Representation of Hamiltonian Formalism. in Dissipative Mechanical System

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1 Ale Mahemacal ceces Vol o Rereseao of amloa ormalsm Dssave Mechacal ysem M. aer Al Bswas Tm ahra aue M. Elyas Karm a M. Ashkur Rahma Mahemacs Dscle Khula Uversy Khula-908 Baglaesh mhabswas@yahoo.com Absrac amloa mechacs s he roo of classcal mechacs. amloa fuco s he mofe verso of agraga fuco whch s of he frs orer ffereal euaos wh geeralze cooraes geeralze momeum a me. o amloa formulaos lay a mora role classcal mechacs as well as mechacal sysems. I hs suy we have esablshe amloa formalsm for ssave sysem. We have emosrae ha wheher he class of ssave mechacal sysem has a aalycal soluo or o ca be reresee as a amloa formalsm. Dssave sysem eals wh he Damg force Mechacal eergy Prcle of leas aco rs egral Jacoba marx a he No-coservave sysem eals wh racoal ervaves amloa sysems o-coservave sysems a alace rasform. Mahemacs ubec Classfcao: 9 Keywors: Classcal mechacs agraga fuco amloa fuco Nocoservave sysem Jacoba marx alace rasform. Irouco amloa mechacs frs rouce by Wllam Rowa amlo 833 s oe of he mos esseal ools of classcal mechacs []. I s reae as he reformulao of classcal mechacs whch s eveloe form of agraga mechacs classcal mechacs. I s observe ha he agraga a he amloa formulaos ca be cosruce for ffere ks of Dssave a No-coservave sysems []. amlo orgae amlo euaos of moo a amloa formulao. I 960s or a Brouwer ulze he classcal amloa formalsm a a

2 93 M. aer Al Bswas e al Perurbao heory o solve o-coservave roblem [6]. They o aem o erve he amloa formalsm of o-coservave roblem [9]. I 990s Tveer mae such a aem a obae a so-calle geeral amlo euao: k r P Q Q k r Q Q P Q P () Q Q where { Q P} are caocal varable a r s he oso vecor ees o he caocal varable se { Q P} a.e. r { Q P }. K s he rasforme amloa; he subscr he aral ervave exressos caes he fucoal eeecy of K a r. If he varable se {Q P} s rasforme o he varable se { } where he oso vecor r ees o a oes o ee exlcly o.e. r () Euao () ca be reuces o r Q Q () r Where eoes geeralze force reco s he amloa. Ye boh euao () a euao () are o amloa formalsms because here s o a coservave amloa uay (frs egral). The resulg amloa s u-hyscal: s uboue from below a uer me reversal he oscllaor s rasforme o s mrrormage. By hs arbrary rck ssave sysems ca be hale as hough hey were coservave. Vuaovc B.D. vesgae ssave sysems from he vso of varaoal mehos. Tarasov suggese a geeralzao of caocal uazao ha mas a yamcal oeraor o a yamcal sueroeraor. Tarasov clame ha hs aroach allows efg cosse uazao roceure for o-amloa a ssave sysems. Keh cosere ha ssao effecs may be clue by coserg he ssave sysems for whch he close egral of aco s a arameereee coformal vara of he moo [5]. e ale hs ea o hyroyamcs..g. Raeev cosere ha a large class of ssave sysems ca be brough o a caocal form by roucg comlex cooraes hase sace a a comlex value amloa [0]. I hs suy he eergy rae from he ssave sysem s cosere. I geeral a examle of a ame oscllaor was gve o emosrae her aroach. Dervao amloa ormalsm for Dssave ysem Geerally we coser s a amg force whch ees o he geeralze cooraes where varable coorae a s geeralze velocy []. Aga we coser s he comoes of he geeralze force

3 Rereseao of amloa formalsm 933 r Q (3) Where r s he oso vecor whch ees o he a. rom he o coservave sysem he geeral form of amloa euao s Q Q Q r Q k P Q Q P r P k Q (4) Where } { P Q are caocal varable a r s he oso vecor ees o he caocal varable se } { P Q a.e. } { P Q r a K s he rasforme amloa. The subscr he aral ervave exresso caes he k a r are eeecy. If he varable se } { P Q s rasforme o he varable se } { where he oso vecor r ees o he oes o ees o. The he euao (4) becomes r (5) rom euao (5) by he hel of (3) we ge (6) e he ew amloa uay Ĥ a we o o chage he efo of caocal momea hece he amloa euao becomes (7) Comarg he euaos (6) a (7) we have (8) I classcal mechacs he amloa Ĥ s mechacal eergy. c (9) Where eoes he hase flow resee by he euao (6) a c s he cosa whch ees o he al coo. Now f he Ĥ hrough le egral alog he hase flow curve usg by euao (6).

4 934 M. aer Al Bswas e al ( ) Ĥ (0) I euao (9) oly use amloa relace by he ew amloa Ĥ we ge c () ubsug he value of euaos (0) a () euao (9) we have c Ĥ ( ) The euao (7) s kow as hase flow curve. Thus accorg o he Newo- alace rcle of eermacy we ca assume ece we ca reasoable assume as ( ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( ))).e. ( ) s a fuco of aloe hus we have 0 w ( ) Now subsug he euao (3) o he euao () we oba w ( ) (4) where w eoes he egave work oe by he amg force. The we mus show ha he amloa resee by he euao (4) sasfes he euao (8). I oher wors he euao (7) s euvale o a Newoa moo euao. ubsug he euao (4) o he euao (7) we ge ( ) w w (5) Where a are cosere as sc varable amloa mechacs a we coser a as eee varable he rocess of cosruco of. () (3)

5 Rereseao of amloa formalsm 935 Thus we have w 0 [ ] 0 Aga w 0 [ ] w 0 w (6) ubsug he euao (6) o he euao (5) we ge (7) The euaos (7) a (8) are ecal. ere we ca coser ha he amloa uay Ĥ sasfy he amlo s euao (8). o he amloa uay Ĥ euao (4) s reresee as r. Proof of he frs egral Accorg o he law of coservao of eergy escrbe by he exae amloa uay s a vara of he hase flow. The roof s as follows: I classcal mechacs he amloa Ĥ s mechacal eergy c (8) Where c s a cosa whch ees o he al coo. The euao (8) ervave of s (9) Accorg o he Newo-alace eermacy we ge w (0)

6 936 M. aer Al Bswas e al uose ha Ĥ s a fuco of a aloe herefore 0 o he euao (8) becomes () I he amloa geeral euao he caocal se } { P Q s rasforme o he ew se } { a s geeralze force he a () uose he ew amloa Ĥ a we o o chage he efo of he caocal momea so we ca wre he ew amloa euao (3) Comarg he euaos () a (3) we ge Ĥ (4) ubsug he euaos () a (4) o he euao () we oba 0 Thus he exae amloa Ĥ s me eee.

7 Rereseao of amloa formalsm 937 The scusso abou geeralze momeum a mechacal momeum The geeralze momeum assocae wh he coorae shall be efe as (5) I geeral he geeralze momeum s ecal wh he usual mechacal momeum. I ervao we see Ĥ ffers from corresog ffers from orgal. Accorg o he euao (5) he geeralze momeum shall vary corresog. Bu ervao abou oes o vary corresogly. We ca exla he heomeo from a ervao wh aoher aroach. I hs ar we eoe he ew amloa wh a eoe he ew agraga wh. I a exra amg force s ae o he sysem oe ees o solve he agraga euao (6) Where he subscr caes he fucoal eeece he aral ervave oerao. If we clue he effecs of he force he agraga we may wre (7) Where s he corbuo o he agraga from. I hs case we woul have (8) ubracg he euao (6) from he euao (8) we ge (9) The amloa s efe as

8 938 M. aer Al Bswas e al (30) Where s he caocal momeum a s efe as (3) If he amg force was o resee we woul have (3) Where he momeum s efe as euao (5). By usg euao (7) a comarg he euaos (30) a (3) we oba (33) Aga usg euaos (30) a (3) we ge (34) The corbuo o amloa form he amg force s gve by (35) If we ake a me ervave of Δ a egrag we have el of (9) (36) We observe ha he creme he value of creme Δ from he amg force s eual o he egave work oe by he force o he sysem. Accorgly Δ ca be reresee as fuco of whou loss of geeraly. We ca coser as a fuco eee of. o he euao (35) becomes Δ (37)

9 Rereseao of amloa formalsm 939 Aga we kow from (7) Δ [ hel of (37)] (38) Now we ca erve euao (33) we ge (39) We see hs ar he geeralze momeum s euvale o mechacal momeum. The relao bewee he wo amloa sysems: If we coser a ew amloa sysem from above ar (40) Where. Aga we cover he amg force o a fuco of aloe. If we rasformao a se { } o a se{ }. o he euao (5) becomes (4) Ths rasformao ca o affec he value of he amloa. Aga we kow f a are geeralze momeum a mechacal momeum he we ca wre (4) e M be he Jacoba marx of he rasformao he M 0 I O (43) I Where I s a ey marx a O s a ull marx.

10 940 M. aer Al Bswas e al O I uose J I O (44) ubsug (43) he euao M T JM we ge M T JM J (45) where he subscr T eoes rasose marx. Thus he rasformao s uary caocal. I ca be show by aoher way ha he value of he ew amloa s o affece by he roose rasformao e ς { } T a { } T Z The we have ς J ξ a Z J Z ς ς Z ς ς MJ T ς Z MJ T Z (46) Z Accorg o he euao (4) we kow ha oes o eee exlcly o ς ς T Thus 0. The we have MJ Z (47) T T ς Because Z ξ ( M ) T T ξ M T ς (48) Z Aga accorg o he euao (45) we have M T JM J hus euao (47) ca be T reresee as ς J ς (49) Comarg euaos (47) a (49) hel of (48) we oba MJ T T J M Z T Z T ς ς ece he rasformao oes o affec he value of he amloa. Cocluso ally we may coclue ha amloa formulao or mechacs s a evelog form of agraga mechacal sysem. Above all we beleve ha hs suy wll lay oeer role exlorg furher suy he fel of Mahemacal Physcs. Ths suy eals wh he scusso of amloa formulao for Dssave

11 Rereseao of amloa formalsm 94 mechacal sysem. The frs egral geeral s o aalycally egrable wh he exceo of he orgal mechacal sysem whch are egrable. The reaso s ha he work oe by amg force ees o he hase flow. If he sysem s egrable he he hase flow ca be exlc wre ou he sysem has a aalycal soluo a herefore he work oe by amg force ca be exlcly egrae coseuely ca be exlcly reresee. owever from he o of vew of hyscs he exae amlo uay s sll a frs egral amely oal eergy. I ca be coclue ha wheher he class of ssave mechacal sysem has a aalycal soluo or o ca be reresee as a amloa formalsm. Refereces [] Bores A. K amloa formalsm uaum mechacs Joural of o lear mahemacal hyscs Vol. 5 No [] Eab M. R. a Tare O amloa formalsm of ocoservave sysems Vol. 3-. [3] eosa. J. B. a Basea B Dssave sysem: Raao loss Quaamze Revsa Braslerae fsca Vol. 4 No.. [4] Karz A Classcal Mechacs Quaum Mechacs el heory Acaemc Press New York.U..A. [5] Keh R.M A exeso of amlo s rcle o clue ssave sysems. J. Mah. Phys. 5(): 9. [6] uo T. a Guo Y amloa ormalsm of Dssave ysem Mah- Phy ssue- Vol. 50. [7] eo A amlo ormalsm for Omzao. Mahemacal sysems Esmao a Corol. Vol. 5 No [8] Nersesya A. P amloa formalsm for arcles wh a geeralze rgy Joural of heorecal a mahemacal hyscs Vol. 7 o.. [9] Rewe.997. O amloa ormalsm of No-Coservave ysem. Physcal Revew E [0] Raeev. G A caocal formulao of ssave mechacs usg comlex value amloas aals of hyscs 3(3):

12 94 M. aer Al Bswas e al [] Thomas. a Tveer J Dervg he amlo euaos of moo for a o-coservave sysem usg a varaoal rcle. Mah. Phys 39(3): Receve: July 009