On the use of the autonomous Birkhoff equations in Lie series perturbation theory

Abstract

In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff–Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff’s equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space.

Appendix: The expression for the averaged Birkhoffian (the first example)

The resulting averaged Birkhoffian is represented as follows:

$$\begin{aligned} B^*=B_{00}^*+B_{01}^*+B_{02}^*+B_{03}^*+B_{04}^*+B_{05}^*,\quad B_{00}^*=B_{00},\quad B_{01}^*= B_{01}, \end{aligned}$$

where

$$\begin{aligned} B_{02}^*= & {} \frac{1}{16}\nu ^2 a^2 (((2+3 e^2)(1-3 \gamma ^2+3(-1+\gamma ^2)\cos (2 h))\\&-15 e^2 \cos (2 g)(1-\gamma ^2+(1+\gamma ^2) \cos (2 h)+30 e^2\gamma \sin (2 g) \sin (2 h)),\\ B_{03}^*= & {} 0,\\ B_{04}^*= & {} \frac{1}{4096}\frac{\nu ^4 a^2}{n^2}(8(47+282 \gamma ^2 +63\gamma ^4)+63 e^4 (239+170\gamma ^2+143 \gamma ^4)\\&-\,72 e^2 (377{+}190\gamma ^2{+}209 \gamma ^4)+2592 \cos (2 h) {+}168 \cos (4 h){-}24 e^2 \cos (2 g) (1{-}\gamma ^2\\&+\,(1+\gamma ^2) \cos (2 h)(27(2+e^2){+}5(78{-}37 e^2)\gamma ^2{+}5({-}78{+}37 e^2)({-}1 {+}\gamma ^2) \cos (2 h)\\&-\,410 e^2\gamma \sin (2 g)\sin (2 h)+3((56\gamma ^2({-}2 {+}\gamma ^2){-}1672 e^2 ({-}1{+}\gamma ^2)^2 \\&+\,1001 e^4(-1 {+}\gamma ^2))\cos (4 h){+}205 e^4 \cos (4 g) (-3(-1 {+}\gamma ^2)^2+4({-}1{+}\gamma ^4)\cos (2 h)\\&-\,(1+6\gamma ^2+\gamma ^4)\cos (4 h))+16 e^2\gamma (27(2+e^2)\\&+\, 5(78-37 e^2)\gamma ^2)\sin (2 g)\sin (2 h)+4\cos (2 h)(-8\gamma ^2(20+7\gamma ^2)\\&+\,152 e^2(-13+2\gamma ^2+11\gamma ^4)-7 e^4(-199+56\gamma ^2+143\gamma ^4)\\&+\, 20 e^2({-}78{+}37 e^2)\gamma ({-}1{+}\gamma ^2)\sin (2g)\sin (2h))),\\ B_{05}^*= & {} \frac{1}{128}\frac{\nu ^4 a^2}{n^2}m\,\eta (\gamma (176-2775 e^2+870 e^4 +(212-1895 e^2+675 e^4)\gamma ^2\\&-\,(212-1895 e^2+675 e^4)(-1+\gamma ^2)\cos (2 h)+4 e^2(-101 +17e^2)\gamma \cos (2 g)(3\\&-\,3\gamma ^2+(-1+3\gamma ^2)\cos (2 h))-8e^2(-101+17 e^2)(-1 +2 \gamma ^2)\sin (2 g) \sin (2 h)). \end{aligned}$$

The above expressions were checked by comparison with the results of Hori (1963), under the condition $\gamma =1$ and $h=0$. Complete coincidence was found with the analytical expressions for the functions $B_{02}^*$, $B_{03}^*$, $B_{04}^*$. For the function $B_{05}^*$, only the secular part coincided. This discrepancy can be explained by the use of different methods (Lie transformations and von Zeipel method) in solving the problem.

Note Expressions for the $B_{0i}$ functions in the traditional form adopted by the perturbation theory can be obtained in the Mathematica package using the function $Expand[TrigReduce[B_{0i}]]$. Examples of these expressions for the planar version of the problem can be found in Sect. 3.1.