试讨论 Lagrange 形式下的一维理想磁流体力学方程组 (5. 33)-(5. 39) 的类型.
解答: 由 (5. 33), (5. 39) 知 $$\bex 0=\cfrac{\p p}{\p \tau}\sex{\cfrac{\p \tau}{\p t'}-\cfrac{\p u_1}{\p m}}+\cfrac{\p p}{\p S}\cfrac{\p S}{\p t'} =\cfrac{\p p}{\p t'}-p'(\tau)\cfrac{\p u_1}{\p m}, \eex$$ 而 $$\bex \cfrac{-1}{p'(\tau)}\cfrac{\p p}{\p t'}+\cfrac{\p u_1}{\p m}=0. \eex$$ 于是 (5. 33)-(5. 39) 为 $$\beex \bea \cfrac{-1}{p'(\tau)}\cfrac{\p p}{\p t'}+\cfrac{\p u_1}{\p m} &=0,\\ \cfrac{\mu_0}{\rho}\cfrac{\p H_2}{\p t'} +\mu_0H_2\cfrac{\p u_1}{\p m}-\mu_0H_1\cfrac{\p u_2}{\p m} &=0,\\ \cfrac{\mu_0}{\rho}\cfrac{\p H_3}{\p t'} +\mu_0H_3\cfrac{\p u_1}{\p m} -\mu_0H_1\cfrac{\ pu_3}{\p m} &=0,\\ \cfrac{\p u_1}{\p t'} +\cfrac{\p\rho}{\p m} +\mu_0H_2\cfrac{\p H_2}{\p m} +\mu_0H_3\cfrac{\p H_3}{\p m}&=F_1,\\ \cfrac{\p u_2}{\p t'}-\mu_0H_1\cfrac{\p H_2}{\p m}&=F_2,\\ \cfrac{\p u_3}{\p t'}-\mu_0H_1\cfrac{\p H_3}{\p m}&=F_3,\\ \cfrac{\p S}{\p t'}&=0; \eea \eeex$$ 其可化为 $$\bex A(U)\cfrac{\p U}{\p t'}+B(U)\cfrac{\p U}{\p m}=C, \eex$$ 其中 $$\beex \bea U&=(p,H_2,H_3,u_1,u_2,u_3,S)^T,\\ A(U)&=\diag\sex{\cfrac{-1}{p'(\tau)},\cfrac{\mu_0}{\rho},\cfrac{\mu_0}{\rho}, 1,1,1,1},\\ B(U)&=\sex{\ba{ccccccc} 0&0&0&1&0&0&0\\ 0&0&0&\mu_0H_2&-\mu_0H_1&0&0\\ 0&0&0&\mu_0H_3&0&-\mu_0H_1&0\\ 1&\mu_0H_2\mu_0H_3&0&0&0&0\\ 0&-\mu_0H_1&0&0&0&0&0\\ 0&0&-\mu_0H_1&0&0&0&0\\ 0&0&0&0&0&0&0 \ea},\\ C&=(0,0,0,F_1,F_2,F_3,0)^T. \eea \eeex$$ 故 Lagrange 形式下的一维理想磁流体力学方程组 (5. 33)-(5. 39) 是一阶对称双曲组.
