HOLONOMIC方程在正则点处的完全可积性条件

ON CONDITIONS FOR COMPLETE INTEGRABILITY OF HOLONOMIC EQUATIONS AT REGULAR POINTS

本文中,一个Holonomic方程E是指(2n+1)接触流形的一个(n+1)维子流形。设F=(F_1,F_2,…,F_n);(J^1(1R^n,1R),z_0)→(1R^n,0)是一个Holonomic方程E的表示,A(F)=((?)F_i/(?)r_i+p_j(?)F_i/(?)u),B(F)=((?)F_i/(?)p_j),其中(x,u,p)=(x_1,…,x_n;u;p_1,…,p_n)是J^1(1R^n,1R)中的典则坐际。我们的主要结果如下: 若z_0是E的π—正则点,则F可以适当选取使得B(F)=I_n而且E在z_0处完全可积的充要条件为A(F)=(A(F))~t; 若z_0是E的接触正则点,则F可以适当选取使得A(F)=I_n,而且E在z_0处完全可积的充要条件为B(F)=(B(F))~t。

In this paper a holonomic equation is an (n+1)-dimensional submani fold in acontact manifold of dimension (2n+1). Let F=(F_1, F_2,…, F_n):(J′(IR_~n, IR)z_0)→(1R^n, O) be a representative of a holonomic equation E, and A(F)=(()F_i/()xj+p_j()F_i/()u),B(F)=(()F_i/()p_j). Our main result is as follows: If Z_0 is a π-regular point of E, then F can be chosen s. t. B(F)=I_n and E iscompletely integrable at z_0 iff A(F)=(A(F))~l; if z_0∈∑π is a contact regular point,then F can be chosen s. t. A(F)=I_n and E is completely integrable at z_0 iff B(F)=(B(F)~l.