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Next: Transformation of Lagrange brackets Up: Derivation of Lagrange planetary Previous: Preliminary analysis

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

$\displaystyle [c_j,c_k] \equiv \sum_{l=1,3}\left(\frac{\partial X_l}{\partial c...
...ac{\partial X_l}{\partial c_k}\,\frac{\partial \dot{X}_l}{\partial c_j}\right),$ (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)
and   $\displaystyle [c_j,c_k]$ $\displaystyle = -[c_k,c_j].$     (G.34)

Let

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

or

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

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

because

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


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Next: Transformation of Lagrange brackets Up: Derivation of Lagrange planetary Previous: Preliminary analysis
Richard Fitzpatrick 2016-03-31