euations (5) then corresponding symmetries are called ie symmetries of agrangian systems on time scales (6). 4. ie symmetrical structural euations and conserved uantities on time scales ie symmetries don t always generate conserved uantities. The subseuent propositions give the condition under which ie symmetries generate conserved uantities and the form of conserved uantities with delta derivatives. Definition. Quantity I t t I t t t 0 euation (6). is said to be a conserved uantity if and only if is preserved along all (t) that satisfy the Euler-agrange Theorem. For the infinitesimal generators τ and ξ satisfying the determining E. (5) if there is a gauge function G Gt satisfying the following euation G 0 (6) then the system possesses a conserved uantity with delta derivatives t t t I t t t t t =const. (7) Proof. Using the Euler-agrange euation (4) the DuBois-Reymond euation (5) and the structure euation (6) we obtain t t t t t t t t t t t t t G t t t t t t t t t t t t G t t t t t t t t t t t G t t t Gt 0 t. t E. (6) is called the structure euation for the agrangian systems on time scales. 8

9 If the transformations (0) satisfy Noether s identity t t t t t t t Gt t t (8) then the transformations (0) are called the Noether symmetrical transformations of the system (6). We have: Theorem 4. The structure euation with delta derivatives (6) of the ie symmetry is euivalent to Noether s identity (8). The method of solution of the direct problem of the ie symmetries on time scales is the following: firstly establish the determining euation with delta derivatives (5) and see the generator τ ξ from these euations; secondly substitute the generator obtained into the structure euation (6) to determineg t ; finally substitute andg t into the formula (7) to get the conserved uantities of the symmetries on time scales. 5. Discussion In the continuous time ( ) previous results are reduced to the classical results of the ie symmetries of the agrangian systems [4]. Here we will consider the ie symmetries of discrete agrangian systems on discrete-time ( ). The time t = is a discrete variable:. The horizon consists of N periods t ==MM+...M+N- where M and N are fixed integers instead of a continuous interval. The purpose of the following wor is to deduce the discrete euations of motion the discrete structure euations and the discrete conserved uantities on discrete-time = from the results we have obtained on an arbitrary time scale. We refer to the literatures for further reading on ie symmetries of the mechanical systems for the discrete case [9]. When problem () is reduced to a one-freedom discrete calculus of variations in which the fundamental problem is to select among all finite seuence the one which minimizes the sum M N M t I (9) 9

10 where is the difference operator. Es. (4) (5) can be written in the form of discrete Euler-agrange euations: 0 (40) (4) where. M M... M N The discrete euations of motion can be put in the form h (4) where we abbreviate by. The vector field of generators () turns out to be 0 (4) t which can be prolonged to the two- and three-point schemes (44) t. (45) The invariance of discrete Euler-agrange euations (40) under the infinitesimal transformations (0) leads to the satisfaction of the following discrete determining euations: (46) where. We hereinafter give the ie symmetries of the discrete agrangian systems on discrete-time ( ). Corollary. If generator satisfies the discrete determining euations (46) then 0

11 corresponding symmetries are called ie symmetries of the discrete agrangian systems (4). Corollary. If the infinitesimal generators satisfy E. (46) and in addition there exists a discrete gauge function G G such that the identity 0 G (47) holds then the discrete agrangian systems possesses the discrete conserved uantities I =const. (48) Proof. Using the discrete Euler-agrange euations (40) (4) and the discrete structure euation (47) we obtain I 0 G whence E. (48) holds. This completes the proof. E. (47) is called the discrete structure euation corresponding to the ie symmetries of discrete agrangian systems and E. (48) is called discrete conserved uantities associated with the systems. Furthermore if the transformations (0) satisfy Noether s identity 0 G (49) then the discrete transformations (0) are called the discrete Noether symmetrical transformations of the system (4). We also have:

12 Corollary. The discrete structure euation (47) of the ie symmetry is euivalent to the discrete Noether identity (49). 6. Example Example We first consider an example of a conservation law of a agrangian system on a discrete but nonhomogeneous time scale (graininess is not conatant). The time scale and the agrangian of the system are { n : n {0}} (50) and t t Euation (8) gives the euation of motion of the system (5). (5) t The determining euation of the ie symmetries of E. (5) under the infinitesimal transformation ξ=ξ(t) τ=τ(t) is. (5) t 4t 4 8t We can obtain the following solution of E. (5): lnt 0. (54) ln The structure euation with delta derivatives (6) gives t G. (55) Substituting the generators (54) into the structure euation (55) yields ln t G. (56) ln According to Theorem substituting the generator (54) and the gauge function (56) into the formula (7) we get the following conserved uantity I t ln t t ln =const. In this example the transformation is also Noether symmetrical and such fact is easily

13 verified by direct application of Definition : t t t 0. I t t t t 7. Conclusion In this wor the ie symmetries of agrangian systems on an arbitrary time scale are investigated. This is a significant wor which unifies and extends the previous formulations of ie s method in the discrete-time and continuous domains. We have obtained the condition under which ie symmetry can lead to a conserved uantity and we have also found the corresponding conserved uantities with delta derivatives from a nown ie symmetry. The results indicate that it is also a promising approach to see the ie symmetries of discrete agrangian systems on discrete-time ( ). Using this approach it might also be possible to obtain ie symmetries of the nonconservative and the nonholonomic mechanical systems with delta derivatives. Acnowledgments The authors would lie to express their sincere thans to referee for the valuable advice. This wor is supported by the Natural Science Foundation of China (Grant No. 078). References [] B. Aulbach S. Hilger Qualitative Theory of Differential Euations Szeged 988. [] M. Bohner Calculus of variations on time scales Dynam. Syst. Appl. (004) [] R. Hilscher V. Zeidan Calculus of variations on time scales J. Math. Anal. Appl. 89 (004) [4] N. Martins D. F. M. Torres Noether's symmetry theorem for nabla problems of the calculus of variations Appl. Math. ett. (00) [5] R. Almeida D. F. M. Torres Isoperimetric problems on time scales with nabla derivatives J. Vib. Control 6 (009) [6] A. B. Malinowsa D. F. M. Torres Necessary and sufficient conditions for local Pareto optimality on time scales J. Math. Sci. 6 (009) [7] R.P. Agarwal M. Bohner Basic calculus on time scales and some of its applications Results Math. 5 (999). [8] Z. Bartosiewicz E. Pawluszewicz Realizations of nonlinear control systems on time scales IEEE. T. Automat. Contr. 5 (008) [9] M. Bohner G. S. Guseinov Double integral calculus of variations on time scales

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