RELATIVE MOTION OF SYSTEMS IN A PARAMETRIC FORMULATION OF MECHANICS UDC Đorđe Mušicki

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1 FCT UNIVESITTIS Sees: Mechancs, uoac Conol an obocs Vol.3, N o,, pp ELTIVE MOTION OF SYSTEMS IN PMETIC FOMULTION OF MECHNICS UDC 53. Đođe Mušck Faculy of Physcs, Unvesy of Belgae, an Maheacal Insue Seban caey of Scences an s, Belgae, Yugoslava bsac. paaec foulaon of echancs, foulae by he auho hself, s base on he sepaaon of he ouble ole of e by he a of a faly of vae pahs, so ha he e as nepenen vaable eans unchange, whle as a paaee s ansfoe no a new paaee, whch epens on a chosen pah an s aken as an aonal genealze coonae. In hs pape hs paaec foulaon of echancs has been exene o he elave oon of abay heonoc syses. In hs way, he coesponng Lagangan an Halonan equaons, as well as he enegy change law fo such syses have been foulae an analse, an he obane esuls ae llusae by a sple exaple.. INTODUCTION ecenly, n he suy of he heonoc syses, V. Vujčć [-3] gave a ofcaon of he analycal echancs of such syses, wh he a o nclue he nfluence of nonsaonay consans o he laws of oon. Supposng ha hese consans always can be wen n he fo F [, f ], he nouce hs funcon f as an aonal genealze coonae q f, an on hs bass foulae an exene syse of he Lagangan an Halonan equaons, wh he aonal equaons coesponng o q. s a consequence of hese equaons, he enegy consevaon law fo such syses s obane n he fo E T U P cons, whee P, so-calle heonoc poenal, aose fo he nonsaonay consans. ffeen appoach o hs poble was gven by he auho hself Đ. Mušck [4-5] n he fo of a paaec foulaon of echancs. I s base on he sepaaon of he ouble ole of e nepenen vaable an a paaee, usng a faly of vae pahs, an on he anson o a new paaee, whch epens on he chosen pah, an whch s aken as an aonal genealze coonae. In hs way, he an geneal pncples of echancs an he enegy elaons wee foulae, an he obane esuls ae n accoance wh he coesponng ones of Vujčć, because of he foal eceve Ocobe,

2 3 Đ. MUŠICKI slay of he oles of e an he nouce paaee, bu wh ohe appoach an ffeen nepeaon. ll hese esuls efe o he absolue oon of he heonoc syses, bu s known ha he Lagangan an Halonan foals ae also applcable o he elave oon of hese syses. In hs case, he Lagangan equaons can be use ehe wh espec o an absolue fae of efeence, o ecly wh espec o a ovng nonneal fae of efeence, when all he quanes wll be elave. Ths s eonsae, fo exaple, n he faous exbooks on he heoecal physcs by Lanau an Lfschz [6] an on he heoecal echancs by P. ppell [7], whee s shown how hs eho can be apple o oban he funaenal equaon of he elave oon n he veco fo, an o suy he elave oon n boh ways hough seveal exaples. The exenson of hs applcaon of he Laganges eho o a syse of pacles n he elave oon was gven by L. Pas [8], who foulae he Lagangan of such syses n he geneal fo as a funcon of he elave genealze coonaes an he elave veloces, bu whou analysng he coesponng Lagangan equaons. Ths analyss was cae ou by. Luje [9], who sue hese Lagangan equaons n eals, an foulae he coesponng Halonan equaons on he bass of he assocae canoncal ansfoaons. In hs analyss wo ypes of hese Halonan equaons ae nouce, accong o whehe he es whch epesen he anspo an Coolss foce wee pone ou explcly o no. ecenly, M. Lukačevć an O. Jeeć [] obane he Halonan equaons fo he scleonoc syses ecly, sang fo he Pass Lagangan of he syse of pacles n s elave oon, an elnang cean foales n he Lujes wok. Naely, nsea of hs foal efnons of wo ypes of genealze oena, hese auhos efne he elave genealze oena n he usual way, as P Λ /, whee Λ s he Lagangan of a syse n s elave oon. In such a way, sang fo he Halons pncple fo he elave oon as a consane vaaonal poble, hey obane he coesponng Halonan equaons, an explane he bee fo he pon of vew of he canoncal ansfoaons, afe fnng he coesponng geneang funcon. In hs pape we shall exen hs paaec foulaon of echancs o he elave oon of he heonoc syses of pacles, an suy he coesponng Lagangan an Halonan equaons, as well as he geneal enegy change law fo such syses.. PMETIC FOMULTION OF MECHNICS Le us conse he oon of a echancal syse of N pacles wh espec o a ovng, nonneal fae of efeence XYZ, whose poson a each nsan s eene by he gven e funcons of he pole velocy an of he angula velocy ω. Suppose ha hs oon s le by k nonsaonay holonoc consans f [, ϕ ],,,..., k;,,..., N. whee we assue ha e n such consans always appeas hough cean funcon ϕ, an eene he poson of hs syse wh espec o he consee nonneal fae of efeence by a se of genealze coonaes q,,...,n, whee n 3N k. In he case of such echancal syses he e has a ouble ole: on one han s he

3 elave Moon of Syses n a Paaec Foulaon of Mechancs 3 nepenen vaable, as n he echancs n geneal, an on he ohe han, n he equaons of nonsaonay consans has he chaace of a paaee. The pncpal ea of hs paaec foulaon of echancs s base on he sepaaon of hs ouble ole of e wh he a of a faly of vae pahs, wha eans val fo he consee elave oon as well. In oe o avo any possble confuson, we shall enoe e as a paaee by p, Fg.. whle e as nepenen vaable - evoluon e whou any nex. Le us agne a faly of vae pahs of he pacles, sang fo he posons M,,...,N n he nsan Fg. an efne n he usual way,.e. vey close o he acual pah an n accoance wh he consans. These vae pahs can be eene by he equaons of oon n he veco fo, o n he genealze coonaes:, λ,,..., N q q, λ,,,..., n. whee λ s a vaable paaee. Fo each pacle M le us pesen wo possble splaceens: along he acual pah an along a vae pah, as veco sus of he coesponng eleenay splaceens n soe fne e neval, Δ. The ffeences epesen he coesponng vual splaceens δ ef, λ, λ, ξ.3 whch accong o he efnon of he vae pahs us be vey sall quanes, even fo an abay lage e neval Δ. If we wsh o pesen so efne vae pahs n genealze coonaes, he poson of he syse n he nsan wll be eene by a se q,,..., n, an n he nsan Δ on he acual pah by a se q, an on he vae pah by q,,...,n. In hs case, he evoluon of he syse along he acual an along he vae pah wll be pesene by a se of agas, pesene on he Fg., n whch he coesponng cuves q q,λ an q q,λ ξ ae close o each ohe. Theefoe, he coesponng Fg.. vaaons δq a each nsan us be vey sall quanes, wha shows cean slay wh he vae pahs n he Halons pncple. Insea of e as a paaee, le us nouce a new paaee τ, whch epens on he chosen pahs by he followng elaon, solvable fo p τ τ, λ τ, λ,.4 p p p

4 3 Đ. MUŠICKI an ean he evoluon e as nepenen vaable. Then, fo any value of hs paaee λ λ ξ, hee exss soe funcon of e τ τ p,λ ξ τ ξ p, assocae wh he coesponng vae pah. Theefoe, he values of hs paaee whch coespon o he posons of he pacle on he acual pa M an on he vae pah M a he sae nsan Δ ae uually ffeen, naely τ τ an τ τ. In oe o ajus hs paaee o he consans, le us choose hs funcon τ p,λ so ha fo λ λ,.e. fo he acual pah, conces wh he funcon ϕ p, whch appeas n he equaons of consans., an le us ake so efne quany τ as an aonal genealze coonae q τ p, λ ϕ p.5 In all he elaons whee e has he ole of a paaee, one can eplace e by so nouce paaee, whch eans he sae value along any fxe pah, acual o a vae pah. So, fo λ λ,.e. fo he acual pah, he consans. can be expesse n he fo f [, ϕ ] f [, τ]..6 p Slaly, by noucng an exene se of genealze coonaes, he Lagangan L,, wll be ansfoe no, * L L[ q, q, q, λ] L q, q,.7 whee he suaon ove he epeae nces s unesoo. In hs paaec foulaon of echancs, he oal wok of he eal eacon foces along abay vual placeens can be foun n he followng way. Sang fo he efnon of he eal eacon foces fo he holonoc syses, an fo he conon fo he vual splaceens f f δ δτ, τ whch s a consequence of.6, one obans whee s gven by δ λ f δ f λ τ δτ,..8.9 Ths quany, whch ases fo he nonsaonay consans an only fo he heonoc syses s ffeen fo zeo, epesens he genealze eacon foce coesponng o he aonal genealze coonae q. The popey ha hs wok δ fo he heonoc syses s ffeen fo zeo s an essenal chaacesc of hs paaec foulaon of echancs, n conas o s usual foulaon, whee δ.

5 elave Moon of Syses n a Paaec Foulaon of Mechancs 33 By usng hs quany, he nfluence of nonsaonay consans can be nclue n he geneal pncples of echancs an he coesponng ffeenal equaons of oon. So, sang fo he funaenal equaon of ynacs n espec o an absolue fae of efeence, an beang n n.8, one can oban he lebe-laganges pncple n hs paaec foulaon as [4] * F a δ δτ,. * whee s he noneal eacon foce, an a he acceleaon fo he -h pacle. Ths pncple can be also expesse n he genealze coonaes, pung δ / δq, an ecoposng he genealze foces no he poenal an * nonpoenal ones Q U / Q. In hs way, by goupng he sla es, can be pesene n he fo * Q δq,. ~ whee L an Q * ae gven by ~ * * * q, q L P T U P, Q Q L. * * Hee T an U ae he knec an he poenal enegy of he syse, Q an he genealze nonpoenal acve an noneal eacon foces especvely, an P s efne by ef P P q..3 q Snce all he vaaons δq ae nepenen, fo. follows eaely ha all he expessons beween paenhess us be equal o zeo, an heefoe Q *,,,..., n.4 These ae he coesponng Lagangan equaons n hs paaec foulaon of echancs, he nube of whch s n, an so nouce Lagangan L hough he e P copses he nfluence of nonsaonay consans. The quany P, efne by.3 was nouce by Vujčć [] n hs ofcaon of he echancs of heonoc syses, an nae he heonoc poenal. I can be obane by pevous fnng, ehe eaely fo s efnon.9 afe eenng he ulples λ n he usual way, o fo he Lagangan equaons.4, by fnng he soluons q q,,...,n of he fs n of he an nseng hose ones no he las Lagangan equaon. In hs way, one obans hs quany n he fo Fq,, an pung hee hese soluons fo q, wh he a of.4 becoes cean funcon only of q, hen P can be foun by negaon.3, also as soe funcon of q.

6 34 Đ. MUŠICKI 3. LGNGIN OF SYSTEM IN ITS ELTIVE MOTION Le us exane how one can apply he Lagangan equaons.4 n hs paaec foulaon of echancs ecly o he elave oon of a echancal syse of pacles wh espec o a ovng, nonneal fae of efeence. Fs, we us effecuae he ansfoaon of he coonaes fo an absolue fae of efeence o hs nonneal one, whch epesens a puncual ansfoaon. Snce he fo of he Laganges equaons eans nvaan n any ansfoaon of hs kn, hese Lagangan equaons can be ulze ecly n he sae fo n he consee elave oon of he syse. Then, all he quanes an he changes us be eae as he elave ones,.e. n a way as hey ae seen by an obseve n hs nonneal fae of efeence, as n he usual foulaon of echancs, whch wll be enoe hee by, fo exaple. In hs a, he coesponng Lagangan of hs syse us be expesse as a funcon of he elave coonaes an elave veloces of he pacles, as well as of he gven e funcons an ω, whch eene he poson of hs nonneal fae of efeence n each nsan. In oe o oban so foulae Lagangan, le us sa fo hs Lagangan n an neal syse of efeence, an subsue he absolue veloces of he pacles by he su of he coesponng elave an anspo veloces ω el 3. Then, hs Lagangan n ou paaec foulaon, accong o. wll be P U P U ] [, L 3. The su of he fouh an he ffh e, on he bass of a geneal elaon beween he absolue an he elave evave of any veco funcon, 3.3 whee he sybol / enoes he elave evave, can be ansfoe no ] [ Hee all he es whch can be expesse as he oal e evave of any funcon can be oe, snce all he Lagangans whch ffe by a oal e evave ae equvalen. In hs way, afe cean encal ansfoaons, ulzng he known popees of he veco an xe veco poucs, hs Lagangan obans he fo,,, P U a ω ω ω L 3.4

7 elave Moon of Syses n a Paaec Foulaon of Mechancs 35 whee a s he acceleaon of he pole. In oe o pon ou he physcal eanng of soe es n hs expesson, le us ecopose n he followng way L, T ω L Π P 3.5 whee T, L an Π ae gven by T, Π U, a c L J ω, J ω 3.6 an he heonoc poenal P, specfc fo hs paaec foulaon of echancs, s efne by.3. So foulae T s he elave knec enegy, L he elave angula oenu of he consee syse, an J s he elave oen of nea of hs syse n s oaon aoun he axs hough he pole, all hese quanes copue wh espec o he nonneal fae of efeence n he nsan. The secon an he h e n Π have he chaace of cean poenal enegy, beng epenen only on he vaable, he fs of he asng fo he anslaon, an he secon fo he oaon of hs fae of efeence, so-calle cenfugal poenal enegy. Ths Lagangan of a syse n s elave oon ffes fo he coesponng one n he usual foulaon of echancs, obane by L. Pas [8], only by he specfc e P fo hs paaec foulaon of echancs, whch expesses he nfluence of he nonsaonay consans o hs poble. Ou poof epesens a genealzaon of he coesponng one, pesene n he veco fo by Lanau an Lfschz [6] o a syse of pacles, n accoance wh he ce Pass esul, exenng o ou paaec foulaon of echancs. 4. LGNGIN EQUTIONS FO THE ELTIVE MOTION Now, le us expess hs Lagangan n he genealze coonaes q,,,...,n, whch eene he poson of hs syse wh espec o he nonneal fae of efeence, n he sense of hs paaec foulaon of echancs. In hs a, pung / q, he fs e n 3.5,.e. he elave knec enegy of he syse can be pesene as β T βq q,, β,,,... n 4. whee he coesponng ec enso n hs nonneal fae of efeence s gven by β β 4. In a sla way, he secon e n 3.5 becoes a lnea funcon of q

8 36 Đ. MUŠICKI ω L B ω q, B, 4.3 an he es Π an P ae cean funcons only of he genealze coonaes q Π Π q,,,..., n, P P q 4.4 Theefoe, hs Lagangan n he genealze coonaes wll be pesene as a quaac funcon of he vaables q β L q, q βq q Bq Π q P q 4.5 Snce he coesponng Lagangan equaons.4 n ou applcaon ae eae ecly wh espec o he nonneal fae of efeence, hey us be use n he fo ~ * Q,,,..., n 4.6 These equaons, accong o he fo, conce wh he ones obane by M. Lukačevć an O. Jeeć [], bu n hs case he nube of hese equaons an he oan of he valy ae geae han n he usual foulaon, an hey have que ffeen eanng. Ou esuls efe o he oe geneal, heonoc syses wh abay nonsaonay consans, an ae gven n ou paaec foulaon of echancs, wh an aonal Lagangan equaon, wha wll be of specal nees fo he coesponng enegy elaons. If we wan o pon ou explcly he es specfc fo he elave oon,.e. he neal foces whch epesen he so-calle anspo an Cools s ones, le us we he Lagangan 4.5 as L q, q L L P, 4.7 whee L T U, Z ω L c a J ω 4.8 ll he necessay paal evaves of so ecopose Lagangan can be foun beang n n ha L epens on q an q hough all he vaables v an v v,,...,n especvely. In hs way, afe fnng he coesponng paal evaves on he bass of 3.4, by nseng he obane expessons no he Lagangan equaons 4.6, an goupng he sla es, hese equaons can be pesene n he fo ~ * co Q Q Q δ,,,,..., n co whee Q an Q ae gven by Q F [ a ] co co Q F

9 elave Moon of Syses n a Paaec Foulaon of Mechancs 37 co These quanes Q an Q epesen he genealze foces whch coespon o he esulans of all he anspo an Cools s foces, acng on he pacles of he syse n hs nonneal fae of efeence, whle he quany s a chaacesc of hs paaec foulaon of echancs. The obane esuls ae n accoance wh he ones foulae by. Luje [9], p , bu gven hee n a ffeen fo an wh annohe appoach an nepeaon of hese esuls, accong o ou paaec foulaon of echancs. 5. HMILTONIN EQUTIONS FO THE ELTIVE MOTION The anson fo he Lagangan o he coesponng Halonan foals fo he consee elave oon of a syse of pacles n hs paaec foulaon can be cae ou n a sla way as n he usual foulaon. Naely, le us nouce he genealze oena by ef p,,,,..., n 5. he nube of wch s n, whee L s he coesponng Lagangan gven by 4.5. Then, he genealze oena wll have he fo β p βq B,, β,,,..., n 5. fo whch, afe ulplyng hese equaons by he conjugae ec enso γ, sung ove epeae nex, an applyng he popey β γ γ δ β, follows γ γ q p B, β,,,..., n 5.3 Snce he ax β fo he egula echancal syses s nonsngula, he ax γ always exss, an heefoe he syse of equaons 5. can always be solve fo he β vaables q, wha allows us he anson o he Halonan foals. The coesponng canoncal, Halonan equaons can be obane as usually, fo exaple sang fo he vaaon of he Lagangan, applyng he Lagangan equaons 4.6, an passng fo he vaables q o he genealze oena, wha can be wen as ~ * δ pq L Q p δq q δp 5.4 Theefoe, he expesson n he paenhess can be consee as a funcon of he vaables q an p, fo whee by copason wh he vaaon of hs expesson one obans p H ~ * Q, p whee he Halonan s gven by H q, p q H, p,,,..., n 5.5 p q L q, q 5.6 Thee ae he coesponng Halonan equaons n hs paaec foulaon of echancs, wh n nepenen ones. Ths Halonan can be foe by

10 38 Đ. MUŠICKI elnang he genealze veloces by eans of 5.3, whch s fo he egula syses always possble. In hs case we can also ephasze he es chaacesc fo he elave oon, slaly as n he Lagangan equaons. Theefoe, we us nouce he genealze oena assocae o L T U ef p T U,,,,..., n 5.7 an ecopose he Halonan no hee pas H q, p H H P, 5.8 whee H p q L, H q L 5.9 Then, afe fnng he coesponng paal evaves, an nseng he no he Halonan equaons 5.5, by goupng he sla es hese equaons can be pesene n he fo whee Q an p q H ~ Q H, p co Q ae gven by 4.. * Q Q co,,,..., n δ 5. The physcal eanng of hese Halonans can be peceve n a sla way as fo he usual Halonan. Naely, on he bass of 4.5 an 5. we have β β H q, p βq B q βq q Bq Π P an, afe cancelng he lnea es wh espec o q, one obans ex H q, p E Π P. 5. Thus, he Halonan of a syse of pacles n s elave oon wh espec o a nonneal fae of efeence epesens he genealze elave echancal enegy of hs syse, exene by he heonoc poenal P, whch ases fo he nonsaonay consans. Howeve, he eanng of he Halonan H efne by 5.9, s sple, snce hee all he es chaacesc fo he elave oon cancel ou, so ha hs Halonan epesens he elave oal enegy E T U of he syse, exene also by he heonoc poenal P. The obane esuls an equaons have he sae fo as he ones pesene by. Luje [9], an n he ce pape by M. Lukačevć an O. Jeeć [] n he usual foulaon of echancs. Bu, hee hese equaons epesen an exene syse of Halonan equaons, wh n of he, an hey have que ffeen eanng n hs paaec foulaon of echancs, nclung he nfluence of he nonsaonay consans, whch s absen n he usual foulaon of echancs.

11 elave Moon of Syses n a Paaec Foulaon of Mechancs ENEGY CHNGE LW FO THE ELTIVE MOTION In oe o exane he coesponng enegy elaons n he elave oon of a syse of pacles wh espec o a nonneal fae of efeence, le us fs pesen he enegy change law n he usual foulaon of echancs, n he veco fo an n he genealze coonaes. a In hs a, le us sa wh he funaenal equaon of he elave oon fo each pacle of he syse,,...,, * N a a F co 6. n whch he eacon foces ae ecopose no he eal an he noneal ones, whle he anspo an Cools s acceleaons ae gven by, co a a a 6. If we ulply hese equaons by, an su ove he epeae nex, one obans he coesponng knec enegy law * a a F T co. 6.3 The secon e epesens he eleenay oal wok of he eal eacon foces along he possble splaceens, whch s of specal nees fo hs analyss of he enegy elaons, an can be foun n he followng way. By applyng he conon fo hese splaceens f f, whch s a consequence of., hs wok wll be equal o f f λ λ, 6.4 whch s ffeen fo zeo only fo he syses wh nonsaonay consans. The ohe es can be pesene n a oe suable fo, ecoposng he acve foces no he poenal an he nonpoenal ones, an copung he eleenay woks of all hese foces, ulzng he known ules n he veco calculus, fo exaple ] [ ω ω ω ω In hs way, he knec enegy law can be ansfoe o * * λ Π F f U T E, 6.5 whee Π s gven by 3.6.

12 3 Đ. MUŠICKI Ths s he geneal enegy change law fo he elave oon n he usual an veco foulaon, whee he secon e on he gh-han se expesses he nfluence of he * * nonsaonay consans. I U /, F, an ω cons, fo hee follows ha f T U λ cons, 6.6 wha can be nepee as a kn of he enegy consevaon law.ths enegy change law epesens soe genealzaon of he sla, bu usually sple law fo he elave oon see f.e. [], p. 86- o he syses wh abay nonsaonay consans. Ths negal fo of he enegy consevaon law, oppose o he usual foulaon, conans a e asng fo hese consans. b Le us sa fo he Lagangan equaons fo he elave oon n he usual foulaon [] Q ~ *,,,..., n 6.7 whee L q, q, ω L U c a J ω, 6.8 ulply by q q, an su ove he epeae nex. In hs way, afe ansfong he fs e, one obans L L L L q q q q q q an beang n n ha L q q, he pevous elaon can be wen as E L L ~ q L Q * q q, 6.9 Snce he elave knec enegy of a syse of pacles, accong o s epenence on q, n he geneal case has hee es: T T T T, he expesson n paenhess on he bass of 6.8 an Eules heoe s equal o E q L T T ~ Q * q Π. 6. Ths fo of he enegy change law ffes fo he pevous one 6.5, hee he nfluence of he nonsaonay consans s absen, an heefoe he coesponng consevaon laws ae also ffeen. Une he conon ha all he es on he ghhan se ae equal o zeo, n he fs case we have E T Π cons, an n he secon one E T T U cons analogous o he Jacobs enegy negal.

13 elave Moon of Syses n a Paaec Foulaon of Mechancs 3 c Howeve, f we sa fo he exene syse of Lagangan equaons n hs paaec foulaon of echancs, anohe enegy chage law wll be obane, as s shown by V. Vujčć [] n hs ofcaon of he echancs of heonoc syses, bu wh a que ffeen nepeaon. In ou case, le us sa fo he coesponng Lagangan equaons 4.6 fo he elave oon, ulply he by q q, an su ove he epeae nex, wha can be wen as q q q ~ Q * q, 6. whee he Lagangan L s gven by 4.5. Is oal ffeenal n hs case wll be L q q, hee he explc epenence of he Lagangan on e s nclue n he fs e hough he vaable q, so ha by goupng he sla es, hs elaon obans he fo ex E ~ * q L Qq, 6. Havng n n he physcal eanng of he Halonan 5., can be pesene explcly as ex E ~ * T Π P Qq, 6.3 whee, accong o 3.6 Π U c a J ω, 6.4 Ths s he coesponng geneal enegy change law fo he elave oon of a syse of pacles wh espec o a nonneal fae of efeence n hs paaec ~ foulaon of echancs. Fo hee s even ha f * Qq,.e. f he effec of all he genealze nonpoenal acve an noneal eacon foces s equal o zeo, he enegy consevaon law s val n he fo E ex T Π P U a c J ω P cons, 6.5 Ths enegy change law an he coesponng consevaon law ffe fo he ones n he pevous case. Howeve, n essence hey ae equvalen o he coesponng laws n he usual veco foulaon 6.5 an 6.6, snce he las e n he secon elaon accong o.9 an.3 s equal o he heonoc poenal P fo q f P λ q. 6.6 In hs way, as n he case of he absolue oon, fo he pon of vew of he enegy

14 3 Đ. MUŠICKI elaons hs paaec foulaon of echancs s equvalen o he aseble of he usual Lagangan foals an he law fo he wok of he eal eacon foces, whch s no conane n hs usual foulaon. 7. N EXMPLE Le us llusae he obane esuls by he elave oon of a pacle wh espec o a nonneal fae of efeence XYZ, whch oaes wh a consan angula velocy ω ϕ/ aoun he Z-axs of a oble fae of efeence OXYZ wh he sae ogn of coonaes Fg. 3. Le hs pacle ove n he consan Eah s gavaonal fel along an nclne sooh lne OB, aache o hs nonneal fae of efeence, an splacng unfoly wh he velocy V along he Y-axs. If we enoe by he angle beween he econ of he oon of hs pacle an he Y-axs, hen n each nsan us be Fg.3 z g V y an heefoe, hs oon s esce by a nonsaonay consan f y, z, V ysn zcos, f x x. 7. The oon of hs pacle has one egee of feeo, an ake fo he genealze coonae q ξ, whch eenes he poson of he pacle a each nsan wh espec o hs nonneal fae of efeence XYZ. Le us choose fo a paaee,.e. an aonal genealze coonae q V, pesene on he fgue, hen he elaons beween hese genealze an ecangula coonaes ae x, y q ξcos, z ξsn 7. In hs case he elave angula oenu L s noal o he plane OOB, an hus also noal o he ω, so ha ω L, an snce he pole whch conces wh he ogn of coonaes s oble, s acceleaon s a. Theefoe, he Lagangan of hs pacle n he consee elave oon, accong o 3.5 an 3.6 s euce o L, T Π P T U J ω P 7.3 The elave knec an poenal enegy of hs pacle wh espec o hs nonneal fae of efeence, expesse n he choosen genealze coonaes ae T x y z ξ q ξ q cos 7.4 U gz gξsn,

15 elave Moon of Syses n a Paaec Foulaon of Mechancs 33 an so-calle cenfugal poenal enegy, conanng he elave oen of nea of hs pacle n s oaon abou Z-axs, accong o 7. s J ω x y ω q ξcos ω 7.5 In hs anne, he pevous expesson fo he Lagangan wll be pesene as L q, q ξ q ξ q cos gξsn q ξcos ω P 7.6 whee he heonoc poenal P can be foun fo he Lagangan equaons. In ou case, he coesponng exene syse of Lagangan equaons 4.6 has only wo equaons, an snce Q * an *,, we have, ξ ξ 7.7 o n explc fo, snce accong o.3 P/q ξ q q cos g sn q ξ cos q ξcos cos ω ξcos ω. 7.8 In oe o fn, le us elnae ξ / fo hese equaons, havng n n ha q V cons. The fs of hese ones gves ξ g sn cos q ξ ω cos an afe nseng hs expesson no he secon equaon, we ge cos ξ q g sn cos q ξcos ω ξcossn ω. 7.9 Bu, n a o oban as a funcon only of q, one us fn he soluon ξ ξ of he fs equaon 7.8 an nse no hs expesson fo. Ths equaon explcly has he fo o, oe concsely whee ξ cos ω ξ g sn V cos ω ξ k ξ a b, 7. k cos ω, a V cos ω, b g sn 7. The geneal soluon of hs ffeenal equaon s equal o he su of such soluon of he coesponng hoogeneous equaon an one pacula soluon of hs ffeenal

16 34 Đ. MUŠICKI equaon ξ C ξ sh k C p, whee C an C ae he consans of negaon, eene by he nal conons. If we seek one pacula soluon n he fo ξ p c, hese consans can be foun fo he conon ha hs expesson us sasfy he ffeenal equaon 7., an so hs geneal soluon obans he fo a b ξ Csh k C. 7. k k By nseng no 7.9, hs quany becoes a funcon of e a b sn cos ω Csh k C k k g sn cos sn ω V, an snce q V, hs expesson can be pesene also n he fo of a funcon of q k Bq Csh q C, 7.3 V whee because of he concseness he coesponng consans ae nouce. Then, he heonoc poenal P can be obane, accong o.3, by negaon wh espec o q k P q q Bq C ch q C, 7.4 v whee C CV / k. Ths esul epesens a genealzaon of he coesponng one obane by Vujčć [], expane hee o he elave oon of a pacle. Fo hee one can see ha n he case of he elave oon he heonoc poenal can epen also on he quanes chaacesc fo such oon, as he angula velocy ω of he nonneal fae of efeence. In he consee case all he conons fo he valy of he enegy consevaon law ae sasfe, so ha accong o 6.5 we have ex E T Π P U J ω P cons 7.5 If we subsue hese quanes by he obane expessons, hs enegy consevaon law ges he followng explc fo ex E ξ q ξ q cos gξsn q ξcos ω 7.6 k q Bq Cch q C cons V Ths enegy negal n he paaec foulaon of echancs ffes fo he coesponng one fo he absolue oon n he usual foulaon by he cenfugal poenal enegy an by he heonoc poenal, he fs beng specfc fo he elave oon, an he secon one fo hs paaec foulaon.

17 elave Moon of Syses n a Paaec Foulaon of Mechancs 35 EFEENCES. Vujčć V.: The ofcaon of analycal ynacs of heonoc syses, Tenso N.S., vol. 46, 987, p Vujčć V.: On Halons pncple fo he heonoous syse, Bull. ca. Seb. Sc. s, Cl. Sc. ah., no 6, 988, p Vujčć V.: Dynacs of heonoc syses - onogaph, Maheacal Insue, Belgae 99, pp Mušck Đ.: paaec foulaon of echancs of heonoc syses, Theo. an ppl. Mechancs Belgae, vol. 8, 99, p Mušck Đ.: Consevaon laws an Noehes heoe n a paaec foulaon of echancs, Theo. an ppl. Mechancs Belgae, vol. 9, 993, p Lanau L. an Lfschz E.: Theoecal physcs, oe I Mechancs, Iz. "Nauka", Moscow 973, p n ussan. 7. ppell P.: Taé e écanque aonnelle, oe II, Gauhe Vllas, Pas 953, p Pas L.: ease on analycal ynacs, Heneann, Lonon 964, p Luje.: nalycal echancs, Fz. Ma. Gz., Moscow 96, p , n ussan. Lukačevć M. an Jeeć O.: On he Halons equaons fo elave oon, Theo. an ppl. Mechancs Belgae, vol., 995, p Olhowsky J.L.: Couse of heoecal echancs fo physcss, Iz. Moskov. Unv., Moscow 974, p. 86- n ussan. ELTIVNO KETNJE SISTEM U PMETSKOJ FOMULCIJI MEHNIKE Đođe Mušck Paaeaska foulacja ehanke, foulsana o saog auoa, zasnva se na azvajanju vosuke uloge veena za eonone ssee nezavsno poenljva paaea pooću zvesne falje vaanh puanja. P oe vee kao nezavsna poenljva osaje nepoenjeno, ok se ueso veena kao paaea uvo nov paaea, koj zavs o zabane puanje z ove falje uza se kao opunska genealsana koonaa. U ovo au ova paaeaska foulacja ehanke pošena je na elavno keanje pozvoljnh eononh ssea. Na aj načn, foulsan su analzan ogovaajuć pošen sse Lagange-ovh Halon-ovh jenačna, kao opš zakon poene enegje za ovakve ssee, a objen ezula su lusovan jen pos peo.