arXiv:1601.02602v3 [math.OC] 18 May 2016 ieti oka:Dsrt y.Nt o.21 21) r.I 80 ID Art. (2016), 2016 Soc. Nat. Dyn. Discrete as: work 28-Ap this Cite Accepted 21-Feb-2016; Revised 23-Jan-2016; Submitted speeti l h opttos uhprmtri certai is parameter Such [1 of computations. calculus the quantum all the in Indeed, present diffi 18]. some is [17, however, it presents, 4 improved [15] [3, Greff 2005 (see t H¨older fu of authors and of different calculus physics several set Cresson’s in by a developed applications further on its been to calculus ca due quantum quantum attention symmetric his attracted the 2005 and in [2], introduced calculus quantum power the u acls[8 4,Hh’ unu acls[0 9 30], 29, [10, calculus quantum Hahn’s 34], [28, calculus tum hsi rpito ae hs nladdfiiefr spbihdin published is form definite at and Society online final and available whose Nature paper in a Dynamics of Discrete preprint a is This ibeHmloinsses nes rbe ftecalculu the of problem inverse systems, Hamiltonian tiable e od n phrases. and words Key eea ye fqatmcluu r vial nteliter the in available are calculus quantum of types Several 2010 etrfrRsac n eeomn nMteaisadAppli and Mathematics in Development and Research for Center ahmtc ujc Classification. Subject Mathematics eateto ahmtc,Uiest fAer,3810-193 Aveiro, of University Mathematics, of Department iecs,w ieteascae Hamiltonian. associated the give we we form case, Precisely, Hamiltonian tive a calculus. admitting quantum equations, nondifferentiable Cresson’s of framework the Abstract. EMOT HOE O NONDIFFERENTIABLE FOR THEOREM HELMHOLTZ AITNA YTM NTEFRAMEWORK THE IN SYSTEMS HAMILTONIAN ntttd ´cnqeCeet td acldes Calcul de M´ecanique C´eleste et de Institut http://dx.doi.org/10.1155/2016/8073023 edrv h emot hoe o odffrnibeHamil nondifferentiable for theorem Helmholtz the derive We FCESNSQATMCALCULUS QUANTUM CRESSON’S OF rso’ unu acls odffrnibecluu of calculus nondifferentiable calculus, quantum Cresson’s bevtied ai,704Prs France Paris, 75014 Paris, de Observatoire [email protected] EFMF .TORRES M. F. DELFIM FR 1. ED 94,70S05. 49N45, SN 0602 Pit,IS:10-8X(Online), 1607-887X ISSN: (Print), 1026-0226 ISSN: , ´ [email protected] Introduction RCPIERRET ERIC ´ 1 fvariations. of s lto.Mroe,i h affirma- the in Moreover, ulation. ril-2016. . ecluu fvrain n has and variations of calculus he ieatermcharacterizing theorem a give ute,adi 01Cesnand Cresson 2011 in and culties, 32,8p.DI 10.1155/2016/8073023 DOI: pp. 8 73023, 4 6 n eeecstherein). references and 16] 14, , tr,icuigJcsnsquan- Jackson’s including ature, h time-scale the l icl oitrrt The interpret. to difficult nly ]ltafe aaee,which parameter, free a let 5] Eph´em´erides, ´ cin 1] hscalculus This [15]. nctions cls[1 2 3.Cresson 13]. 12, [11, lculus vio Portugal Aveiro, ain (CIDMA), cations oinssesin systems tonian aitos nondifferen- variations, q cluu 8 33], [8, -calculus 2 F. PIERRET AND D. F. M. TORRES new calculus of [17, 18] bypasses the problem by considering a quantity that is free of extra parameters and reduces to the classical derivative for differentiable functions. It is this new version of 2011 that we consider here, with a brief review of it being given in Section 2. Along the text, by Cresson’s calculus we mean this quantum version of 2011 [17, 18]. For the state of the art on the quantum calculus of variations we refer the reader to the recent book [31]. With respect to Cresson’s approach, the quantum calculus of variations is still in its infancy: see [3, 5, 6, 17, 18, 23]. In [17] nondifferentiable Euler–Lagrange equations are used in the study of PDEs. Euler–Lagrange equations for variational functionals with Lagrangians containing multiple quantum derivatives, depending on a parameter or containing higher-order quantum derivatives, are studied in [5]. Variational problems with constraints, with one and more than one independent variable, of first and higher-order