Lagrange brackets

Six new equations can be derived from Equations (G.24)–(G.29) by multiplying them successively by $-\partial \dot{X}/\partial c_j$, $-\partial \dot{Y}/\partial c_j$, $-\partial \dot{Z}/\partial c_j$, $\partial X/\partial c_j$, $\partial Y/\partial c_j$, and $\partial Z/\partial c_j$, and then summing the resulting equations. The right-hand sides of the new equations are

$\displaystyle \frac{\partial {\cal R}}{\partial X}\,\frac{\partial X}{\partial ...
...\,\frac{\partial Z}{\partial c_j}\equiv
\frac{\partial {\cal R}}{\partial c_j}.$ (G.30)

The new equations can be written in a more compact form via the introduction of Lagrange brackets, which are defined as

$\displaystyle [c_j,c_k] \equiv \sum_{l=1,3}\left(\frac{\partial X_l}{\partial c...{\partial X_l}{\partial c_k}\,\frac{\partial \dot{X}_l}{\partial c_j}\right),$ (G.31)

where $X_1\equiv X$, $X_2\equiv Y$, and $X_3\equiv Z$. Thus, the new equations become

$\displaystyle \sum_{k=1,6}[c_j, c_k]\,\frac{dc_k}{dt} = \frac{\partial {\cal R}}{\partial c_j},$ (G.32)

for $j=1,6$. Note, incidentally, that

$\displaystyle [c_j,c_j]$ $\displaystyle = 0,$ (G.33)
$\displaystyle [c_j,c_k]$ $\displaystyle = -[c_k,c_j].$ (G.34)


$\displaystyle [p,q] = \sum_{l=1,3}\left(\frac{\partial X_l}{\partial p}\,\frac{...
... \frac{\partial X_l}{\partial q}\,\frac{\partial \dot{X}_l}{\partial p}\right),$ (G.35)

where $p$ and $q$ are any two orbital elements. It follows that

$\displaystyle \frac{\partial}{\partial t} [p,q] = \sum_{l=1,3}\left(\frac{\part...
...l}{\partial q}\,\frac{\partial^{\,2} \dot{X}_l}{\partial p\,\partial t}\right),$ (G.36)


$\displaystyle \frac{\partial}{\partial t} [p,q]= \sum_{l=1,3}\left[
...\partial X_l}{\partial p}\,\frac{\partial \dot{X}_l}{\partial t}\right)\right].$ (G.37)

However, in the preceding expression, $X_l$ and $\dot{X}_l$ stand for coordinates and velocities of Keplerian orbits calculated with $c_1,\cdots,c_6$ treated as constants. Thus, we can write $\partial X_l/\partial t\equiv \dot{X}_l$ and $\partial\dot{X}_l/\partial t\equiv \ddot{X}_l$, giving

$\displaystyle \frac{\partial}{\partial t} [p,q]= \sum_{l=1,3}\left[
...rac{\partial F_0}{\partial X_l}\,\frac{\partial X_l}{\partial p}\right)\right],$ (G.38)


$\displaystyle \ddot{X}_l = \frac{\partial F_0}{\partial X_l},$ (G.39)

where $F_0 = \mu/r$. Expression (G.38) reduces to

$\displaystyle \frac{\partial}{\partial t}[p,q] = \frac{1}{2}\frac{\partial^{\,2...
...partial q\,\partial p} + \frac{\partial^{\,2} F_0}{\partial q\,\partial p} = 0,$ (G.40)

where $\varv^{\,2}=\sum_{l=1,3}\dot{X}_l^{\,2}$. Hence, we conclude that Lagrange brackets are functions of the osculating orbital elements, $c_1,\ldots,c_6$, but are not explicit functions of $t$. It follows that we can evaluate these brackets at any convenient point in the orbit.