The C. Neumann problem as a completely integrable system on an adjoint orbit
The C. Neumann problem as a completely integrable system on an adjoint orbit
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DOI：
10.1090/s0002-9947-1981-0603766-3
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发表时间：
1981-02
影响因子：
1.3
通讯作者：
T. Ratiu
中科院分区：
文献类型：
--
作者：
T. Ratiu
用纯李代数方法证明了C.Neumann问题是一个完全可积的Euler-Poisson方程组，它是一个在李代数的半直积的最小维轨道上的Euler-Poisson方程组。1.C.Neumann问题。在二次势U(X)=‘Ax*x，xER’，A=diag(al，…，An)的影响下，球面S‘1上一点的运动是完全可积的哈密顿系统。对于n=3，C.Neumann在1859年[12]证明了这一点；对于任意n，K.Uhlenbeck[16]，R.Devaney[3]，J.Moser[10]，[11]，M.Adler和P.van Moerbeke[2]都证明了这一点。本文在Euler-Poisson方程[41，[5]，[14]，[17]的框架下证明了C·Neumann问题是李代数的半直积中最小维伴随轨道上的哈密顿系统。因此，它的完全可积性将完全源于李代数的考虑。运动方程为i=-ax1+Axx，i=1，…，n，(1.1)，其中拉格朗日乘数A=Ax‘x 11x112在运动过程中被选择为x E S n-i。设x=y，得到等价系统为(1.1)xi=yi，yi=-a1xi+(Ax.X 11y112)xi，llxll=1，x y=0。(1.2)促使本次调查的以下关键评论是K.Uhlenbeck说的，可以毫不费力地加以核实。引理1.1。将X=(Xixj)，P=(Yi Xj Xiyj)。系统(1.2)等价于X=[P，X，P=[X，A]，lIxl=1，xy=O。(1.3)请注意，如果用XID/n和A(Tr(A))ID/n分别替换X和A，其中ID是n×n单位矩阵，则等式(1.3)保持不变。从现在开始，我们将假设在(1.3)中进行了这样的改变，使得X，P，A，E，sl(N)。下一节将对这些方程进行李代数解释。由编辑于1980年3月12日收到。1980年《数学学科分类》。小学58F07，70H05；中学53C15，17B99。
It is shown by purely Lie algebraic methods that the C. Neumann problem-the motion of a material point on a sphere under the influence of a quadratic potential-is a completely integrable system of Euler-Poisson equations on a minimal-dimensional orbit of a semidirect product of Lie algebras. 1. The C. Neumann problem. The motion of a point on the sphere S`1 under the influence of a quadratic potential U(x) = 'Ax * x, x E R', A = diag(al,... , an) is a completely integrable Hamiltonian system. For n = 3 this has been shown by C. Neumann in 1859 [12] and for arbitrary n by K. Uhlenbeck [16], R. Devaney [3], J. Moser [10], [11], M. Adler, and P. van Moerbeke [2]. In this paper we show how this problem fits naturally in the framework of Euler-Poisson equations [41, [5], [14], [17] proving that the C. Neumann problem is a Hamiltonian system on a minimaldimensional adjoint orbit in a semidirect product of Lie algebras. Thus its complete integrability will follow entirely from Lie algebraic considerations. The equations of motion are i= -ax1 + AXx, i = 1,...,n, (1.1) where the Lagrange multiplier A = Ax' x 11x112 is chosen such that x E Sn-I during the motion. Set x = y and get the equivalent system to (1.1) xi = yi, yi = -a1xi + (Ax. x 11y112)xi, llxll = 1, x y = 0. (1.2) The following crucial remark that motivated the present investigation is due to K. Uhlenbeck and can be verified without any difficulties. LEMMA 1.1. Put X = (xixj), P = (yixj xiyj). System (1.2) is equivalent to X=[P,X, P =[X,A], lIxl = 1, x y = O. (1.3) Remark that if one replaces X and A by X Id/n and A (Tr(A))Id/n respectively, where Id is the n X n identity matrix, equations (1.3) remain unchanged. From now on we shall assume that in (1.3) this change has been made so that X, P, A E sl(n). The next section gives a Lie algebraic interpretation to these equations. Received by the editors March 12, 1980. 1980 Mathematics Subject Classification. Primary 58F07, 70H05; Secondary 53C15, 17B99.