Lagrangian Analysis of a Class of Quadratic Liénard-Type Oscillator Equations with Exponential-Type Restoring Force function
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1 agrangian Analysis of a Class of Quadratic iénard-ty Oscillator Equations wit Eonntial-Ty Rstoring Forc function J. Akand, D. K. K. Adjaï,.. Koudaoun,Y. J. F. Komaou,. D. onsia. Dartmnt of Pysics, Univrsity of Abomy-Calavi, Abomy-Calavi,.B.P. 56, Cotonou, BENIN.. Dartmnt of Industrial and Tcnical Scincs, ENSET-okossa, Univrsity of okossa, okossa, BENIN Abstract Tis rsarc work rooss a agrangian and amiltonian analysis for t uniqu class of osition-dndnt mass oscillator caractrizd by a armonic riodic solution and arabolic otntial nrgy and its invrtd vrsion admitting a osition-dndnt mass dynamics.. Analysis of t class of quadratic iénard-ty armonic nonlinar oscillator quations Tis sction is dvotd to t analysis of a class of quadratic iénard-ty nonlinar dissiativ oscillator quations tat admits act analytical armonic riodic solutions. Considr t quation [, ] ' tat rrsnts t class of quations undr analysis. and ar arbitrary aramtrs, and is an arbitrary function of tim, and rim olds for diffrntiation wit rsct to and, t quation,yilds. T dot ovr a symbol mans diffrntiation wit rsct to. By rstriction of ln f f f ' f wr f, is an arbitrary function of. T quation is of t gnral form F G 3 for wic t agrangian is givn by [3,4] 4, V wr F d 5 Corrsonding autor. addrss : janakand7@gmail.com
2 and V G d 6 dsignat t osition dndnt mass and t otntial function rsctivly. T agrangian of t quation bcoms, Alying t Eulr-agrang quation formula in [4] ' V 7 8 to t quation 7, givs t quation. By rstricting V to t armonic otntial, tat is V m, wit unit mass, quation quation m, t quation, wit t osition-dndnt mass function f 8 bcoms idntical to t. In tis rgard, t rrsnts t uniqu class of osition-dndnt mass oscillators ibiting not only act armonic riodic solution but also a armonic otntial function. Now, using [3], V 9 on may dduc from 5 and 6 t amiltonian, t us now considr, as illustration, som scific amls of bcoms. t. Tn T quation admits t osition dndnt mass and t otntial, and V rsctivly, wic rovids t agrangian function
3 , T alication of t Eulr-agrang quation rgard t amiltonian associatd to 8 to taks t form 3 givs, as ctd, 3. In tis, So, t amilton quations 4 5 yild for 4 6 T licit rssion for t conjugat momntum t form, as a function of and taks tn 7 Putting now, into, on may obtain as quation 8 T osition dndnt mass and t otntial of 8 tak tn t form V and 9 rsctivly. T associatd agrangian bcoms, T alication of t Eulr-agrang quation 8 to givs wit satisfaction 8. So, t associatd amiltonian may b writtn as
4 , suc tat t amilton quations tak t form T rlation btwn and rads in tis rsctiv 3. Analysis of invrtd vrsions Considr now t invrtd vrsion of ' 4 wic givs for, t following quation 5 T osition dndnt mass and otntial function of 5 may b tn dducd from 4 as 4 4 and V rsctivly. Trfor, t agrangian for 5 may b writtn in t form 4 4, In tis rsctiv, it may b vrifid tat t alication of t Eulr-agrang quation to 7 yilds, as ctd, 5. T amiltonian for 5 may also b comutd as 8 4 4, wic givs t amiltonian quations 4 from wic t conjugat momntum bcoms 9
5 3 By analysis, otr forms of quations ar also suggstd by t rvious studid quations. So, t following quations may also b considrd in t rsctiv of tis study, tat is 3 or in gnral 3 33 Finally on may considr t following mor gnralizations Ts quations will b invstigatd in a subsqunt work. Rfrncs []. D. onsia, J. Akand, D. K. K. Adjaï,.. Koudaoun, Y. J. F. Komaou, A class of osition-dndnt mass iénard diffrntial quations via a gnral nonlocal transformation, vixra:68.6v.6. []. D. onsia, J. Akand, D. K. K. Adjaï,.. Koudaoun, Y. J. F. Komaou, Eact Analytical Priodic Solutions wit Sinusoidal Form to a Class of Position-Dndnt ass iénard-ty Oscillator Equations, vixra: v.6. [3] V. C. Ruby, V. K. Candraskar,. Sntilvlan and. aksmanan, Rmoval of ordring ambiguity for a class of osition dndnt mass quantum systms wit an alication to t quadratic iénard ty nonlinar oscillators, arxiv:4.75v 4. [4] Omar ustafa, Position-dndnt mass agrangians: nonlocal transformations, Eulr- agrang invarianc and act solvability, arxiv: 4.445v3 5.
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