ON SOME PROPERTIES OF THE SYMPLECTIC AND HAMILTONIAN MATRICES

Transcription

1 Proceedings of he Inernaional Conference on Theory and Applicaions of Mahemaics and Informaics - ICTAMI 2004, Thessaloniki, Greece ON SOME PROPERTIES OF THE SYMPLECTIC AND HAMILTONIAN MATRICES by Dorin Wainberg Absrac: In he firs par of he paper he symplecic and Hamilonian marices are defined and some properies are poined, and in he second par a relaion beween hose wo ses of marices is proved. 1. Proprieies of symplecic and Hamilonian marices For he beginning i will be inroduced he square mari J M 2 n 2n ( ) defined by 0n I n J = (1) I n 0n where: 0n M n n ( ) zero mari I M n ( ) ideniy mari n n Remark 1 I is no very complicaed o prove ha he ne properies of he mari J are real (properies ha will be very useful o prove some proposiions ha follow): i) J = -J J ii) J -1 = iii) J J = I 2 n iv) J J = - I 2 n 2 v) J = - I 2 n vi) de J = ± 1 Now he definiion of he symplecic and Hamilonian marices are given: Definiion 1 A mari A M 2 n 2n ( ) is called symplecic if: A J A = J (2) 442

2 Dorin Wainberg - On some properies of he symplecic and Hamilonian marices where J M 2 n 2n ( ) is from (1). We will denoe by SP(n, ) no = { A M 2 n 2n ( ) A J A = J } he se of 2n 2n real symplecic marices. Definiion 2 A mari A M 2 n 2n ( ) is called Hamilonian if: A J + JA = 0 (3) where J M 2 n 2n ( ) is from (1). We will denoe by sp(n, ) no = { A M 2 n 2n ( ) A J + JA = 0 } he se of 2n 2n real Hamilonian marices. In he ne par some properies of hose ses of marices will be proved, for eample: Proposiion 1 Le A, B SP(n, ). The ne relaions are rue: a) A is nonsingular; b) A -1 = - J A J; c) A, A -1, AB SP(n, ). a) From he definiion of symplecic marices we have A J A = J de( A JA) = de J de de J de A = de J de A de A = 1 2 (de A ) = 1 de A = ± 1 0 A is nonsingular. A b) A SP(n, ) A J A = J J A J = A -1 A A J = J A -1 J -1 A J = A -1 J =J J =J A -1 = - J A J. More, we have: A = - J A -1 J. c) A SP(n, ) A J A = J J A J A J = 2 J = - I 2 n A A J A J A = - A Now, muliplying a he lef side by (A ) -1, we will obain: J A J A = - I 2 n. Muliplying again a he righ side by J : A J A = - J = -(-J) = J A J A = J (A ) J A = J A SP(n, ) We will proof now ha A -1 SP(n, ). Using he relaion A -1 = - J A J (from he poin b)) we have successive: (A -1 ) J A -1 = (- J A J) J (- J A J) = 443

3 Dorin Wainberg - On some properies of he symplecic and Hamilonian marices = J (J A) J J A J = J 2 = - I 2 n = J A J (- I 2 n ) A J = = - J A J A J = A J A = (A J A) = = - J J J = J J = - I 2 n = J A -1 SP(n, ) We will proof now ha AB SP(n, ): (AB) J AB = B A J A B = A J A = J = B J B = = J AB SP(n, ) Consequence. The se of symplecic marices SP(n, ) is a subgroup of he se of nonsingular marices GL(n, ) repored o muliplicaion. Indeed we have AB SP(n, ), A, B SP(n, ), and AB -1 SP(n, ), A, B SP(n, ). Proposiion 2 Le A SP(n, ) and p A () - he characerisic polynomial of he mari A. If p A (c) = 0, hen p A ( 1 ) = p (c) c A = p A ( 1 ) = 0, where c. c p A () = de(a I 2 n ) = A = - J A -1 J = de(- J A -1 J I 2 n ) = I 2 n = - J 2 = de( J( - A -1 ) J + J 2 ) = = de( J( - A -1 + I 2 n ) J ) = = de J de( - A -1 + I 2 n ) de J = de J = ± 1 = de( - A -1 + I 2 n ) = = de( - A -1 + AA -1 ) = = de( - I 2 n + A ) de( A -1 )= = ± de(a - I 2 n ) = ± 2n de(a - 1 I 2 n ) = ± 2n p A ( 1 ) 2n p A ( 1 ) = 0 p ) A ( 1 = 0 p ) A ( 1 = 0 p (c) c A = 0 p A ( 1 ) = 0. c Proposiion 3 The ne relaions are equivalen: a) A is a Hamilonian mari; b) A = JS, where S = S ; c) ( JA ) = JA. 444 J

4 Dorin Wainberg - On some properies of he symplecic and Hamilonian marices a b A = J J J = I2n J A A = J(- J)A A sp( n, R) ( J(- JA)) J + JA = 0 (- JA) = - JA (- JA) = - JA J(- JA) = A If - JA = S A = J(- JA) = J(- JA) A = JS = J S S = a c A J + JA = 0 A J = - JA JA = - ( JA ) ( JA ) = JA S. J J ( ) (A J) = (- JA) J A = - ( JA ) - Proposiion 4 Le A, B sp(n, ). The ne relaions are rue: a) A + B sp(n, ); b) α A sp(n, ), α ; c) [A, B] sp(n, ), where [A, B] def = AB BA a) Because A and B are Hamilonian marices i resuls ha A J + JA = 0 respecively B J + JB = 0. By adding hose wo relaions we will obain: (A + B ) J + J(A + B) = 0 (A + B) J + J(A + B) = 0 A + B sp(n, ). ii) A J + JA = 0 α A J α + JA α = 0 A α J + J(A α ) = 0 (A α ) J + J(A α ) = 0 α A sp(n, ). iii) We will prove ha [A, B] = J M, where M = M. We know ha A = JS and B = JR, where S = S and R = R. [A, B] = AB BA = JSJR JRJS = J(SJR RJS), from where, making he noaion SJR RJS = M we will obain [A, B] = J M. M Now we will show ha M = M = (SJR RJS) = (SJR) (RJS) = R J S S J R = - RJS + SJR = SJR RJS = M Consequence. ( sp(n, ), [, ] ) is a Lie algebra.. We will prove he necessary properies of he bracke [, ] : bilineariy, anisymmery and Jacobi s relaion. 445

5 Dorin Wainberg - On some properies of he symplecic and Hamilonian marices i) [ αa + βb, C] = α[a, C] + β[b, C] - evidenly he operaion is bilinear. ii) [A, B] = AB BA = - (BA AB) = - [B, A] he operaion is anisymmeric. iii) Jacobi s relaion is saisfied: [[A, B], C] + [[C, A], B] + [[B, C], A] = [AB BA, C] + [CA AC, B] + [BC CB, A] = =ABC BAC (CAB CBA) + CAB ACB (BCA BAC) + BCA CBA (ABC ACB) = 0 Proposiion 5 Le A sp(n, ) and p A () - he characerisic polynomial of he mari A. Then: a) p A () = p A ( ) b) if p A (c) = 0, hen p A ( c) = p A (c) = p A ( c) = 0, where c. a) p A () = de(a I 2n ), bu A = J A J p A () = de(j A J I 2n ) A = - J A -1 J = de(j A J J J) = = de( J( A + I 2n )J ) = = de J de(a + I 2n )de J = de J = ± 1 = de(a + I 2n ) = de(a + I 2n ) = = de(a + I 2n ) = = de(a + I 2n ) = de(a ( ) I 2n ) = p A ( ) b) p A (c) = 0 a) p A ( c) = 0. p A () is a real coefficiens polynomial p A (c) = 0 p A ( c) = 0. 2.A relaion beween he ses sp(n, ) and SP(n, ) Theorem: Le A M 2 n 2n ( ). The ne relaions are equivalen: a. A sp (n, ) b. ep (A) SP (n, ) a b A sp (n, ) A J + JA = O 446 a)

6 Dorin Wainberg - On some properies of he symplecic and Hamilonian marices Le U() = (ep(a)) J ep(a) Bu, on he oher hand we have: du d = [ ep(a)] J ep(a) + [ep(a)] d J [ ep(a)] d d d = [A ep(a)] J ep(a) + [ep(a)] J A (ep(a)) = [ep(a)] A J ep (A) + [ep(a)] JA ep(a) = [ep(a)] [A J + J A] ep (A) = 0. U() consan. ( * ) Bu U(0) = (ep(a Α O)) J ep(a Α O) = J ( ** ) From ( * ) and ( ** ) U () = 0 (ep(a)) J ep(a) = J ep(a) SP (n, ). b a ep (A) sp (n, ) (ep (A)) Α J Α ep (A) = J. d [(ep(a)) J Α (ep (A))] = 0 d d [ ep (A)] J ep (A) + (ep (A)) d J [ ep (A)] = 0 d d (A ep(a)) J ep(a) + (ep(a)) J A ep(a) = 0 (ep(a)) A J ep(a) + (ep(a)) J A ep(a) = 0 (ep(a)) [A J + JA] ep(a) = 0 Muliplying a he righ by ((ep (A)) ) -1 and a he lef by (ep (A)) -1 we will obain: A J + JA = 0 A sp (n, ). References: [1] Mircea Pua Calcul mariceal, Timişoara, Ediura Miron, 2000 [2] Mircea Pua Varieăţi diferenţiabile - probleme, Timişoara, Ediura Miron, 2002 [3] Mircea Pua Hamilonian Mechanical Sysems and Geomeric Quaninizaion, Kluwer, [4] R. Abraham, J. Marsden and T. Raiu - Manifolds, Tensor Analysis and Applicaions, Second Ediion, Applied Mah. Sciences 75, Springer Verlag, Auhor: 447

7 Dorin Wainberg - On some properies of he symplecic and Hamilonian marices Dorin Wainberg, 1 Decembrie 1918 Universiy of Alba Iulia, address: wainbergdorin@yahoo.com 448