Lagrangian vector field on Kähler manifold (Q1367409)

The notion of Lagrangian vector field is introduced in Kähler manifolds in order to solve the complex-valued problems arising in mechanics. Denoting by \(U(z,\overline z)\) the potential, the Lagrangian function is given by \(L= {1\over 2} h_{jk} \dot z^j {\dot {\overline z}}^k -U(z,\overline z) =T-U\) and the total energy by \(E=T+U\), from which the author is able to define the Lagrangian vector field in the same way taken by \textit{W. D. Curtis} and \textit{F. R. Miller} [`Differential manifolds and theoretical physics' (Pure and Applied Math. 116, Academic Press, Orlando) (1985; Zbl 0566.57001)] in the Riemannian case before. Then the Newtonian and Lagrangian equations of motion are shown to be of the form \(\ddot{\overline z}^j+ h^{ \overline ji} {\partial h_j \overline K \over \partial \overline z^j} \dot {\overline z}^j \dot {\overline z}^k= -2h^{\overline ji} {\partial U\over \partial z^j}\), (conj.), and \({d\over dt} {\partial L \over \partial \dot z^i}- {\partial L\over \partial z^i} =0\), (conj.), respectively. Thus the integral curve of a Lagrangian vector field is nothing but the path of motion of a particle that satisfies these two systems of equation.