Evaluation of the Lagrange Brackets

Abstract

The Lagrange brackets, $\left [{c_j},{c_k}\right ]$, must be evaluated in terms of the orbital elements and a particular coordinate frame so that

$$\displaystyle \sum \limits _{k=1}^{6}{\left [ {{c}_{j}},{{c}_{k}} \right ]}{{\dot {c}}_{k}}=\mathbf {F}\cdot \frac {\partial \mathbf {r}}{\partial {{c}_{j}}}, \hspace {0.5in} j=1,2,\dots ,6 $$

may be solved for the ${{\dot {c}}_{k}}$ (i.e., to find the differential equations). (See Eq. 3.41.)