D t2 71 dai I q

because of Eqn (9.19) and, since q is the solution to the two-body problem, it

Quoted from Wilson (1995b).

follows that

Equations (9.21) and (9.22) represent six linear algebraic equations forthe functions a j (t). Considered as six first-order differential equations for the position and velocity, they are equivalent to the three second-order Eqns (9.19).

In order to write these equations in a more elegant form, we dot Eqn (9.21) with -dp/daj, and Eqn (9.22) with dq/daj, and add the resulting equations. Since R is a function of position only, we obtain 6

daj daj daj daj

The quantities (aj, a,) are known as 'Lagrange brackets' and Lagrange showed that d dt'

so that t does not enter explicitly into the expression for (aj, a,) but only

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