Energy-Work Connection Integration Scheme for Nonholonomic Hamiltonian Systems

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1 Commun. Theor. Phy. Beiing China pp Chinee Phyial Soiety Vol. 50 No. 5 November Energy-Wor Connetion Integration Sheme for Nonholonomi Hamiltonian Sytem WANG Xian-Jun 1 and FU Jing-Li 2 1 Department of Phyi Henan Intitute of Eduation Zhengzhou China 2 Intitute of Mathematial Phyi Zheiang Si-Teh Univerity Hangzhou China Reeived January ; Revied June Abtrat Thi paper foue on tudying a new energy-wor relationhip numerial integration heme of nonholonomi Hamiltonian ytem. The ignal-tage numerial multi-tage and parallel ompoition numerial integration heme are preented. The high-order energy-wor relation heme of the ytem i ontruted by a parallel onnetion of n multi-tage heme of order 2 it order of auray i 2n. The onnetion whih i direte analogue of uual ae between the hange of energy and wor of nonholonomi ontraint fore i obtained for nonholonomi Hamiltonian ytem. Thi paper alo give that there i maller error of the heme when taing a large number of tage than a le one. Finally an applied example i diued to illutrate thee reult. PACS number: Hh Ef Key word: numerial integration differential equation high-order heme energy-wor relationhip nonholonomi Hamiltonian ytem 1 Introdution Reently there have been a great number of tudie on the o-alled geometri numerial integration heme whih preerve the truture of ytem. [13] Leimuler and Reih pointed out that the geometri numerial integrator are time-tepping method deigned uh that they exatly atify onervation law ymmetrie or ympleti propertie of a ytem of differential equation. [1] Hairer Lubih and Wanner preented the ympleti integration of eparable Hamiltonian ordinary and partial differential equation. In thi way the ympleti tep i performed prior to the patial tep a oppoed to the tandard approah of patially direditing the PDE to form a ytem of Hamiltonian ODE to whih a ympleti integrator an be applied. [2] An energy-onerving heme i one of uh a geometri numerial integration heme. [48] It i well nown that a high-order heme an be ontruted by onneting low-order heme in erie hereafter we hall all it erie ompoition. [13] Now the high-order energy-onerving heme ha been ontruted with thi method. [9] Thi paper i to preent a new numerial integration heme whih i an energy-wor relation integration heme for nonholonomi Hamiltonian ytem. Thi wor alo tudie that a high-order energywor relation heme an be ontruted by onneting low-order heme in erie. Thi high-order energy-wor relation heme ha a truture onneting the order 2 multi-tage heme in parallel hereafter we hall all it parallel ompoition heme. 2 Numerial Integration for Nonholonomi Hamiltonian Sytem Let the onfiguration of mehanial ytem be deribed by the n generalized oordinate q 1...n. Suppoe the motion of the ytem be ubeted to the g ideal nonholonomi ontraint of Chetaey type f β t q q 0 β 1...g. 1 Here the ontraint funtion f β and the virtual variation δq atify the Appell Chetaey ondition [1011] f β δq 0. 2 q The equation of motion of the ytem an be written in the form d L L Q f β + λ β n 3 dt q q q where L Lq q i the Lagrangian of the ytem Q the non-potential generalized fore and λ β the ontraint multiplier. Suppoe 2 L det 0. 4 q q By uing Eq. 1 and 3 we an expre the ontraint multiplier λ β a the funtion of t q and q before integrating the differential equation 3 namely λ β λ β t q q. 5 Introdue the generalized momentum and Hamiltonian of the ytem a p L q Hq p [p q Lq q] q qtqp. 6 Equation 3 i expreed by where q H p ṗ H q + Q + Λ 7 H Hq 1...q n p 1... p n The proet upported by National Natural Siene Foundation of China under Grant No and the Natural Siene Foundation of Henan Provine under Grant No Correponding author qfuingli@163.om

2 1042 WANG Xian-Jun and FU Jing-Li Vol. 50 Q Q t q p f β Λ λ β q Λ t q p. 8 q q tqp Equation 7 i alled the generalized Hamilton anonial equation of the orreponding holonomi ytem for the nonholonomi ytem 1 and 3. Novoelov pointed out that the motion of the nonholonomi ytem i given by the olution of the holonomi ytem provided the initial ondition atify the ontraint equation. [12] Therefore in order to tudy the olution of nonholonomi ytem we firt tudy the olution of the orreponding holonomi ytem 7 then impoe the retrition of the ontraint on the initial ondition. In Eq. 8 H repreent the total energy. The relationhip between the hange of energy and the power of nonholonomi ontraint fore i eaily verified a dh dt n 1 H q q + H p ṗ [ H H H H + H ] Q + Λ q p p q p 1 1 H p Q + Λ. 9 The numerial integration i onidered a the diretization of t +1 q t +1 q t Hq 1...q n p 1 p n + dt t p t +1 p t +1 Hq 1...q n p 1 p n p t t q + Q + Λ t dt t t whih are obtained by integrating both ide of Eq. 7 on the interval [t t +1 ] where t i the tep ize. 3 Seond-Order Sheme of Numerial Integration for Nonholonomi Hamiltonian Sytem 3.1 Single-Stage Sheme Let p and q be the numerial approximation of p t and q t repetively. Then a one-tage heme i given by q +1 q + I 10 p p I 10 q + Q + Λ t n 11 with Iq ab Ip ab a b tδ ab q µ ab q H a b tδ ab p µ ab p H H Hq 1...q n p 1...p n Q + Λ Q + Λ q 1...q n p 1... p n. 12 denote the partial differene quotient operator with repet to p and q repetively whih are defined a The notation δ ab p and δ ab q δq ab 1...q l n n p p n n Ea q Eq b F Eq a Eq b q l δp ab 1...q l n n p p n n Ea p Ep b F Ep a Ep b p 13 where E a q and E a p are the hift operator defined a E a q 1...q l 1 1 ql q l ql n n p p n n 1...q l 1 1 ql +a q l ql n n p p n n E a p 1...q l n n p p 1 1 p p p n n 1...q l n n p p 1 1 p +a p p n n. 14 The notation µ ab q and µ ab p denote the mean differene operator with repet to all variable exept for q and p repetively whih are defined a with µ ab q 1... q l n n p p n n M ab E q1...e q1 E q+1...e qn E p1...e pn µ ab p 1... q l n n p p n n M ab E q1...e qn E p1...e p1 E p+1...e pn 15 M ab x 1 x 2...x r1 1 r! x a 1 x a 2 x a } r1 r r l x per a 1 x a 2 x a r1 x b 1 x b 2 x b r1 } l 1 x b 1 x b 2 x b r1 16 where per A denote the permanent or plu determinant of a matrix A. [13] For example in the ae d 1 we have µ ab q 1 M ab E q1 1 2 Ea q 1 + E b q 1 µ ab p 1 M ab E p1 1 2 Ea p 1 + E b p The operator δq ab δp ab µ ab q and µ ab p expreed a δ ab q δ ba q δ ab p δ ba p µ ab q µ ba q have ymmetry µ ab p µ ba p. 18

3 No. 5 Energy-Wor Connetion Integration Sheme for Nonholonomi Hamiltonian Sytem Relation Between Change of Energy and Wor of Nonholonomi Contraint Fore for Nonholonomi Hamiltonian Sytem Propoition 1 The relation between the hange of energy and the wor of nonholonomi ontraint fore for nonholonomi Hamiltonian ytem hold: H +a H +b [δq ab µ ab 1 q H q +a q +b + δp ab µ ab p H p +a p +b ]. 19 Proof For impliity we et Ē a E a q 1...E a q n E a p 1...E a p n T. 20 We firt note the identity n Eq a Ep a 1 2n! per Ē a.. }{{.Ēa. 21 } 2n It follow that n H +a H +b Eq a Ep a 1 1 2n! 2 2 n 1 Eq b Ep b H 1 [ Ēa per.. 2n! }{{.Ēa pere b... E b ] H }}{{} 2n 2n [ Ēa ] per Ēa... }{{ Ēa E b...e b per E b }}{{} Ēa... Ēa E b...e b H }{{}}{{} 2nl l1 2nl l1 1 per Ē a E b 2n! Ēa... }{{ Ēa E b...e b H }}{{} 2nl l1 [Eq a Eq b M ab E q1...e qn E p1...e p1 E p+1...e pn 1 + Ep a Ep b M ab E q1...e q1 E q+1...e qn E p1...e pn ] H [δq ab µ ab q H q +a q +b + δp ab µ ab p H p +a p +b ] 22 1 where we have ued the propertie of the permanent and the definition of operator [13] Propoition 2 The heme 11 and 12 atify the relation between the hange of energy and the wor of nonholonomi ontraint fore for the ytem. Proof: We ee from the hain rule 19 that the hange of energy H i equivalent to the wor of nonholonomi ontraint fore: H +1 H [δq 10 µ 10 q H q +1 q 1 + δp 10 µ 10 p H p ] t [δq 10 µ 10 q H δp 10 µ 10 p H 1 + δp 10 µ 10 p H δq 10 µ 10 q H + Q + Λ ] tδp 10 µ 10 p H Q + Λ 23 1 whih i a direte analogue of Eq Order of Auray The loal error involved in the determination of q +1 } n 1 from {p q } n 1 are O t 3 that i { Iq 10 + Q + Λ and Ip 10 in the heme 11 are the eond-order approximation of the integral in Eq. 10 repetively. Although thi an be proved by Taylor expanion it i obviou beaue the heme i ymmetri ee Sube Multi-tage Sheme A -tage heme i ontruted by onneting the eond-order heme with mall integration interval of length t/ in erie: P +m/ Q +m/ P +m1/ I m/m1/ Q + Q + Λ Q 1...Q n P 1...P n t Q +m1/ + I m/m1/ P P +1 P p Q +1 q +1 Q q n m with I ab Q a b tδ ab Q µ ab Q HQ 1...Q n P 1...P n I ab P a b tδ ab P µ ab P HQ 1...Q n P 1...P n 25 where P +m/ and Q +m/ are the internal tage variable. It hould be noted that the above heme i equivalent to the heme p I l/l1/ Q + Q + Λ q1 qn p 1...p n t q +1 q I l/l1/ P

4 1044 WANG Xian-Jun and FU Jing-Li Vol. 50 P +m/ Q + m m p m q + I l/l1/ Q I l/l1/ P + m + Q + Λ t + m q +1 lm+1 + I l/l1/ P lm+1 I l/l1/ Q + Q + Λ t P +1 P p Q +1 q +1 Q q n m The latter heme 26 will be ued in the next etion to ontrut a higher-order heme. It i obviou for the -tage that the relationhip between the hange of energy and the wor of nonholonomi ontraint fore i exatly equivalent and the order of auray i 2. We point out here that the loal error i expreed a O[ t/ 3 ] 2 O t 3. 4 Higher-Order Sheme of Numerial Integration for Nonholonomi Hamiltonian Sytem 4.1 Parallel Compoition Sheme Let n be arbitrary poitive integer atifying 1 < 2 < < n 27 then a new heme i ontruted by onnetion 1 -tage 2 -tage... n -tage heme of order 2 in parallel: P +m/ Q +m/ p d m p m q + I l/ l1/ Q + Q + Λ q 1...q n p 1... p n tq +1 q + I l/ l1/ Q + Q + Λ t + m I l/ l1/ Q + m q +1 + lm+1 lm+1 I l/ l1/ Q d I l/ l1/ Q I l/ l1/ P i + Q + Λ t P +1 P p Q +1 q +1 Q q n u m with the weight 1 for u 1 2u2 d u 2 2 l for u u l where I ab P a b tδ ab P µ ab P H I ab Q a b tδ ab Q µ ab Q H H HQ 1...Q n P 1...P n 29 Q + Λ Q + Λ Q 1...Q n P 1... P n Relation Between Change of Energy and Wor of Nonholonomi Contraint Fore Propoition 3 The heme 29 with the ondition d 1 31 whih atifie relationhip between the hange of energy and wor of nonholonomi ontraint fore for nonholonomi Hamiltonian ytem Proof We firt note H +1 H +1 H H u 32 we ee from Propoition 1 that H +a H +b 1 δ ab P µ ab P H P +a It follow from Eq that H +1 H d H +1 H [ d m1 1 + δ m/ m1/ Q d t m1 1 d m1 δ m/ m1/ P P +b + δ ab Q µ ab Q H Q +a Q +b. 33 H +m/ H +m1/ µ m/ m1/ P H +m/ P µ m/ m1/ Q H +m/ Q [ IP +m/ +m1/ P +m/ Q +m1/ P +m1/ P +m1/ ]

5 No. 5 Energy-Wor Connetion Integration Sheme for Nonholonomi Hamiltonian Sytem 1045 We obtain from Eq. 28 P +m/ Q +m/ + I Q +m/ +m1/ P +m1/ 1 p + 1 r1 Q +m1/ 1 q +1 q Q +m/ r I l/ rl1/ r Q r + Q +m1/ ]. 34 I l/ l1/ Q I m/ m1/ Q + Q + Λ t I m/ m1/ Q + Q + Λ t 1 r r1 I l/ rl1/ r P r I l/ l1/ Q I l/ l1/ P + I m/ m1/ P Q + Λ t I l/ l1/ P + I m/ m1/ P. 35 Subtituting Eq. 35 into Eq. 34 yield n H +1 H d m1 1 I m/ m1/ P Q + Λ 36 whih i a direte analogue of that relation between the hange of energy and the wor of nonholonomi ontraint fore for the ytem Order of a Symmetri Sheme Propoition 4 Conider the heme 28 a mapping φ t : q 1... q n p 1...p n q qn n 37 and let φ 1 t be the invere mapping of φ t. Then we have φ 1 t φ t. 38 That i the heme i ymmetri. Proof The invere φ 1 t i obtained by exhanging p q and q +1. Replaing t by t and rearranging term in φ 1 t lead to the mapping φ1 t. For thi φ1 t etting P +m/ P +1m/ n u m and omitting the tilde we an obtain φ t. Therefore form 37 hold. Propoition 5 If we hooe the weight d 1 d 2...d n a Eq. 29 the auray of the heme 28 i at leat of order 2n. Proof It i nown that if a one-tep heme i ymmetri it order of auray i even. [12] Therefore the loal error of the heme φ t i O t 2r+1 with a poitive integer r. We firt hooe {d } u uh that d Sine the error of I m/ m1/ P I m/ m1/ Q and Q + Λ are O[ t/3 ] the error of φ t i expreed a [ t 3 ] d O 2 d O t 3. If we hooe {d } u uh that 2 d then the O t 3 -term in the error of φ t vanihe. Sine the error of φ t i of odd order it beome O t 5. The O t 5 -term in the error {d } u -term in the error of φ t i expreed a [ t 5 ] d O 4 d O t 5. If in addition to the ondition 40 and 41 we hooe {d } u uh that 4 d 0 42 then the O t 5 -term in the error of φ t vanihe and the error beome O t 7. Thee proedure an be repeated. The final ondition for {d } u i 2u1 d Therefore if we hooe {d } u uh that they atify the n imultaneou linear equation { 1 for l 0 2l d 44 0 for l u 1. Then the error of φ t i O t 2u+1. Sine the olution of Eq. 44 i given by Eq. 28 the order of auray i 2u. 5 A Numerial Example Conidering the motion of a Appel Hamel problem with ma m it Lagrangian i L 1 2 m q 1 + q 2 + q 3 mgq 3 45

6 1046 WANG Xian-Jun and FU Jing-Li Vol. 50 and the motion of the ytem i ubeted to nonholonomi ontraint f q q 2 2 q Introdue the generalized momentum and Hamiltonian of the ytem a p 1 L q 1 m q 1 p 2 L q 2 m q 2 p 3 L q 3 m q 3 47 H p 1 q 1 + p 2 q 2 + p 3 q m q2 1 + q q mgq 3 1 2m p2 1 + p p mgq The equation of motion inluding ontraint multiplier of the ytem i ṗ 1 2λ p 1 m q 1 p 1 m ṗ 2 2λ p 2 m q 2 p 2 m ṗ 3 mg 2λ p 3 m q 3 p 3 m. 49 We an from Eq. 47 and 48 obtain the ontraint multiplier p 3 λ 2p p2 2 + g m2 g 50 p2 3 m2 4p 3 then the equation of motion of the ytem i ṗ 1 p 1 2p 3 mg q 1 p 1 m ṗ 2 p 2 2p 3 mg q 2 p 2 m ṗ mg q 3 p 3 m 51 whih give the following olution p 1 1 t + 3 q m 2 t2 + 3 t + 4 p 1 2 t + 3 q m 2 t2 + 3 t + 5 p mgt + 3 q mgt where 0 < t <. We tae the initial ondition p 1 0 p 2 0 p q 1 0 q q and the alulation time t T. The parallel ompoition heme with u 54 wa ued. We alulated the global error given by et p K 1 p 1T 2 + p K 2 p 2T 2 + p K 3 p 3T 2 + q1 K q 1T 2 + q2 K q 2T 2 + q3 K q 3T 2 where K T/ t. Sine the global error et i about T/ t time the loal error et i expreed a et O t 2n. The loal error of the parallel ompoition i expreed a 2u d O t 2n O t 2n n We hould point out that the lager the number of tage of the heme i the maller the error of the heme for nonholonomi Hamilton ytem i. 6 Conluion In thi paper the new numerial integration heme of nonholonomi Hamilton ytem are etablihed. Thi tudy how that the numerial onnetion between the energy of the ytem and the wor of the nonholonomi ontraint fore i an analog of uual energy-wor onnetion and the numerial onnetion between the high-order energy-wor i alo ontained. Numerial reult how that there i a maller error of the heme when taing the large number of tage. Referene [1] B. Leimuler and S. Reih Simulating Hamiltonian Dynami Cambridge Univerity Pre Cambridge [2] E. Hairer C. Lubih and G. Wanner Geometri Numerial Integration Struture-Preerving Algorithm for Ordinary Differential Equation Springer Berlin [3] J.M. Sanz-Serna and M.P. Calvo Numerial Hamiltonian Problem Chapman & Hall London [4] D. Greenpan J. Comput. Phy [5] D. Greenpan N-body Problem and Model World Sientifi Singapore [6] I. Itoh T and K. Abe J. Comput. Phy [7] C.W. Gear Phyia D [8] T. Matuo J. Comput. Appl. Math [9] I. Yui Phy. Lett. A [10] J.G. Papatavridi Analytial Mehani Oxford Univerity Pre New Yor [11] F.X. Mei Appl. Meh. Rev. ASME [12] V.S. Novoelov Variational Method in Mehani LGU Leningrad Pre 1966 in Ruian. [13] H. Min Permanent Addion-Weley Reading Ma

696 Fu Jing-Li et al Vol. 12 form in generalized coordinate Q ffiq dt = 0 ( = 1; ;n): (3) For nonholonomic ytem, ffiq are not independent of

696 Fu Jing-Li et al Vol. 12 form in generalized coordinate Q ffiq dt = 0 ( = 1; ;n): (3) For nonholonomic ytem, ffiq are not independent of Vol 12 No 7, July 2003 cfl 2003 Chin. Phy. Soc. 1009-1963/2003/12(07)/0695-05 Chinee Phyic and IOP Publihing Ltd Lie ymmetrie and conerved quantitie of controllable nonholonomic dynamical ytem Fu Jing-Li(ΛΠ±)