High-Order Hamilton s Principle and the Hamilton s Principle of High-Order Lagrangian Function

Transcription

1 Commun. Theor. Phys. Bejng, Chna pp c Chnese Physcal Socety Vol. 49, No., February 15, 008 Hgh-Orer Hamlton s Prncple an the Hamlton s Prncple of Hgh-Orer Lagrangan Functon ZHAO Hong-Xa an MA Shan-Jun College of Physcs an Communcaton Electroncs, Jangx Normal Unversty, Nanchang 3300, Chna Receve January 10, 007; Revse March 6, 007 Abstract In ths paper, base on the theorem of the hgh-orer velocty energy, ntegraton an varaton prncple, the hgh-orer Hamlton s prncple of general holonomc systems s gven. Then, three-orer Lagrangan equatons an four-orer Lagrangan equatons are obtane from the hgh-orer Hamlton s prncple. Fnally, the Hamlton s prncple of hgh-orer Lagrangan functon s gven. PACS numbers: 0.30.Jr, 45.0.Jj Key wors: Hamlton s prncple, hgh-orer velocty energy, ntegraton an varaton prncple, Lagrangan functon 1 Introucton Traonal Hamlton s prncple of general holonomc systems s use to get the two-orer fferental equatons when we know the external force actng on the mechancal system. But for many practcal problems, a motve mechancal system s not acte by a resultant external force, but the tme rate of change of force,.e. the one-orer ervatve of force wth respect to tme. Corresponng to the tme rate of change of force, jerk, [13 acceleraton energy [4 an jumpulse equaton [57 were gven. Due to the ntroucton of these concepts, the stues on traonal ynamcal equatons, whch are lmte n the scope of two-orer fferental equatons, have been change. The three- or hgher orer ynamcal equatons were obtane. [4,811 In recent years, a lot of works on stuy of the threeorer Lagrangan equatons an more than three-orer ynamcal equatons have been one, conclung the ntal stues on the theores an applcaton of moton of varable acceleraton, the ynamcs of varable acceleraton, jumpulse equaton; [57 the stues on the fferent forms of the three-orer Lagrangan equatons for fferent conons, [1,13 the symmetry an conserve quantty of the three-orer Lagrangan equatons, [1416 the threeorer pseuo-hamlton s canoncal equatons, [17 an the stues on the eucton of the hgh-orer ynamcal moton equatons. [911 In ths paper, base on the theorem of the hgh-orer velocty energy, the hgh-orer Hamlton s prncple for general holonomc systems s euce by use of ntegraton an varaton prncple. Then, threeorer Lagrangan equatons an four-orer Lagrangan equatons have been obtane from hgh-orer Hamlton s prncple. Fnally, the Hamlton s prncple on hgh-orer Lagrangan functon L n q n1, q n,..., q, q, q, t s gven. From the Hamlton s prncple on hgh-orer Lagrangan functon, hgh-orer Lagrangan equatons [11 can be obtane. Hgh-orer Hamlton s Prncple for General Holonomc System Accorng to the theorem of the hgh-orer velocty energy, [10 for general ynamcal system we have F m δr m δs m1, 1,,...n; m 1,,..., 1 where S m1 1 n m r m r m s the m 1- orer velocty energy of the -th partcle n the system. From the efnton of S m1 one has δs m1 m r m δr m. For the system, we conser that there are many moton paths n the lmtaton of constrant. All paths have the same startng pont an en pont n space. When t an t t, the system s locate at startng pont an en pont. Integratng Eq. wth respect to tme along an arbtrary path from to t, we have δs m1 m r m δr m. 3 For the sochronous varaton δt 0, the followng commutatve relaton can be use δ δ. 4 Usng ntegraton by parts, one has The project supporte by the Natural Scence Founaton of Jangx Provnce an the Founaton of Eucaton Department of Jangx Provnce uner Grant No. [ Corresponng author, E-mal: shanjun@nc.jx.cn

2 98 ZHAO Hong-Xa an MA Shan-Jun Vol. 49 an δs m1 [ m r m m r m δr m1 δr m1 t t m r m3 δr m1 m r m3 m1 δr. 5 On the other han, the m 1-orer ervatve of the coornate of the -th partcle can be wrtten as follows: r m1 r m1 q, q,...,q m1, t, 6 δr m1 1 r m1 q m1 δq m1 rm1 δq m q m rm1 q rm1 δt, 7 t where q 1,,...,s s the generalze coornates. Conserng the case of sochronous varaton, we have the followng conons of startng an en ponts, δq m1 tt1 δq m1 tt 0, δq m tt1 δq m tt 0,...,δq tt1 δq tt 0. 8 Usng Eq. 8, equaton 5 can be smplfe as Usng ntegraton by parts agan, we have δs m1 Smlarly we fnally get or δs m1 δs m1 [ m r m3 m r m m r m3 δr m m r m4 δr m δr m1. 9 δr m m r m4 δr m. 10 [ 1 m1 m r m4 δr, 11 [ δsm1 1 m m r m4 δr 0. 1 Equaton 1 s the hgh-orer Hamlton s prncple for general holonomc system. Especally when m 1, equaton 1 can be wrtten as δs0 m r δr 0, 13 where S 0 1 m ṙ ṙ T, m r δr Q δq are knetc energy an generalze force n analytcal mechancs respectvely. In analytcal mechancs, f the generalze force has a force functon, equaton 13 s Hamlton s prncple for general holonomc system. From Eq. 13 the two-orer Lagrangan equatons for holonomc system can be obtane. If m 0, equaton 1 can be wrtten as 1 δs m r 4 δr 0, 14 where S 1 1 n m r r s acceleraton energy. Next we shall try to get the three-orer Lagrangan equatons from Eq. 14. Conserng S 1 S 1 q, q, q, t an sochronous varaton Eq. 8, equaton 14 can be expresse as S1 δ q S 1 δ q S 1 δq m r 4 r S 1 t S 1 t δq δ q δq q q q q q q 1 S 1 δ q S1 δq S 1 δq q q q 1 m r 4 1 r q δq 0, 1

3 No. Hgh-Orer Hamlton s Prncple an the Hamlton s Prncple of Hgh-Orer Lagrangan Functon 99 where the ntegraton by parts an the commutatve relaton have been use. Substtutng Eq. 8 nto the above equaton, one can get [ S1 δ q q n m r r m r r m r 4 r δq q q q 1 [ S1 n δ q q 1 [ S1 n δ q q 1 [ 1 1 n m r ṙ q 1 S1 q δ q S1 q δ q m r 3 m r 3 m r 3 n [ n r q ṙ q m r ṙ q m r ṙ q r q δ q } m r ṙ m r 3 r t δq q q 1 S1 1 S 1 q q m r 3 m r r q m r r q m r 3 1 m r 3 m r r q m r 4 r q δq r q δq m r 4 r q δq S1 n δ q m r ṙ q q r q δq m r 3 r q δ q } r q } δ q Snce δ q s epenent, f equaton 15 s satsfe, the three-orer Lagrangan equatons for holonomc system [8 have been gven, S1 1 S 1 m r 3 r 0, 1,,...,s. 16 q q q If m 1, equaton 1 can be wrtten as δs m r 6 δr 0, 17 where S 1 n m r 3 r 3 s the two-orer velocty energy. From Eq. 17 we can euce four-orer Lagrangan equatons. Conserng S S q, q, q, q, t, an usng the conons of sochronous varaton δt 0 an the conons of startng an en ponts Eq. 8, we have δs m r 6 S δr δq S δ q S δ q S δq m r 6 r δq q q q q q [ S δ q q S δ q q S δ q S δ q S δq q q q S δq S δq q q 1 m r 6 S δ q q S δ q S t δq q q 1 r q δq } S δ q S δq S δq q q q 1 [ S δ q q 1 m r 6 r q δq } [ S δ q q S δq S q δq q

4 300 ZHAO Hong-Xa an MA Shan-Jun Vol. 49 S δq S δq q q 1 S t δq q 1 S δ q q m r r q m r 6 1 r q δq } [ S S δ q q S S q q q m r 5 m r r q r 6 q m r 6 m r 4 r q 3 where the ntegraton by parts an commutatve relatons Eq. 4 have been use. On the other han, we have the expresson [11 an the expresson [9,10 r n1 Conserng Eqs. 19 an 0, one has 3 m r 4 m r 3 r q m r 6 r q δq } m r 4 r q r q δq } 0, 18 r n n 1 0, 19 q m m q m1 m r m r Substtutng Eq. 1 nto Eq. 18, fnally we have δs m r 6 δr 1 S δ q q m r 6 1 n 1 r q m r 5 S δ q q m r 5 r q n m r 5 ṙ q n n S δ q q m r 4 [ n r m m ṙ q m1 q. 0 r q r 3 q m r 4 r 3 q ṙ q m r 4 m r 5 m r 4 ṙ q ṙ q ṙ q δq } m r m r r, q r, q r. 1 q r q n m r 4 m r 3 m r 4 r q r q δq } r q m r 3 m r 4 ṙ m r 3 r m r 5 r m r 4 ṙ δq q q q q m r 4 ṙ m r 3 r m r 5 r m r 4 ṙ δ q } q q q q 1 1 S δ q q S δ q q [ n [ n m r 3 ṙ m r 4 r δ q } q q m r 3 ṙ m r 4 r δ q q q r q

5 No. Hgh-Orer Hamlton s Prncple an the Hamlton s Prncple of Hgh-Orer Lagrangan Functon 301 m r 3 ṙ m r 4 r δ q } q q 1 1 S δ q q [ S 1 S q 3 q m r 3 ṙ m r 4 r δ q } q q m r 4 r q δ q } 0. Snce δ q s epenent, f equaton s satsfe, the four-orer Lagrangan equatons for holonomc system are gven as follows: S 1 S m q r 4 r 0 1,,...,s. 3 3 q q The equatons are the same wth the results whch are obtane from hgh-orer Lagrangan equatons for holonomc system. [911 3 Hamlton s Prncple of Hgh-Orer Lagrangan Functon Defnton The ntegraton of the hgh-orer Lagrangan functon [11 L n q n1, q n,..., q, q, q, t wth respect to tme from to t s calle Hamlton s acton I n whch s corresponng wth hgh-orer Lagrangan functon L n q n1, q n,..., q, q, q, t,.e. I n L n. 4 The Hamlton s prncple of hgh-orer Lagrangan functon can be expresse as follows: In all possble movements of the system, the real movement of the system wll make the Hamlton s acton I n have an extreme value when all movements start at the same pont an en at the same pont,.e. δi n δ L n q n1, q n,..., q, q, q, t 0. 5 If n 1, L L q, q, q, t, the Hamlton s prncple, whch s corresponng to one-orer Lagrangan functon, s gven as follows: L δi 1 δ L δ q L δ q L δq q q q L q δ q t 1 L q δ q 1 L t t δ q q 1 L δ q L δq L δq L δq q q q q L t t δq q 1 [ 1 [ L L L δq } q q q L δq L δq q L δq L } δq q q q [ L L L δq } 0, 6 q q q where the ntegraton by parts, commutatve relaton Eq. 4, an the conons of startng an en ponts Eq. 8 have been use. Substtutng the equaton [11 L n1 Ln q m q L n m1 q m nto Eq. 6, we have δi 1 [ L L L L L δq } q q q q q 1 1 [ L L L L L L L δq } q q q q q q q

6 30 ZHAO Hong-Xa an MA Shan-Jun Vol L q 3 L q 3 L q δq } 0. 7 Snce δq s epenent, f equaton 7 s satsfe, one has L 3 L 3 L 0. 8 q q q On the other han, we know the Lagrangan equatons of the hgh-orer Lagrangan functon [11 L n1 L n n 0. 9 q m m q m1 Let n, m an n 1, m 0 respectvely, we have L L 0, 30 q q L 3 L q q Subtractng Eq. 31 from Eq. 30, we can get the result of Eq. 8. Here we have prove that the Hamlton s prncple of hgh-orer Lagrangan functon s correct. 4 Example Conser that a projectle s launche wth ntal spee V 0 at an angle wth the horzontal, the mass of boy s m, the sspatve force actng on the boy s rect proportonal to the velocty of the boy,.e. fv kv. Now let us fn the coornates at any tme. We have r ẍ ÿj, m r F, 3 S 1 1 m r r 1 mẍ ÿ. 33 Substtutng Eqs. 3 an 33 nto Eq. 14 yels m ẍδẍ ÿδÿ r 4 δr 0, 34 or [m xδẋ y δẏ F x δẋ F y δẏ Conserng the force F mgj, fv k ẋ ẏ an substtutng them nto Eq. 35, one has x ωẍ 0, y ωÿ 0, 36 where ω k/m. The egenvalues of Eqs. 35 an 36 are λ 1 0, λ 0, λ 3 ω, 37 an the functon relatons of x, y an t are x a 0 a 1 t a e ωt, 38 y b 0 b 1 t b e ωt. 39 Substtutng the ntal conon,.e. when t 0, there are x 0, y 0, ẋ V 0 cos, ẏ V 0 sn, ẍ ωv 0 cos, ÿ g ωv 0 sn, fnally, we have x V 0 ω 1 eωt cos, g y ω V 0 ω sn 1 e ωt g ω t. 40 The results 40 are the same as the results of the problem that comes from Newton s law of moton. It s easy to obtan the solutons of the problem f we use the hghorer Hamlton s prncple for general holonomc system. 5 Dscusson In ths paper, the hgh-orer Hamlton s prncple for general holonomc mechancal system an the Hamlton s prncple of hgh-orer Lagrangan functon have been gven. Usng these prncples, we can get the three-orer Lagrangan equaton, the four-orer Lagrangan equaton an the Lagrangan equaton of the hgh-orer Lagrangan functon. These two prncples not only prove us a metho to obtan the ynamcal equaton of the system, but also enrch our theory on the ynamcal varyng acceleraton. References [1 S.H. Schot, Am. J. Phys [ M. Zhu, Mech. Practce n Chnese. [3 K.F. Tan, Y.K. Zhao, an X.D. Guo, Mech. Practce n Chnese. [4 F.X. Me, D. Lu, an Y. Luo, Avance Analytcal Mechancs, Bejng Insttute of Technology Press, Bejng 1991 n Chnese. [5 P.T. Huang, Physcs n Chnese. [6 P.T. Huang, W. Huang, an L.Y. Hu, J. Jangx Normal Unversty n Chnese. [7 P.T. Huang, S.J. Ma, an L.Y. Hu, J. Jangx Normal Unv n Chnese. [8 S.J. Ma, X.X. Xu, P.T. Huang, an L.Y. Hu, Acta Phys. Sn n Chnese. [9 X.W. Zhang, Acta Phys. Sn n Chnese. [10 X.W. Zhang, Acta Phys. Sn n Chnese. [11 Y. Sh an S.J. Ma, Acta Phys. Sn n Chnese. [1 S.J. Ma, M.P. Lu, an P.T. Huang, Chn. Phys [13 S.J. Ma, W.G. Ge, an P.T. Huang, Chn. Phys [14 X.H. Yang an S.J. Ma, Chn. Phys [15 S.J. Ma, X.H. Yang, R. Yan, an P.T. Huang, Commun. Theor. Phys. Bejng, Chna [16 S.J. Ma, X.H. Yang, an R. Yan, Commun. Theor. Phys. Bejng, Chna [17 S.J. Ma, P.T. Huang, R. Yan, an H.X. Zhao, Chn. Phys