ON THE LAGRANGIAN RHEONOMIC MECHANICAL SYSTEMS

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1 THE PUBISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Vlume 1, Number 1/9, pp. ON THE AGRANGIAN RHEONOMIC MECHANICA SYSTEMS Radu MIRON*, Tmak KAWAGUCHI**, Hrak KAWAGUCHI*** * Rmaa Academy ** Presdet f Tesr Scety, Tky, Japa *** Tesr Scety, Japa Crrespdg authr: Radu MIRON, e-mal: radu.mr@uac.r Oe defes the t f Rhemc agraga Mechacal system Σ = (M,, F e) where = ( M, ) s a tme -depedet agrage space ad F e are the exteral frces. The evlut equats f are the agrage equats (17). The mst mprtat result s gve by the fllwg Therem: there exsts a cacal semspray S, determed ly by the system, whse tegral curves are evlut curves f. S s a dyamcal system the phase space Rx TM. The gemetry f the par (Rx TM, S) s the Gemetry f agraga rhemc mechacal system. Key wrds: agraga rhemc mechacal system; semspray; lear cect. 1. INTRODUCTION The prblem f gemetrzat f classcal -cservatve mechacal systems s a ld e [1, 6, 14]. Essetal ctrbuts t slved ths prblem have bee de by R. Abraham ad J. Marsde [1], J. Kle, F. Pra, M. de e, O. Krupkva [11], R. Mr ad M. Aastase [9,11], V. Vukvc [14], K. (Stevavc) Hedrh [6] et alt. Recetly, t the 4th Sympsum Fsler Gemetry, rgazed by H. Shmada ad S. Sabau Sappr 5, the frst authr f the preset paper slved the prblem fr the sclermc -cservatve mechacal system [11]. Wth ths ccas have bee defed the Fslera ad agraga mechacal systems ad has bee determed the cacal evlut semspray fr such systems. I the preset paper we csder rhemc cservatve mechacal systems by usg the gemetrcal thery f tme-depedet agrage spaces realzed by M. Aastase ad H. Kawaguch, []. I the partcular case f the rhemc Fslera mechacal systems we refer t the paper f C. Frgu [5]. Nw, we study the rhemc agraga mechacal systems: Σ = ( M, ( t, x ), F ( t, x e )) where M s the cfgurat space, ( t, x ) s a tme-depedet regular agraga ad F e ( t, x ) are the exteral frces. We fd the cacal semspray S f where ts tegral curves are the evlut curves f. Therefre: the gemetry f the vectr feld S the phase space Rx TM s the gemetry f the mechacal system. But S s a dyamcal system Rx TM determed ly by system. S we ca study the glbal prpertes f such as the stablty f sluts curves f the evlut equats f.

2 Radu MIRON, Tmak KAWAGUCHI, Hrak KAWAGUCHI. RHEONOMIC AGRANGE SPACES. PREIMINARIES et M be a real -dmesal smth mafld ad ( TM, M ),π ts taget budle. Csder the mafld E=Rx TM wth lcal crdates ( t, x, y ), (,, k, = 1,,,). These crdates trasfrm by rule: wth: Usually, we csder ~ ~ 1 x = x ( x,..., x ), Φ () t = at+ b, a x = x y y, t = Φ() t (1) x dφ rak ( ) =,. () x dt. The a rhemc agrage space [] s a par: = ( M, ( t, y)) whch (t, y) s a tme-depedet regular agraga. The fudametal tesr f the space s as fllws: 1 g (t, y) =, (3) assumg that g has a cstat sgature ad t s verfed the cdt: Therefre we ca csder the ctravarat tesr g. rak( g ) =. (4) Nw let us vestgate the tegral f act f the agraga alg a smth curve c:,1 M : It leads t the Euler-agrage equats []: These equats are equvalet t: 1 dx Ic () = (, τ x(), τ ) dτ dτ d ( ) =, x dτ [ ] dx y =. (5) d τ where: d x dx dx G x G x + ( τ,, ) + ( τ,, ) = dτ dτ dτ 1 k dx G = g ( y ), y =, k k x x dτ G = 1 g k k t (6) (7) Clearly, G s a d-vectr feld []. Wth respect t (1) the equats (5) ad (6) have a gemetrcal meag. et us csder the vectr feld S E = Rx TM:

3 3 O the agraga rhemc mechacal systems S = y ( G + G ) x (8) The we have []: Therem.1. The fllwg prpertes hld: 1. S s a semspray the mafld E.. S depeds the agrage space ly. 3. The tegral curves f S are gve by Euler-agrage equats f. Csequetly S s the dyamcal system the phase space E f the tme-depedet agraga. Csder the eergy f agraga : E = y, (9) ad the Pcaré 1-frm: ω = dx E dt (1) Applyg the exterr peratr d f dfferetat we bta the Carta -frm: A vectr feld X E wth the prperty X Oe prves [, 9]: θ = dω = d dt { dx y dt} x θ = s called characterstc fr the -frm θ. Therem.. The sem-spray S s a characterstc vectr feld fr the Carta -frm θ.,...,, 1 t Remarkg that the system f vectr felds { } E we ca csder a splttg f the taget space Tu E: u u u (11) determe the vertcal dstrbut V T E = N V, u E. (1) Therefre the hrztal dstrbut N s a lear cect the mafld E ad the adapted bass f δ the drect decmpst (1) s (,, ), where: δx t The par N, N (, ) δ = N N δx x t. (13) s the system f ceffcets f the lear cect N. Therem. 3. Gve the semspray S f the rhemc agrage space wth the ceffcets G G frm (7) the there exsts a lear cect N determed ly by. The ceffcets f N are expressed by:

4 Radu MIRON, Tmak KAWAGUCHI, Hrak KAWAGUCHI 4 N G =, N = g G. (14) y Remark. I the partcular case f tme-depedet Fsler space gemetry f rhemc Fsler spaces []. = F the prevus thery leads t the 3. RHEONOMIC AGRANGIAN MECHANICA SYSTEMS A rhemc agraga mechacal system s a trple: Σ= ( M, txy (,, ), F( txy,, )), (15) e where = (M, (t, y)) s a rhemc agrage space ad F e are the exteral frces, a prr gve as a vertcal vectr feld, the frm: Fe (, txy, ) = F(, txy, ) Csequetly, F (t, y) s a d-vectr feld the mafld E. Pstulate: The evlut equats f the mechacal system the phase space E = TMxR are the fllwg agrage equats: wth: (16) d dx = F, y =, (17) x dt dt F (t, y) = g F (t, y). (18) The prevus d-cvectr feld s the d-cvarat vectr f exteral frces. Obvusly, f the tme t des t explctly eters the system we bta the sclermc cservatve mechacal system, studed by the frst authr the paper [11]. Tw remarkable partcular cases are gve by: a. s a Remaa (r pseud-remaa) space whe Σ s the classcal cservatve mechacal systems ad (17) are ts evlut equats. b. s a tme-depedet Fsler space F = ( M, F( t, y)) studed by M. Aastase ad H. Kawaguch [, I, II, III]. The s a rhemc Fslera mechacal system. Mre geeral, whe s a tme-depedet agraga we have the rhemc agraga mechacal system ad whe stead f we csder the geeralzed agrage space G = (M, g (t, y)) the we have geeralzed rhemc mechacal systems, [,1,11]. Returg t the rhemc agraga mechacal systems (15) we remark that the agrage equats (17) are equvalet t the system f dfferetal equats: where: d x dt 1. G ( t, x) G ( t, x. x) F ( t, x), (19) = 1 k 1 k G = g ( y ), G k k = g k x x t Therefre the gemetry f a rhemc agraga mechacal system Σ s the gemetry f the semspray S whse the tegral curves are gve by the equats (19) ad (). ()

5 5 O the agraga rhemc mechacal systems Example 3.1. The rhemc agraga mechacal system f tme-depedet electrdyamcs s gve by the agraga: e txy (,, ) = mcg (, txxx ) + A (, txx ) + Utx (, ) (1) mc where m, c, e are the kw physcal cstats, g (t, x) are the tme depedet gravtatal ptetals, A (t, x) are the electrmagetc tme depedet ptetals ad U(t, x) s a tme depedet ptetal fuct, []. The evlut equats (17) ca be wrtte wthut dffculty. I ths case the exteral frces ca be gve by F = h(t)y, h(t) beg a fuct depedg by t ly. Returg t the geeral thery f the mechacal system, we remark the gemetrcal meag f the agrage equats (17) r (18) whch ca be easly demstrated. The mst mprtat result the rhemc agraga mechacal systems s gve by the fllwg therem: Therem 3.. The fllwg prpertes hld: 1. There exsts a semspray S the phase space E=RxTM depedg ly the rhemc mechacal system frm (15).. S s gve by: 1 S = y ( G + G ) + F x () 3. The tegral curves f semspray S are gve by the agrage equats (17). Prf. Wrtg S the frm: 1 S = S+ F y, (3) where S s the cacal semspray f rehmc agrage space all prpertes expressed the prevus Therem ca be prved wthut dffcultes. Frm these reass we ca call S the cacal semspray f rhemc mechacal system. It ca be develped by the same methds as the sclermc case, [11]. REFERENCES 1. ABRAHAM, R., MARSDEN, J., Fudats f Mechacs, Beam, New-Yrk, ANASTASIEI, M., KAWAGUCHI, H., A gemetrcal thery f tme depedet agragas: I, Nlear cects. Tesr, N. S., 48, pp. 73-8, 1989, II. M-cects, Tesr, N. S., 48, pp , 1989, III, Applcats. Tesr, N. S., 49, pp , ANTONEI, P.., (Ed): Hadbk f Fsler Gemetry, Kluwer Acad. Publ., BUCATARU, I., MIRON, R., Fsler-agrage Gemetry. Applcats t Dyamcal Systems, Ed. Academe Rmae, FRIGIOIU, C., agraga gemetrzat Mechacs, Tesr, N. S., 65, pp. 5-33, HEDRIH (Stevavc), K., Rhemc crdate methd appled t lear vbrat systems wth heredtary elemets, Facta Uverstats, 1, pp ,. 7. KAWAGUCHI, A., O the thery f lear cects I, II, Tesr, N. S.,, pp , 195, 6, pp , MIRON, R., Cmpedum the Gemetry f agrage Spaces. I Dlle F. J. T.ad. C. A. Verstraele (Eds), Hadbk f Dfferetal Gemetry, Vl. II, 6, pp MIRON, R., ANASTASIEI, M., The Gemetry f agrage Spaces: Thery ad Applcats, Kluwer Acad.Publ., FTPH 59, MIRON, R., KAWAGUCHI, T., Relatvstc gemetrcal Optcs, Iteratal Jural f Theretcal Physcs, 3, pp , 1991.

6 Radu MIRON, Tmak KAWAGUCHI, Hrak KAWAGUCHI MIRON, R., Dyamcal Systems f agraga ad Hamlta Mechacal Systems, Advaced Studes Pure Mathematcs 48, Fsler Gemetry, Sappr, 5, Eds. H. Shmada ad S. Sabau, pp , MIRON, R., HRIMIUC, D., SHIMADA, H., SABAU S. V., The gemetry f Hamlt ad agrage Spaces, Kluwer Acad. Publ., FTPH 118, de EON, M., RODRIGUES, P. R., Methds f Dfferetal Gemetry Aalytcal Mechacs, Nrth-Hllad, VUJICIC, V. A., HEDRIH (Stevavc) K., The rhemc cstrats chage frce, Facta Uverstats, 1, pp , Receved Octber 1, 8