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# Transformation of Lagrange brackets

The most common set of orbital elements used to parameterize Keplerian orbits consists of the major radius, ; the mean longitude at epoch, ; the eccentricity, ; the inclination (relative to some reference plane), ; the longitude of the perihelion, ; and the longitude of the ascending node, . (See Section 4.12.) The mean orbital angular velocity is [see Equation (4.117)].

Consider how a particular Lagrange bracket transforms under a rotation of the coordinate system , , about the -axis (if we look along the axis). We can write

 (G.41)

where

 (G.42)

Let the new coordinate system be . A rotation about the -axis though an angle brings the ascending node to the -axis. (See Figure 4.6.) The relation between the old and new coordinates is (see Section A.6)

 (G.43) (G.44) and (G.45)

The partial derivatives with respect to can be written

 (G.46) (G.47) (G.48) and (G.49)

where

 (G.50) (G.51) (G.52) and (G.53)

Let , , , and be the equivalent quantities obtained by replacing by in the preceding equations. It thus follows that

 (G.54) and (G.55)

Hence,

 (G.56)

Now,

 (G.57)

Similarly,

 (G.58)

Let

 (G.59)

Because and , it follows that

 (G.60)

However,

 (G.61)

because the left-hand side is the component of the angular momentum per unit mass parallel to the -axis. Of course, this axis is inclined at an angle to the -axis, which is parallel to the angular momentum vector. Thus, we obtain

 (G.62)

Consider a rotation of the coordinate system about the -axis. Let the new coordinate system be , , . A rotation through an angle brings the orbit into the - plane. (See Figure 4.6.) Let

 (G.63)

By analogy with the previous analysis,

 (G.64)

However, and are both zero, because the orbit lies in the - plane. Hence,

 (G.65)

Consider, finally, a rotation of the coordinate system about the -axis. Let the final coordinate system be , , . A rotation through an angle brings the perihelion to the -axis. (See Figure 4.6.) Let

 (G.66)

By analogy with the previous analysis,

 (G.67)

However,

 (G.68)

so, from Equations (G.62) and (G.65),

 (G.69)

It thus remains to calculate .

The coordinates and --where represents radial distance from the Sun, and is the true anomaly--are functions of the major radius, , the eccentricity, , and the mean anomaly, . Because the Lagrange brackets are independent of time, it is sufficient to evaluate them at ; that is, at the perihelion point. It is easily demonstrated from Equations (4.86) and (4.87) that

 (G.70) (G.71) (G.72) and (G.73)

at small . Hence, at ,

 (G.74) (G.75) (G.76) (G.77) (G.78) and (G.79)

because . All other partial derivatives are zero. Because the orbit in the , , coordinate system only depends on the elements , , and , we can write

 (G.80)

Substitution of the values of the derivatives evaluated at into this expression yields

 (G.81) (G.82) (G.83)

and

 (G.84)

where . Hence, from Equation (G.69), we obtain

 (G.85)

Next: Lagrange planetary equations Up: Derivation of Lagrange planetary Previous: Lagrange brackets
Richard Fitzpatrick 2016-03-31