Time scale differential, integral, and variational embeddings of Lagrangian systems

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1 Computers nd Mtemtics wit Applictions ( ) Contents lists vilble t SciVerse ScienceDirect Computers nd Mtemtics wit Applictions journl omepge: Time scle differentil, integrl, nd vritionl embeddings of Lgrngin systems Jcky Cresson,b, Agnieszk B. Mlinowsk c,d,, Delfim F.M. Torres d,e Lbortoire de Mtémtiques Appliquées de Pu, Université de Pu, Pu, Frnce b Institut de Mécnique Céleste et de Clcul des Épémérides, Observtoire de Pris, Pris, Frnce c Fculty of Computer Science, Biłystok University of Tecnology, Biłystok, Polnd d R&D Unit CIDMA, University of Aveiro, Aveiro, Portugl e Deprtment of Mtemtics, University of Aveiro, Aveiro, Portugl r t i c l e i n f o b s t r c t Keywords: Coerence Embedding Lest-ction principle Discrete clculus of vritions Difference Euler Lgrnge equtions We introduce differentil, integrl, nd vritionl delt embeddings. We prove tt te integrl delt embedding of te Euler Lgrnge equtions nd te vritionl delt embedding coincide on n rbitrry time scle. In prticulr, new coerent embedding for te discrete clculus of vritions tt is comptible wit te lest-ction principle is obtined Elsevier Ltd. All rigts reserved. 1. Introduction An ordinry differentil eqution is usully given in differentil form, i.e., dx(t) = f (t, x(t)), t [, b], x(t) R n. dt However, one cn lso consider te integrl form of te eqution: x(t) = x() + t f (s, x(s))ds, t [, b]. Te differentil form is relted to dynmics vi te time derivtive. Te integrl form is useful for proving te existence nd unicity of solutions or to study nlyticl properties of solutions. In order to give mening to differentil eqution over new set (e.g., stocstic processes, non-differentible functions, or discrete sets), one cn use te differentil or te integrl form. In generl, tese two generliztions do not give te sme object. In te differentil cse, we need to extend first te time derivtive. As n exmple, we cn look to Scwrtz s distributions [1] or bckwrd/forwrd finite differences in te discrete cse. Using te new derivtive, one cn ten generlize differentil opertors nd ten differentil equtions of rbitrry order. In te integrl cse, one needs to give mening to te integrl over te new set. Tis strtegy is for exmple used by Itô [2] in order to define stocstic differentil equtions, defining stocstic integrls first. In generl, te integrl form imposes fewer constrints on te underlying objects. Tis is lredy true in te clssicl cse, were we need differentible function to write te differentil form but only continuity or weker regulrity to give mening to te integrl form. Corresponding utor t: Fculty of Computer Science, Biłystok University of Tecnology, Biłystok, Polnd. E-mil ddresses: jcky.cresson@univ-pu.fr (J. Cresson),.mlinowsk@pb.edu.pl (A.B. Mlinowsk), delfim@u.pt (D.F.M. Torres) /$ see front mtter 2012 Elsevier Ltd. All rigts reserved. doi: /j.cmw

2 2 J. Cresson et l. / Computers nd Mtemtics wit Applictions ( ) Te notion of embedding introduced in [3] is n lgebric procedure providing n extension of clssicl differentil equtions over n rbitrry vector spce. Embedding is bsed on te differentil formultion of te eqution. Tis formlism ws developed in te frmework of frctionl equtions [4], stocstic processes [3], nd non-differentible functions [5]. Recently, it s been extended to discrete sets in order to discuss discretiztion of differentil equtions nd numericl scemes [6]. In tis pper, we define n embedding contining te discrete s well s te continuous cse using time scle clculus. We use te proposed embedding in order to define time scle extensions of ordinry differentil equtions bot in differentil nd integrl forms. Of prticulr importnce for mny pplictions in pysics nd mtemtics is te cse of Lgrngin systems governed by n Euler Lgrnge eqution. Lgrngin systems possess vritionl structure, i.e., teir solutions correspond to criticl points of functionl, nd tis crcteriztion does not depend on te coordinte system. Tis induces strong constrints on solutions, for exmple te conservtion of energy for utonomous clssicl Lgrngin systems. Tt is, if te Lgrngin does not depend explicitly on te independent vrible t, ten te energy is constnt long pysicl trjectories. We use te time scle embedding in order to provide n nlogy of te clssicl Lgrngin functionl on n rbitrry time scle. By developing te corresponding clculus of vritions, one ten obtins te Euler Lgrnge eqution. Tis extension of te originl Euler Lgrnge eqution pssing by te time scle embedding of te functionl is clled time scle vritionl embedding. Suc extensions re known under te terminology of vritionl integrtors in te discrete setting. We ten ve tree wys to extend given ordinry differentil eqution: differentil, integrl, or vritionl embedding. All tese extensions re priori different. Te coerence problem introduced in [3], in te context of te stocstic embedding, consider te problem of finding conditions under wic tese extensions coincide. Here, we prove tt te integrl nd vritionl embeddings re coerent (see Teorem 11). Te result is new nd interesting even in te discrete setting, providing new form of te Euler Lgrnge difference eqution (see (15)) tt is comptible wit te lest-ction principle. 2. Note on te nottion used We denote by f or t f (t) function, nd by f (t) te vlue of te function t point t. Trougout te text we consistently use squre brckets for te rguments of opertors nd round brckets for te rguments of ll te oter types of function. Functionls re denoted by uppercse letters in clligrpic mode. We denote by D te usul differentil opertor nd by i te opertor of prtil differentition wit respect to te it vrible. 3. Reminder bout time scle clculus Te reder interested in clculus on time scles is refereed to te book [7]. Here, we just recll te necessry concepts nd fix some nottion. A nonempty closed subset of R is clled time scle nd is denoted by T. Tus, R, Z, nd N, re trivil exmples of time scles. Oter exmples of time scles re [ 2, 4] N, Z := {z z Z} for some > 0, q N 0 := {qk k N 0 } for some q > 1, nd te Cntor set. We ssume tt time scle T s te topology tt it inerits from rel numbers wit stndrd topology. Te forwrd jump σ : T T is defined by σ (t) = inf{s T : s > t} for ll t T, wile te bckwrd jump ρ : T T is defined by ρ(t) = sup{s T : s < t} for ll t T, were inf = sup T (i.e., σ (M) = M if T s mximum M) nd sup = inf T (i.e., ρ(m) = m if T s minimum m). Te grininess function µ : T [0, ) is defined by µ(t) = σ (t) t for ll t T. Exmple 1. If T = R, ten σ (t) = ρ(t) = t nd µ(t) = 0. If T = Z, ten σ (t) = t +, ρ(t) = t, nd µ(t) =. On te oter nd, if T = q N 0, were q > 1 is fixed rel number, ten we ve σ (t) = qt, ρ(t) = q 1 t, nd µ(t) = (q 1)t. In order to introduce te definition of delt derivtive, we define new set T κ wic is derived from T s follows: if T s left-scttered mximl point M, ten T κ := T\{M}; oterwise, T κ := T. In generl, for r 2, T κr := (T κr 1 ) κ. Similrly, if T s rigt-scttered minimum m, ten we define T κ := T \ {m}; oterwise, T κ := T. Moreover, we define T κ κ := Tκ T κ. Definition 1. We sy tt function f : T R is delt differentible t t T κ if tere exists number [f ](t) suc tt for ll ε > 0 tere is neigborood U of t suc tt f (σ (t)) f (s) [f ](t)(σ (t) s) ε σ (t) s, for ll s U. We cll [f ](t) te delt derivtive of f t t, nd we sy tt f is delt differentible on T κ provided tt [f ](t) exists for ll t T κ. Exmple 2. If T = R, ten [f ](t) = D[f ](t), i.e., te delt derivtive coincides wit te usul one. If T = Z, ten [f ](t) = 1 (f (t + ) f (t)) =: +[f ](t), were + is te usul forwrd difference opertor defined by te lst eqution. If T = q N 0 f (qt) f (t), q > 1, ten [f ](t) =, i.e., we get te usul derivtive of quntum clculus. (q 1)t

3 J. Cresson et l. / Computers nd Mtemtics wit Applictions ( ) 3 A function f : T R is clled rd-continuous if it is continuous t rigt-dense points nd if its left-sided limit exists t left-dense points. We denote te set of ll rd-continuous functions by C rd (T, R) nd te set of ll delt differentible functions wit rd-continuous derivtive by C 1 rd (T, R). It is known (see [7, Teorem 1.74]) tt rd-continuous functions possess delt ntiderivtive, i.e., tere exists function ξ wit [ξ] = f, nd in tis cse te delt integrl is defined d by f (t)t = ξ(d) ξ(c) for ll c, d T. c Exmple 3. Let, b T wit < b. If T = R, ten f (t)t = f (t)dt, were te integrl on te rigt-nd side is te clssicl Riemnn integrl. If T = Z, ten f (t)t = 1 (1 q) t [,b) tf (t). Te delt integrl s te following properties: (i) if f C rd nd t T, ten t f (τ) τ = µ(t)f (t); k= f (k). If T = q N 0, q > 1, ten f (t)t = (ii) if c, d T nd f nd g re delt differentible, ten te following formuls of integrtion by prts old: d t=d d f (σ (t)) [g](t)t = (fg)(t) [f ](t)g(t)t, c d c f (t) [g](t)t = (fg)(t) t=c c t=d d t=c c [f ](t)g(σ (t))t. (1) 4. Time scle embeddings nd evlution opertors Let T be bounded time scle wit := min T nd b := mx T. We denote by C([, b]; R) te set of continuous functions x : [, b] R. As introduced in Section 3, by C rd (T, R) we denote te set of ll rel-vlued rd-continuous functions defined on T, nd by C 1 rd (T, R) te set of ll delt differentible functions wit rd-continuous derivtive. A time scle embedding is given by specifying te following. A mpping ι : C([, b], R) C rd (T, R). An opertor δ : C 1 ([, b], R) C 1 rd (Tκ, R), clled generlized derivtive. An opertor J : C([, b], R) C rd (T, R), clled generlized integrl opertor. We fix te following embedding. Definition 2 (Time Scle Embedding). Te mpping ι is obtined by restriction of functions to T. Te opertor δ is cosen to be te derivtive, nd te opertor J is given by te integrl s follows: δ[x](t) := [x](t), J[x](t) := x(s)s. Definition 3 (Evlution Opertor). Let f : R R be continuous function. We denote by f te opertor ssocited to f nd defined by f : C(R, R) C(R, R) x f [x] := t f (x(t)). (2) Te opertor f given by (2) is clled te evlution opertor ssocited wit f. Te definition of evlution opertor is esily extended in vrious wys. We give in Definition 4 specil evlution opertor tt nturlly rises in te study of problems of te clculus of vritions nd respective Euler Lgrnge equtions (see Section 5). Definition 4 (Lgrngin Opertor). Let L : [, b] R R R be C 1 function defined for ll (t, x, v) [, b] R 2 by L(t, x, v) R. Te Lgrngin opertor L : C 1 ([, b], R) C 1 ([, b], R) ssocited wit L is te evlution opertor defined by L[x] := t L(t, x(t), D[x](t)). We consider ordinry differentil equtions of te form O[x](t) = 0, t [, b],

4 4 J. Cresson et l. / Computers nd Mtemtics wit Applictions ( ) were x C n (R, R) nd O is differentil opertor of order n, n 1, given by n O = i (D i i ), i=0 were ( i ) (respectively, (i )) is te fmily of evlution opertors ssocited to fmily of functions ( i ) (respectively, (b i )), nd D i is te derivtive of order i, i.e., D i = di dti. Differentil opertors of form (3) ply crucil role wen deling wit Euler Lgrnge equtions. We re now redy to define te time scle embedding of evlution nd differentil opertors. Definition 5 (Time Scle Embedding of Evlution Opertors). Let f : R R be continuous function nd f te ssocited evlution opertor. Te time scle embedding ft of f is te extension of f to Crd (T, R): ft : C rd(t, R) C rd (T, R) x ft [x] := t f (x(t)). Te next definition gives te time scle embedding of te differentil opertor (3). Definition 6 (Time Scle Embedding of Differentil Opertors). Te time scle embedding of te differentil opertor (3) is defined by n O = it ( i it ). i=0 Te two previous definitions re sufficient to define te time scle embedding of given ordinry differentil eqution. Definition 7 (Time Scle Embedding of Differentil Equtions). Te delt-differentil embedding of n ordinry differentil eqution O[x] = 0, x C n ([, b], R), is given by O [x] = 0, x C n rd (Tκn, R). In order to define te delt-integrl nd te delt-vritionl embeddings (see Sections 7 9) we need to know ow to embed n integrl functionl. (3) Definition 8 (Time Scle Embedding of Integrl Functionls). Let L : [, b] R 2 functionl defined by L(x) = t L(s, x(s), D[x](s))ds = t L[x](s)ds. Te time scle embedding L of L is given by L (x) = L(s, x(s), [x](s))s = LT [x](s)s. R be continuous function nd L te 5. Clculus of vritions Te clssicl vritionl functionl L is defined by L(x) = L(t, x(t), D[x](t))dt, were L : [, b] R R R is smoot rel-vlued function clled te Lgrngin (see, e.g., [8]). Functionl (4) cn be written, using te Lgrngin opertor L (Definition 4), in te following equivlent form: L(x) = L[x](t)dt. Te Euler Lgrnge eqution ssocited to (4) is given (see, e.g., [8]) by D[τ 3 [L](τ, x(τ), D[x](τ))](t) 2 [L](t, x(t), D[x](t)) = 0, (5) t [, b], wic we cn write, equivlently, s (D 3 [L])[x](t) 2 [L][x](t) = 0. Still noter wy to write te Euler Lgrnge eqution consists in introducing te differentil opertor EL L, clled te Euler Lgrnge opertor, given by EL L := D 3 [L] 2 [L]. We cn ten write te Euler Lgrnge eqution simply s EL L [x](t) = 0, t [, b]. (4)

5 J. Cresson et l. / Computers nd Mtemtics wit Applictions ( ) 5 6. Delt-differentil embedding of te Euler Lgrnge eqution By Definition 6, te time scle delt embedding of te Euler Lgrnge opertor EL L gives te new opertor (EL L ) := ( 3 [L]) T ( 2 [L]) T. As consequence, we ve te following lemm. Lemm 9 (Delt-Differentil Embedding of te Euler Lgrnge Eqution). Lgrnge eqution is given by (EL L ) [x](t) = 0, t T κ2, i.e., Te delt-differentil embedding of te Euler ( 3 [L] T )[x](t) 2 [L] T [x](t) = 0 (6) for ny t T κ2. In te discrete cse T = [, b] Z, we obtin from (6) te well-known discrete version of te Euler Lgrnge eqution, often written s + L v (t, x(t), +x(t)) L x (t, x(t), +x(t)) = 0, (7) f (t+) f (t) t T κ2, were + f (t) =. Te importnt point to note ere is tt, from te numericl point of view, Eq. (7) does not provide good sceme. Let us see simple exmple. Exmple 4. Consider te Lgrngin L(t, x, v) = 1 2 v2 U(x), were U is te potentil energy nd (t, x, v) [, b] R R. Ten te Euler Lgrnge eqution (7) gives x k+2 2x k+1 + x k 2 + U x (x k) = 0, k = 0,..., N 2, (8) were N = b nd x k = x( + k). Tis numericl sceme is of order 1, mening tt we mke n error of order t ec step, wic is of course not good. In te next section, we sow n lterntive Euler Lgrnge eqution to (7) tt leds to more suitble numericl scemes. As we sll see in Section 9, tis comes from te fct tt te embedded Euler Lgrnge eqution (6) is not coerent, mening tt it does not preserve te vritionl structure. As consequence, te numericl sceme (8) is not symplectic, in contrst to te flow of te Lgrngin system (see [9]). In prticulr, te numericl sceme (8) dissiptes energy rtificilly (see [10, Fig. 1, p. 364]). 7. Discrete vritionl embedding Time scle embedding cn be lso used to define delt nlogue of te vritionl functionl (4). Using Definition 8, nd remembering tt σ (b) = b, te time scle embedding of (4) is L (x) = L(t, x(t), [x](t)) t = LT [x](t)t. (9) A clculus of vritions on time scles for functionls of type (9) is developed in Section 9. Here, we just empsize tt, in te discrete cse T = [, b] Z, functionl (9) reduces to te clssicl discrete Lgrngin functionl N 1 L (x) = L(t k, x k, + x k ), k=0 (10) were N = b, x k = x(+k) nd + x k = x k+1 x k, nd tt te Euler Lgrnge eqution obtined by pplying te discrete vritionl principle to (10) tkes te form L v (t, x(t), +x(t)) L x (t, x(t), +x(t)) = 0, (11) f (t) f (t ) t T κ κ, were is te bckwrd finite-difference opertor defined by f (t) = [6,11]. Te numericl sceme corresponding to te discrete vritionl embedding, i.e., to (11), is clled in te literture vritionl integrtor [6,11]. Te next exmple sows tt te vritionl integrtor ssocited wit te problem in Exmple 4 is better numericl sceme tn (8).

6 6 J. Cresson et l. / Computers nd Mtemtics wit Applictions ( ) Exmple 5. Consider te sme Lgrngin s in Exmple 4 of Section 6: L(t, x, v) = 1 2 v2 U(x), were (t, x, v) [, b] R R. Te Euler Lgrnge eqution (11) cn be written s x k+1 2x k + x k U x (x k) = 0, k = 1,..., N 1, were N = b nd x k = x( + k). Tis numericl sceme possess very good properties. In prticulr, it is esily seen tt te order of pproximtion is now 2 nd not of order 1, s in Exmple 4. We remrk tt te form of L given by (9) is not te usul one in te literture of time scles (see [12 14] nd references terein). Indeed, in te literture of te clculus of vritions on time scles, te following version of te Lgrngin functionl is studied: L usul T (x) = L(t, x(σ (t)), [x](t))t. (12) However, te composition of x wit te forwrd jump σ found in (12) does not seem nturl from te point of view of embedding. 8. Delt-integrl embedding of te Euler Lgrnge eqution in integrl form We begin by rewriting te clssicl Euler Lgrnge eqution into integrl form. Integrting (5), we obtin tt 3 [L](t, x(t), D[x](t)) = t 2 [L](τ, x(τ), D[x](τ))dτ + c, (13) for some constnt c nd ll t [, b] or, using te evlution opertor, 3 [L][x](t) = t 2 [L][x](τ)dτ + c. Using Definition 8, we obtin te delt-integrl embedding of te clssicl Euler Lgrnge eqution (13). Lemm 10 (Delt-Integrl Embedding of te Euler Lgrnge Eqution in Integrl Form). Te delt-integrl embedding of te Euler Lgrnge eqution (13) is given by 3 [L](t, x(t), [x](t)) = 2 [L](τ, x(τ), [x](τ))τ + c (14) or, equivlently, s 3 [L] T [x](t) = were c is constnt nd t T κ. 2 [L] T [x](τ)τ + c, Note tt, in te prticulr cse T = [, b] Z, eqution (14) gives te discrete Euler Lgrnge eqution L t k, x k, x k+1 x k k L = t i, x i, x i+1 x i + c, (15) v x i=0 were t i = + i, i = 0,..., k, x i = x(t i ), nd k = 0,..., N 1. Tis numericl sceme is different from (7) nd (11), nd s not been discussed before in te literture wit respect to embedding nd coerence. Tis is done in Section Te delt-vritionl embedding nd coerence Our next teorem sows tt Eq. (14) cn lso be obtined from te lest-ction principle. In oter words, Teorem 11 sserts tt te delt-integrl embedding of te clssicl Euler Lgrnge eqution in integrl form (13) nd te deltvritionl embedding re coerent. Teorem 11. If ˆx is locl minimizer or mximizer to (9) subject to te boundry conditions x() = x nd x(b) = x b, ten ˆx stisfies te Euler Lgrnge eqution (14) for some constnt c nd ll t T κ. Proof. Suppose tt L s wek locl extremum t ˆx. Let x = ˆx + ε, were ε R is smll prmeter, nd C 1 rd suc tt () = (b) = 0. We consider φ(ε) := L (ˆx + ε) = L(t, ˆx(t) + ε(t), [ˆx](t) + ε [](t))t.

7 J. Cresson et l. / Computers nd Mtemtics wit Applictions ( ) 7 A necessry condition for ˆx to be n extremizer is given by φ (ε) ε=0 = 0 Te integrtion by prts formul (1) gives 2 [L](t, ˆx(t), [ˆx](t))(t)t = ( 2 [L](t, ˆx(t), [ˆx](t))(t) + 3 [L] t, ˆx(t), [ˆx](t) [](t))t = 0. (16) t t=b 2 [L](τ, ˆx(τ), [ˆx](τ))τ (t) t= 2 [L](τ, ˆx(τ), [ˆx](τ))τ [](t) t. Becuse () = (b) = 0, te necessry condition (16) cn be written s 3 [L](t, ˆx(t), [ˆx](t)) 2 [L](t, ˆx(t), [ˆx](t))τ [](t) t = 0 for ll C 1 rd suc tt () = (b) = 0. Tus, by te Dubois Reymond Lemm (see [15, Lemm 4.1]), we ve 3 [L](t, ˆx(t), [ˆx](t)) = 2 [L](τ, ˆx(τ), [ˆx](τ))τ + c for some c R nd ll t T κ. 10. Conclusion Given vritionl functionl nd corresponding Euler Lgrnge eqution, te problem of coerence concerns te coincidence of direct embedding of te given Euler Lgrnge eqution wit te one obtined from te ppliction of te embedding to te vritionl functionl followed by ppliction of te lest-ction principle. An embedding is not lwys coerent, nd nontrivil problem is to find conditions under wic te embedding cn be mde coerent. An exmple of tis is given by te stndrd discrete embedding: te discrete embedding of te Euler Lgrnge eqution gives (7) but te Euler Lgrnge eqution (11) obtined by te stndrd discrete clculus of vritions does not coincide. On te oter nd, from te point of view of numericl integrtion of ordinry differentil equtions, we know tt te discrete vritionl embedding is better tn te direct discrete embedding of te Euler Lgrnge eqution (see Exmple 5). Te lck of coerence mens tt pure lgebric discretiztion of te Euler Lgrnge eqution is not good in generl, becuse we miss some importnt dynmicl properties of te eqution wic re encoded in te Lgrngin functionl. A metod to solve tis defult of coerence d been recently proposed in [6], nd consists in rewriting te clssicl Euler Lgrnge eqution (5) s n symmetric differentil eqution using left nd rigt derivtives. Inspired by te results of [16], ere we propose completely different point of view to embedding bsed on te Euler Lgrnge eqution in integrl form. For tt we introduce new delt-integrl embedding (see Definition 8). Our min result sows tt te delt-integrl embedding nd te delt-vritionl embedding re coerent for ny possible discretiztion (Teorem 11 is vlid on n rbitrry time scle). Acknowledgments Te second nd tird utors were prtilly supported by te Systems nd Control Group of te R&D Unit CIDMA troug te Portuguese Foundtion for Science nd Tecnology (FCT). Mlinowsk ws lso supported by BUT Grnt S/WI/2/11; Torres ws supported by te FCT reserc project PTDC/MAT/113470/2009. References [1] L. Scwrtz, Téorie des distributions, in: Publictions de l Institut de Mtémtique de l Université de Strsbourg, Nouvelle éd., in: Entièrement Corrigée, Refondue et Augmentée, vol. IX X, Hermnn, Pris, [2] K. Itô, On stocstic differentil equtions, Mem. Amer. Mt. Soc. 4 (1951) [3] J. Cresson, S. Drses, Stocstic embedding of dynmicl systems, J. Mt. Pys. 48 (7) (2007) pp. [4] J. Cresson, Frctionl embedding of differentil opertors nd Lgrngin systems, J. Mt. Pys. 48 (3) (2007) pp. [5] J. Cresson, I. Greff, Non-differentible embedding of Lgrngin systems nd prtil differentil equtions, J. Mt. Anl. 384 (2) (2011) [6] L. Bourdin, J. Cresson, I. Greff, P. Inizn, Vritionl integrtors on frctionl Lgrngin systems in te frmework of discrete embeddings, 2011, Preprint. rxiv: [7] M. Boner, A. Peterson, Dynmic Equtions on Time Scles, Birkäuser, Boston, Boston, MA, [8] B. vn Brunt, Te clculus of vritions, in: Universitext, Springer-Verlg, New York, [9] V.I. Arnold, Mtemticl metods of clssicl mecnics, in: K. Vogtmnn, A. Weinstein (Eds.), second ed., in: Grdute Texts in Mtemtics, 60, Springer, New York, 1989, (trnslted from te Russin). [10] J.E. Mrsden, M. West, Discrete mecnics nd vritionl integrtors, Act Numer. 10 (2001)

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