Normalization of Hamiltonian and nonlinear stability of the triangular equilibrium points in non-resonance case with perturbations

Abstract

For the study of nonlinear stability of a dynamical system, normalized Hamiltonian of the system is very important to discuss the dynamics in the vicinity of invariant objects. In general, it represents a nonlinear approximation to the dynamics, which is very helpful to obtain the information as regards a realistic solution of the problem. In the present study, normalization of the Hamiltonian and analysis of nonlinear stability in non-resonance case, in the Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness, and a disc is performed. To describe nonlinear stability, initially, quadratic part of the Hamiltonian is normalized in the neighborhood of triangular equilibrium point and then higher order normalization is performed by computing the fourth order normalized Hamiltonian with the help of Lie transforms. In non-resonance case, nonlinear stability of the system is discussed using the Arnold–Moser theorem. Again, the effects of radiation pressure, oblateness and the presence of the disc are analyzed separately and it is observed that in the absence as well as presence of perturbation parameters, triangular equilibrium point is unstable in the nonlinear sense within the stability range $0<\mu<\mu_{1}=\bar{\mu_{c}}$ due to failure of the Arnold–Moser theorem. However, perturbation parameters affect the values of $\mu$ at which $D_{4}=0$, significantly. This study may help to analyze more generalized cases of the problem in the presence of some other types of perturbations such as P-R drag and solar wind drag. The results are limited to the regular symmetric disc but it can be extended in the future.

Appendix

1.1 A.1 Arnold–Moser theorem (Meyer and Schmidt 1986; Meyer et al. 1992)

Consider a Hamiltonian, which is the function of canonical coordinates $x_{i}, y_{i}, i=1, 2$, expressed as

$$\begin{aligned} H=H_{2}+H_{4}+H_{6}+\cdots+H_{2n}+H^{*}_{2n+1}, \end{aligned}$$

(77)

where

• 1. $H$ is real analytic in the a neighborhood of the origin in $\mathbb{R}^{4}$;

• 2. $H_{2k}, 1\leq k\leq n$, is a homogeneous function of degree $k$ in $I_{i}=\frac{1}{2}(x^{2}_{i}+y^{2}_{i}), i=1, 2$;

• 3. $H^{*}$ has a series expansion which starts with terms at least of order $2n+1$;

• 4. $H_{2}=\omega_{1}I_{1}-\omega_{2}I_{2}$ with $\omega_{i}, i=1, 2$ positive constants;

• 5. $H_{4}=\frac{1}{2} (AI^{2}_{1}-2BI_{1}I_{2}+CI^{2}_{2} ), A, B, C$ constants.

There are several implicit assumptions in stating that Hamiltonian $H$ in the form of (77). As $H$ is at least quadratic in canonical coordinates $x_{i}, y_{i}, i=1, 2$, the origin is assumed to be the equilibrium point in question. Again, $H_{2}=\omega_{1}I_{1}-\omega_{2}I_{2}$ is the Hamiltonian of two harmonic oscillators with frequency $\omega_{1}$ and $\omega_{2}$, the linearization at the origin of the system of equations whose Hamiltonian is $H$, is two harmonic oscillators. Since $H_{2}$ is not sign definite, a simple appeal to the stability theorem of Lyapunov cannot be made. Again, $H_{2}, H_{4}, \dots, H_{2n}$ are functions of only $I_{i}=\frac{1}{2}(x_{i}+y_{i}), i=1, 2$, the Hamiltonian is assumed to be in Birkhoff’s normal form up to terms of degree $2n$. Birkhoff’s normal form usually requires some non-resonance assumptions on the frequencies $\omega_{1}$ and $\omega_{2}$, but in order to state the theorem, we assume that $H$ is in the required form.

Theorem A.1

Arnold–Moser

The origin is stable for the system whose Hamiltonian is (77) provided for some $k, 2\leq k\leq n, D_{2k}=H_{2k}(\omega_{2}, \omega_{1})\neq 0$ or equivalently provided $H_{2}$ does not divide $H_{2k}$.

1.2 A.2 Coefficient in $K_{2}$

$$\begin{aligned} {k_{2020}} =&\frac{1}{\omega _{1} (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} )} \bigl[-4 i {g_{1011}} {g_{1110}} \omega _{1}^{3}-2 i {g_{0120}} {g_{2001}} \omega _{1}^{2} \omega _{2} +12 i {g_{1020}} {g_{2010}} \omega _{1}^{2} \omega _{2}+2 i {g_{0021}} {g_{2100}} \omega _{1}^{2} \omega _{2} \\ &{}+12 i {g_{0030}} {g_{3000}} \omega _{1}^{2} \omega _{2} +4 {g_{2020}} \omega _{1}^{3} \omega _{2}+i {g_{1011}} {g_{1110}} \omega _{1} \omega _{2}^{2}+i{g_{0120}} {g_{2001}} \omega _{1} \omega _{2}^{2}+i {g_{0021}} {g_{2100}} \omega _{1} \omega _{2}^{2} -3 i {g_{1020}} {g_{2010}} \omega _{2}^{3} \\ &{}-3 i {g_{0030}} {g_{3000}} \omega _{2}^{3}-{g_{2020}} \omega _{1} \omega _{2}^{3} \bigr], \end{aligned}$$

(78)

$$\begin{aligned} k_{2020e1} =& \frac{1}{\omega _{1} (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} )} \bigl[-4 i {g_{1011e1}} {g_{1110}} \omega _{1}^{3}-4 i {g_{1011}} {g_{1110e1}} \omega _{1}^{3}-2 i {g_{0120e1}} {g_{2001}} \omega _{1}^{2} \omega _{2} \\ &{}-2 i {g_{0120}} {g_{2001e1}} \omega _{1}^{2} \omega _{2}+12 i {g_{1020e1}} {g_{2010}} \omega _{1}^{2} \omega _{2} +12 i {g_{1020}} {g_{2010e1}} \omega _{1}^{2} \omega _{2}+2 i {g_{0021e1}} {g_{2100}} \omega _{1}^{2} \omega _{2} \\ &{}+2 i {g_{0021}} {g_{2100e1}} \omega _{1}^{2} \omega _{2} + 12 i {g_{0030e1}} {g_{3000}} \omega _{1}^{2} \omega _{2}+12 i {g_{0030}} {g_{3000e1}} \omega _{1}^{2} \omega _{2}+4 g_{2020e1} \omega _{1}^{3} \omega _{2} +i {g_{1011e1}} {g_{1110}} \omega _{1} \omega _{2}^{2} \\ &{}+i {g_{1011}} {g_{1110e1}} \omega _{1} \omega _{2}^{2}+i {g_{0120e1}} {g_{2001}} \omega _{1} \omega _{2}^{2} +i {g_{0120}} {g_{2001e1}} \omega _{1} \omega _{2}^{2}+i {g_{0021e1}} {g_{2100}} \omega _{1} \omega _{2}^{2}+i {g_{0021}} {g_{2100e1}} \omega _{1} \omega _{2}^{2} \\ &{}-3 i {g_{1020e1}} {g_{2010}} \omega _{2}^{3}- 3 i {g_{1020}} {g_{2010e1}} \omega _{2}^{3}-3 i {g_{0030e1}} {g_{3000}} \omega _{2}^{3} -3 i {g_{0030}} {g_{3000e1}} \omega _{2}^{3}-{g_{2020e1}} \omega _{1} \omega _{2}^{3} \bigr], \end{aligned}$$

(79)

$$\begin{aligned} k_{2020A} =&\frac{1}{\omega _{1} (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} )} \bigl[-4 i {g_{1011A}} {g_{1110}} \omega _{1}^{3}-4 i {g_{1011}} {g_{1110A}} \omega _{1}^{3}-2 i {g_{0120A}} {g_{2001}} \omega _{1}^{2} \omega _{2}-2 i{g_{0120}} {g_{2001A}} \omega _{1}^{2} \omega _{2} \\ &{}+12 i {g_{1020A}} {g_{2010}} \omega _{1}^{2} \omega _{2} +12 i {g_{1020}} {g_{2010A}} \omega _{1}^{2} \omega _{2}+2 i {g_{0021A}} {g_{2100}} \omega _{1}^{2} \omega _{2}+2 i {g_{0021}} {g_{2100A}} \omega _{1}^{2} \omega _{2} \\ &{}+ 12 i {g_{0030A}} {g_{3000}} \omega _{1}^{2} \omega _{2}+12 i {g_{0030}} {g_{3000A}} \omega _{1}^{2} \omega _{2}+4 {g_{2020A}} \omega _{1}^{3} \omega _{2} +i {g_{1011A}} {g_{1110}} \omega _{1} \omega _{2}^{2}+i {g_{1011}} {g_{1110A}} \omega _{1} \omega _{2}^{2} \\ &{}+i {g_{0120A}} {g_{2001}} \omega _{1} \omega _{2}^{2} +i {g_{0120}} {g_{2001A}} \omega _{1} \omega _{2}^{2}+i {g_{0021A}} {g_{2100}} \omega _{1} \omega _{2}^{2}+i {g_{0021}} {g_{2100A}} \omega _{1} \omega _{2}^{2} -3 i {g_{1020A}} {g_{2010}} \omega _{2}^{3} \\ &{}- 3 i {g_{1020}} {g_{2010A}} \omega _{2}^{3}-3 i {g_{0030A}} {g_{3000}} \omega _{2}^{3} -3 i {g_{0030}} {g_{3000A}} \omega _{2}^{3}-{g_{2020A}} \omega _{1} \omega _{2}^{3} \bigr], \end{aligned}$$

(80)

$$\begin{aligned} k_{2020e2} =& \frac{1}{\omega _{1} (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} )} \bigl[-4 i {g_{1011e2}} {g_{1110}} \omega _{1}^{3}-4 i {g_{1011}} {g_{1110e2}} \omega _{1}^{3}-2 i {g_{0120e2}} {g_{2001}} \omega _{1}^{2} \omega _{2} \\ &{}-2 i {g_{0120}} {g_{2001e2}} \omega _{1}^{2} \omega _{2}+12 i {g_{1020e2}} {g_{2010}} \omega _{1}^{2} \omega _{2} +12 i {{g1020}} {g_{2010e2}} \omega _{1}^{2} \omega _{2}+2 i {g_{0021e2}} {g_{2100}} \omega _{1}^{2} \omega _{2} \\ &{}+2 i {g_{0021}} {g_{2100e2}} \omega _{1}^{2} \omega _{2} + 12 i {g_{0030e2}} {g_{3000}} \omega _{1}^{2} \omega _{2}+12 i {g_{0030}} {g_{3000e2}} \omega _{1}^{2} \omega _{2}+4 {g_{2020e2}} \omega _{1}^{3} \omega _{2} +i {g_{1011e2}} {g_{1110}} \omega _{1} \omega _{2}^{2} \\ &{}+i {g_{1011}} {g_{1110e2}} \omega _{1} \omega _{2}^{2}+i {g_{0120e2}} {g_{2001}} \omega _{1} \omega _{2}^{2} +i {g_{0120}} {g_{2001e2}} \omega _{1} \omega _{2}^{2}+i {g_{0021e2}} {g_{2100}} \omega _{1} \omega _{2}^{2}+i {g_{0021}} {g_{2100e2}} \omega _{1} \omega _{2}^{2} \\ &{} -3 i {g_{1020e2}} {g_{2010}} \omega _{2}^{3}- 3 i {g_{1020}} {g_{2010e2}} \omega _{2}^{3}-3 i {g_{0030e2}} {g_{3000}} \omega _{2}^{3} -3 i{g_{0030}} {g_{3000e2}} \omega _{2}^{3}-{g_{2020e2}} \omega _{1} \omega _{2}^{3} \bigr], \end{aligned}$$

(81)

$$\begin{aligned} k_{1111} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} ) (\omega _{1}+2 \omega _{2} )} \bigl[-8 i {g_{0201}} {g_{1011}} \omega _{1}^{5}-8 i {g_{0102}} {g_{1110}} \omega _{1}^{5}-16 i {g_{0210}} {g_{1002}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{1020}} {g_{1101}} \omega _{1}^{4} \omega _{2}+16 i {g_{0012}} {g_{1200}} \omega _{1}^{4} \omega _{2}+8 i {g_{0120}} {g_{2001}} \omega _{1}^{4} \omega _{2}+8 i {g_{0111}} {g_{2010}} \omega _{1}^{4} \omega _{2}+8 i{g_{0021}} {g_{2100}} \omega _{1}^{4} \omega _{2} \\ &{}+4 {g_{1111}} \omega _{1}^{5} \omega _{2}+32 i {g_{0210}} {g_{1002}} \omega _{1}^{3} \omega _{2}^{2}+ 34 i {g_{0201}} {g_{1011}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0102}} {g_{1110}} \omega _{1}^{3} \omega _{2}^{2}+32 i {g_{0012}} {g_{1200}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}-4 i {g_{0120}} {g_{2001}} \omega _{1}^{3} \omega _{2}^{2}+4 i {g_{0021}} {g_{2100}} \omega _{1}^{3} \omega _{2}^{2}+4 i {g_{0210}} {g_{1002}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{1020}} {g_{1101}} \omega _{1}^{2} \omega _{2}^{3}-4 i{g_{0012}} {g_{1200}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}-32 i {g_{0120}} {g_{2001}} \omega _{1}^{2} \omega _{2}^{3}- 34 i {g_{0111}} {g_{2010}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0021}} {g_{2100}} \omega _{1}^{2} \omega _{2}^{3}-17 {g_{1111}} \omega _{1}^{3} \omega _{2}^{3}-8 i {g_{0210}} {g_{1002}} \omega _{1} \omega _{2}^{4} \\ &{}-8 i {g_{0201}} {g_{1011}} \omega _{1} \omega _{2}^{4}-8 i {g_{0102}} {g_{1110}} \omega _{1} \omega _{2}^{4} -8 i {g_{0012}} {g_{1200}} \omega _{1} \omega _{2}^{4}+16 i {g_{0120}} {g_{2001}} \omega _{1} \omega _{2}^{4}-16 i {g_{0021}} {g_{2100}} \omega _{1} \omega _{2}^{4} \\ &{}+8 i {g_{1020}} {g_{1101}} \omega _{2}^{5}+8 i {g_{0111}} {g_{2010}} \omega _{2}^{5}+4 {g_{1111}} \omega _{1} \omega_{2}^{5} \bigr], \end{aligned}$$

(82)

$$\begin{aligned} k_{1111e1} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} ) (\omega _{1}+2 \omega _{2} )} \bigl[-8 i {g_{0201e1}} {g_{1011}} \omega _{1}^{5}-8 i {g_{0201}} {g_{1011e1}} \omega _{1}^{5}-8 i {g_{0102e1}} {g_{1110}} \omega _{1}^{5} \\ &{}-8 i {g_{0102}} {g_{1110e1}} \omega _{1}^{5} -16 i {g_{0210e1}} {g_{1002}} \omega _{1}^{4} \omega _{2}-16 i {g_{0210}} {g_{1002e1}} \omega _{1}^{4} \omega _{2}+8 i {g_{1020e1}} {g_{1101}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{1020}} {g_{1101e1}} \omega _{1}^{4} \omega _{2}+16 i {g_{0012e1}} {g_{1200}} \omega _{1}^{4} \omega _{2}+ 16 i {g_{0012}} {g_{1200e1}} \omega _{1}^{4} \omega _{2} +8 i {g_{0120e1}} {g_{2001}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{0120}} {g_{2001e1}} \omega _{1}^{4} \omega _{2}+8 i {g_{0111e1}} {g_{2010}} \omega _{1}^{4} \omega _{2} +8 i{g_{0111}} {g_{2010e1}} \omega _{1}^{4} \omega _{2}+8 i {g_{0021e1}} {g_{2100}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{0021}} {g_{2100e1}} \omega _{1}^{4} \omega _{2} +4 {g_{1111e1}} \omega _{1}^{5} \omega _{2}+32 i{g_{0210e1}} {g_{1002}} \omega _{1}^{3} \omega _{2}^{2}+ 32 i {g_{0210}} {g_{1002e1}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0201e1}} {g_{1011}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}+34 i {g_{0201}} {g_{1011e1}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0102e1}} {g_{1110}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0102}} {g_{1110e1}} \omega _{1}^{3} \omega _{2}^{2}+32 i{g_{0012e1}} {g_{1200}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}+32 i {g_{0012}} {g_{1200e1}} \omega _{1}^{3} \omega _{2}^{2}-4 i {g_{0120e1}} {g_{2001}} \omega _{1}^{3} \omega _{2}^{2}-4 i {g_{0120}} {g_{2001e1}} \omega _{1}^{3} \omega _{2}^{2}+ 4 i {g_{0021e1}} {g_{2100}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}+4 i {g_{0021}} {g_{2100e1}} \omega _{1}^{3} \omega _{2}^{2}+4 i {g_{0210e1}} {g_{1002}} \omega _{1}^{2} \omega _{2}^{3}+4 i {g_{0210}} {g_{1002e1}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{1020e1}} {g_{1101}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}-34 i {g_{1020}} {g_{1101e1}} \omega _{1}^{2} \omega _{2}^{3}-4 i {g_{0012e1}} {g_{1200}} \omega _{1}^{2} \omega _{2}^{3}-4 i {g_{0012}} {g_{1200e1}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0120e1}} {g_{2001}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}- 32 i {g_{0120}} {g_{2001e1}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{0111e1}} {g_{2010}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{0111}} {g_{2010e1}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0021e1}} {g_{2100}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}-32 i {g_{0021}} {g_{2100e1}} \omega _{1}^{2} \omega _{2}^{3}-17 {g_{1111e1}} \omega _{1}^{3} \omega _{2}^{3}-8 i {g_{0210e1}} {g_{1002}} \omega _{1} \omega _{2}^{4}-8 i {g_{0210}} {g_{1002e1}} \omega _{1} \omega _{2}^{4}-8 i {g_{0201e1}} {g_{1011}} \omega _{1} \omega _{2}^{4} \\ &{}- 8 i {g_{0201}} {g_{1011e1}} \omega _{1} \omega _{2}^{4}-8 i {g_{0102e1}} {g_{1110}} \omega _{1} \omega _{2}^{4}-8 i {g_{0102}} {g_{1110e1}} \omega _{1} \omega _{2}^{4}-8 i {g_{0012e1}} {g_{1200}} \omega _{1} \omega _{2}^{4} \\ &{}-8 i {g_{0012}} {g_{1200e1}} \omega _{1} \omega _{2}^{4}+16 i {g_{0120e1}} {g_{2001}} \omega _{1} \omega _{2}^{4}+16 i {g_{0120}} {g_{2001e1}} \omega _{1} \omega _{2}^{4} -16 i {g_{0021e1}} {g_{2100}} \omega_{1} \omega_{2}^{4} \\ &{}-16 i {g_{0021}} {g_{2100e1}} \omega _{1} \omega _{2}^{4}+8 i {g_{1020e1}} {g_{1101}} \omega _{2}^{5}+8 i {g_{1020}} {g_{1101e1}} \omega _{2}^{5}+8 i {g_{0111e1}} {g_{2010}} \omega _{2}^{5}+8i {g_{0111}} {g_{2010e1}} \omega_{2}^{5} \\ &{} +4 {g_{1111e1}} \omega _{1} \omega_{2}^{5} \bigr], \end{aligned}$$

(83)

$$\begin{aligned} k_{1111A} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} ) (\omega _{1}+2 \omega _{2} )} \bigl[-8 i {g_{0201A}} {g_{1011}} \omega _{1}^{5}-8 i {g_{0201}} {g_{1011A}} \omega _{1}^{5}-8 i {g_{0102A}} {g_{1110}} \omega _{1}^{5} \\ &{}-8 i {g_{0102}} {g_{1110A}} \omega _{1}^{5} -16 i {g_{0210A}} {g_{1002}} \omega _{1}^{4} \omega _{2}-16 i {g_{0210}} {g_{1002A}} \omega _{1}^{4} \omega _{2}+8 i {g_{1020A}} {g_{1101}} \omega _{1}^{4} \omega _{2}+8 i {g_{1020}} {g_{1101A}} \omega _{1}^{4} \omega _{2} \\ &{}+16 i {g_{0012A}} {g_{1200}} \omega _{1}^{4} \omega _{2}+16 i {g_{0012}} {g_{1200A}} \omega _{1}^{4} \omega _{2} + 8 i {g_{0120A}} {g_{2001}} \omega _{1}^{4} \omega _{2}+8 i {g_{0120}} {g_{2001A}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{0111A}} {g_{2010}} \omega _{1}^{4} \omega _{2} +8 i {g_{0111}} {g_{2010A}} \omega _{1}^{4} \omega _{2}+8 i {g_{0021A}} {g_{2100}} \omega _{1}^{4} \omega _{2}+8 i {g_{0021}} {g_{2100A}} \omega _{1}^{4} \omega _{2} +4 {g_{1111A}} \omega _{1}^{5} \omega _{2} \\ &{}+32 i {g_{0210A}} {g_{1002}} \omega _{1}^{3} \omega _{2}^{2}+32 i {g_{0210}} {g_{1002A}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0201A}} {g_{1011}} \omega _{1}^{3} \omega _{2}^{2}+ 34 i {g_{0201}} {g_{1011A}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}+34 i {g_{0102A}} {g_{1110}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0102}} {g_{1110A}} \omega _{1}^{3} \omega _{2}^{2}+32 i {g_{0012A}} {g_{1200}} \omega _{1}^{3} \omega _{2}^{2}+32 i {g_{0012}} {g_{1200A}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}-4 i {g_{0120A}} {g_{2001}} \omega _{1}^{3} \omega _{2}^{2}-4 i {g_{0120}} {g_{2001A}} \omega _{1}^{3} \omega _{2}^{2}+4 i {g_{0021A}} {g_{2100}} \omega _{1}^{3} \omega _{2}^{2}+4 i {g_{0021}} {g_{2100A}} \omega _{1}^{3} \omega _{2}^{2}+ 4 i {g_{0210A}} {g_{1002}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}+4 i {g_{0210}} {g_{1002A}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{1020A}} {g_{1101}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{1020}} {g_{1101A}} \omega _{1}^{2} \omega _{2}^{3}-4 i {g_{0012A}} {g_{1200}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}-4 i {g_{0012}} {g_{1200A}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0120A}} {g_{2001}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0120}} {g_{2001A}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{0111A}} {g_{2010}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}- 34 i {g_{0111}} {g_{2010A}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0021A}} {g_{2100}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0021}} {g_{2100A}} \omega _{1}^{2} \omega _{2}^{3}-17 {g_{1111A}} \omega _{1}^{3} \omega _{2}^{3}-8 i {g_{0210A}} {g_{1002}} \omega _{1} \omega _{2}^{4} \\ &{}-8 i {g_{0210}} {g_{1002A}} \omega _{1} \omega _{2}^{4}-8 i {g_{0201A}} {g_{1011}} \omega _{1} \omega _{2}^{4}-8 i {g_{0201}} {g_{1011A}} \omega _{1} \omega _{2}^{4} -8 i {g_{0102A}} {g_{1110}} \omega _{1} \omega _{2}^{4}- 8 i {g_{0102}} {g_{1110A}} \omega _{1} \omega _{2}^{4} \\ &{}-8 i {g_{0012A}} {g_{1200}} \omega _{1} \omega _{2}^{4} -8 i {g_{0012}} {g_{1200A}} \omega _{1} \omega _{2}^{4}+16 i {g_{0120A}} {g_{2001}} \omega _{1} \omega _{2}^{4}+16 i {g_{0120}} {g_{2001A}} \omega _{1} \omega _{2}^{4} \\ &{} -16 i {g_{0021A}} {g_{2100}} \omega _{1} \omega _{2}^{4}-16 i {g_{0021}} {g_{2100A}} \omega _{1} \omega _{2}^{4}+8 i {g_{1020A}} {g_{1101}} \omega _{2}^{5}+8 i {g_{1020}} {g_{1101A}} \omega _{2}^{5}+ 8 i {g_{0111A}} {g_{2010}} \omega _{2}^{5} \\ &{}+8 i {g_{0111}} {g_{2010A}} \omega _{2}^{5}+4 {g_{1111A}} \omega _{1} \omega _{2}^{5} \bigr], \end{aligned}$$

(84)

$$\begin{aligned} k_{1111e2} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) (2 \omega _{1}-\omega _{2} ) \omega _{2} (2 \omega _{1}+\omega _{2} ) (\omega _{1}+2 \omega _{2} )} \bigl[-8 i {g_{0201e2}} {g_{1011}} \omega _{1}^{5}-8 i {g_{0201}} {g_{1011e2}} \omega _{1}^{5}-8 i {g_{0102e2}} {g_{1110}} \omega _{1}^{5} \\ &{}-8 i {g_{0102}} {g_{1110e2}} \omega _{1}^{5} -16 i {g_{0210e2}} {g_{1002}} \omega _{1}^{4} \omega _{2}-16 i {g_{0210}} {g_{1002e2}} \omega _{1}^{4} \omega _{2}+8 i {g_{1020e2}} {g_{1101}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{1020}} {g_{1101e2}} \omega _{1}^{4} \omega _{2}+16 i {g_{0012e2}} {g_{1200}} \omega _{1}^{4} \omega _{2}+ 16 i {g_{0012}} {g_{1200e2}} \omega _{1}^{4} \omega _{2} +8 i {g_{0120e2}} {g_{2001}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{0120}} {g_{2001e2}} \omega _{1}^{4} \omega _{2}+8 i {g_{0111e2}} {g_{2010}} \omega _{1}^{4} \omega _{2} +8 i {g_{0111}} {g_{2010e2}} \omega _{1}^{4} \omega _{2}+8 i {g_{0021e2}} {g_{2100}} \omega _{1}^{4} \omega _{2} \\ &{}+8 i {g_{0021}} {g_{2100e2}} \omega _{1}^{4} \omega _{2} +4 {g_{1111e2}} \omega _{1}^{5} \omega _{2}+32 i {g_{0210e2}} {g_{1002}} \omega _{1}^{3} \omega _{2}^{2}+ 32 i {g_{0210}} {g_{1002e2}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0201e2}} {g_{1011}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}+34 i {g_{0201}} {g_{1011e2}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0102e2}} {g_{1110}} \omega _{1}^{3} \omega _{2}^{2}+34 i {g_{0102}} {g_{1110e2}} \omega _{1}^{3} \omega _{2}^{2}+32 i {g_{0012e2}} {g_{1200}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}+32 i {g_{0012}} {g_{1200e2}} \omega _{1}^{3} \omega _{2}^{2}-4 i {g_{0120e2}} {g_{2001}} \omega _{1}^{3} \omega _{2}^{2}-4 i {g_{0120}} {g_{2001e2}} \omega _{1}^{3} \omega _{2}^{2}+ 4 i {g_{0021e2}} {g_{2100}} \omega _{1}^{3} \omega _{2}^{2} \\ &{}+4 i {g_{0021}} {g_{2100e2}} \omega _{1}^{3} \omega _{2}^{2}+4 i {g_{0210e2}} {g_{1002}} \omega _{1}^{2} \omega _{2}^{3}+4 i {g_{0210}} {g_{1002e2}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{1020e2}} {g_{1101}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}-34 i {g_{1020}} {g_{1101e2}} \omega _{1}^{2} \omega _{2}^{3}-4 i {g_{0012e2}} {g_{1200}} \omega _{1}^{2} \omega _{2}^{3}-4 i {g_{0012}} {g_{1200e2}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0120e2}} {g_{2001}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}- 32 i {g_{0120}} {g_{2001e2}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{0111e2}} {g_{2010}} \omega _{1}^{2} \omega _{2}^{3}-34 i {g_{0111}} {g_{2010e2}} \omega _{1}^{2} \omega _{2}^{3}-32 i {g_{0021e2}} {g_{2100}} \omega _{1}^{2} \omega _{2}^{3} \\ &{}-32 i {g_{0021}} {g_{2100e2}} \omega _{1}^{2} \omega _{2}^{3}-17 {g_{1111e2}} \omega _{1}^{3} \omega _{2}^{3}-8 i {g_{0210e2}} {g_{1002}} \omega _{1} \omega _{2}^{4}-8 i {g_{0210}} {g_{1002e2}} \omega _{1} \omega _{2}^{4}-8 i {g_{0201e2}} {g_{1011}} \omega _{1} \omega _{2}^{4} \\ &{}- 8 i {g_{0201}} {g_{1011e2}} \omega _{1} \omega _{2}^{4}-8 i {g_{0102e2}} {g_{1110}} \omega _{1} \omega _{2}^{4}-8 i {g_{0102}} {g_{1110e2}} \omega _{1} \omega _{2}^{4}-8 i {g_{0012e2}} {g_{1200}} \omega _{1} \omega _{2}^{4} \\ &{}-8 i {g_{0012}} {g_{1200e2}} \omega _{1} \omega _{2}^{4}+16 i {g_{0120e2}} {g_{2001}} \omega _{1} \omega _{2}^{4}+16 i {g_{0120}} {g_{2001e2}} \omega _{1} \omega _{2}^{4} -16 i {g_{0021e2}} {g_{2100}} \omega _{1} \omega _{2}^{4} \\ &{}-16 i {g_{0021}} {g_{2100e2}} \omega _{1} \omega _{2}^{4}+ 8 i {g_{1020e2}} {g_{1101}} \omega _{2}^{5} +8 i {g_{1020}} {g_{1101e2}} \omega _{2}^{5}+8 i {g_{0111e2}} {g_{2010}} \omega _{2}^{5}+8 i {g_{0111}} {g_{2010e2}} \omega _{2}^{5} \\ &{}+4 {g_{1111e2}} \omega _{1} \omega _{2}^{5} \bigr], \end{aligned}$$

(85)

$$\begin{aligned} k_{0202} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) \omega _{2} (\omega _{1}+2 \omega _{2} )} \bigl[-3 i {g_{0102}} {g_{0201}} \omega _{1}^{3}-3 i {g_{0003}} {g_{0300}} \omega _{1}^{3}+i {g_{0210}} {g_{1002}} \omega _{1}^{2} \omega _{2}+i {g_{0111}} {g_{1101}} \omega _{1}^{2} \omega _{2} \\ &{}+i {g_{0012}} {g_{1200}} \omega _{1}^{2} \omega _{2} + {g_{0202}} \omega _{1}^{3} \omega _{2}+12 i {g_{0102}} {g_{0201}} \omega _{1} \omega _{2}^{2}+12 i {g_{0003}} {g_{0300}} \omega _{1} \omega _{2}^{2} -2 i {g_{0210}} {g_{1002}} \omega _{1} \omega _{2}^{2} \\ &{}+2 i {g_{0012}} {g_{1200}} \omega _{1} \omega _{2}^{2}-4 i {g_{0111}} {g_{1101}} \omega _{2}^{3}-4 {g_{0202}} \omega _{1} \omega _{2}^{3} \bigr], \end{aligned}$$

(86)

$$\begin{aligned} k_{0202e1} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) \omega _{2} (\omega _{1}+2 \omega _{2} )} \bigl[-3 i {g_{0102e1}} {g_{0201}} \omega _{1}^{3}-3 i {g_{0102}} {g_{0201e1}} \omega _{1}^{3}-3 i {g_{0003e1}} {g_{0300}} \omega _{1}^{3}-3 i {g_{0003}} {g_{0300e1}} \omega _{1}^{3} \\ &{}+i {g_{0210e1}} {g_{1002}} \omega _{1}^{2} \omega _{2}+i {g_{0210}} {g_{1002e1}} \omega _{1}^{2} \omega _{2}+i {g_{0111e1}} {g_{1101}} \omega _{1}^{2} \omega _{2}+i {g_{0111}} {g_{1101e1}} \omega _{1}^{2} \omega _{2} +i {g_{0012e1}} {g_{1200}} \omega _{1}^{2} \omega _{2} \\ &{}+ i {g_{0012}} {g_{1200e1}} \omega _{1}^{2} \omega _{2}+ {g_{0202e1}} \omega _{1}^{3} \omega _{2}+12 i {g_{0102e1}} {g_{0201}} \omega _{1} \omega _{2}^{2}+12 i {g_{0102}} {g_{0201e1}} \omega _{1} \omega _{2}^{2} \\ &{}+12 i {g_{0003e1}} {g_{0300}} \omega _{1} \omega _{2}^{2}+12 i {g_{0003}} {g_{0300e1}} \omega _{1} \omega _{2}^{2}-2 i {g_{0210e1}} {g_{1002}} \omega _{1} \omega _{2}^{2}-2 i {g_{0210}} {g_{1002e1}} \omega _{1} \omega _{2}^{2} \\ &{} +2 i {g_{0012e1}} {g_{1200}} \omega _{1} \omega _{2}^{2}+ 2 i {g_{0012}} {g_{1200e1}} \omega _{1} \omega _{2}^{2}-4 i {g_{0111e1}} {g_{1101}} \omega _{2}^{3}-4 i {g_{0111}} {g_{1101e1}} \omega _{2}^{3}-4 {g_{0202e1}} \omega _{1} \omega _{2}^{3} \bigr], \end{aligned}$$

(87)

$$\begin{aligned} k_{0202A} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) \omega _{2} (\omega _{1}+2 \omega _{2} )} \bigl[-3 i {g_{0102A}} {g_{0201}} \omega _{1}^{3}-3 i {g_{0102}} {g_{0201A}} \omega _{1}^{3}-3 i {g_{0003A}} {g_{0300}} \omega _{1}^{3}-3 i {g_{0003}} {g_{0300A}} \omega _{1}^{3} \\ &{}+i {g_{0210A}} {g_{1002}} \omega _{1}^{2} \omega _{2}+i {g_{0210}} {g_{1002A}} \omega _{1}^{2} \omega _{2}+i {g_{0111A}} {g_{1101}} \omega _{1}^{2} \omega _{2}+i {g_{0111}} {g_{1101A}} \omega _{1}^{2} \omega _{2} +i {g_{0012A}} {g_{1200}} \omega _{1}^{2} \omega _{2} \\ &{}+ i {g_{0012}} {g_{1200A}} \omega _{1}^{2} \omega _{2}+ {g_{0202A}} \omega _{1}^{3} \omega _{2}+12 i {g_{0102A}} {g_{0201}} \omega _{1} \omega _{2}^{2}+12 i {g_{0102}} {g_{0201A}} \omega _{1} \omega _{2}^{2}+12 i {g_{0003A}} {g_{0300}} \omega _{1} \omega _{2}^{2} \\ &{}+12 i {g_{0003}} {g_{0300A}} \omega _{1} \omega _{2}^{2}-2 i {g_{0210A}} {g_{1002}} \omega _{1} \omega _{2}^{2}-2 i {g_{0210}} {g_{1002A}} \omega _{1} \omega _{2}^{2} +2 i {g_{0012A}} {g_{1200}} \omega _{1} \omega _{2}^{2}+2 i {g_{0012}} {g_{1200A}} \omega _{1} \omega _{2}^{2} \\ &{}- 4 i {g_{0111A}} {g_{1101}} \omega _{2}^{3} -4 i {g_{0111}} {g_{1101A}} \omega _{2}^{3}-4 {g_{0202A}} \omega _{1} \omega _{2}^{3} \bigr], \end{aligned}$$

(88)

$$\begin{aligned} k_{0202e2} =&\frac{1}{\omega _{1} (\omega _{1}-2 \omega _{2} ) \omega _{2} (\omega _{1}+2 \omega _{2} )} \bigl[-3 i {g_{0102e2}} {g_{0201}} \omega _{1}^{3}-3 i {g_{0102}} {g_{0201e2}} \omega _{1}^{3}-3 i {g_{0003e2}} {g_{0300}} \omega _{1}^{3}-3 i {g_{0003}} {g_{0300e2}} \omega _{1}^{3} \\ &{}+i {g_{0210e2}} {g_{1002}} \omega _{1}^{2} \omega _{2}+i {g_{0210}} {g_{1002e2}} \omega _{1}^{2} \omega _{2}+i {g_{0111e2}} {g_{1101}} \omega _{1}^{2} \omega _{2}+i {g_{0111}} {g_{1101e2}} \omega _{1}^{2} \omega _{2} +i {g_{0012e2}} {g_{1200}} \omega _{1}^{2} \omega _{2} \\ &{}+ i {g_{0012}} {g_{1200e2}} \omega _{1}^{2} \omega _{2}+ {g_{0202e2}} \omega _{1}^{3} \omega _{2}+12 i {g_{0102e2}} {g_{0201}} \omega _{1} \omega _{2}^{2}+12 i {g_{0102}} {g_{0201e2}} \omega _{1} \omega _{2}^{2}+12 i {g_{0003e2}} {g_{0300}} \omega _{1} \omega _{2}^{2} \\ &{}+12 i {g_{0003}} {g_{0300e2}} \omega _{1} \omega _{2}^{2}-2 i {g_{0210e2}} {g_{1002}} \omega _{1} \omega _{2}^{2}-2 i {g_{0210}} {g_{1002e2}} \omega _{1} \omega _{2}^{2} +2 i {g_{0012e2}} {g_{1200}} \omega _{1} \omega _{2}^{2} \\ &{}+ 2 i {g_{0012}} {g_{1200e2}} \omega _{1} \omega _{2}^{2}-4 i {g_{0111e2}} {g_{1101}} \omega _{2}^{3}-4 i {g_{0111}} {g_{1101e2}} \omega _{2}^{3}-4 {g_{0202e2}} \omega _{1} \omega _{2}^{3} \bigr]. \end{aligned}$$

(89)

1.3 A.3 Coefficient in $D_{2}$

$$\begin{aligned} D_{20} =&\frac{-i}{4 \omega _{1}^{5} \omega _{2}-17 \omega _{1}^{3} \omega _{2}^{3}+4 \omega _{1} \omega _{2}^{5}} \bigl[-12 ( {g_{1020}} {g_{2010}}+ {g_{0030}} {g_{3000}} ) \omega _{2}^{7}+4 \omega _{1}^{7} (3 {g_{0102}} {g_{0201}} +3 {g_{0003}} {g_{0300}}+i \omega _{2}{g_{0202}} ) \\ &{}- 4 \omega _{1}^{6} \omega _{2} ( {g_{0210}} {g_{1002}}-2 {g_{0201}} {g_{1011}}+ {g_{0111}} {g_{1101}} -2 {g_{0102}} {g_{1110}} +{g_{0012}} {g_{1200}}-i \omega _{2} {g_{1111}} ) \\ &{}+ \omega _{1}^{2} \omega _{2}^{5} (8 {g_{0210}} {g_{1002}}+8 {g_{0201}} {g_{1011}} -4 {g_{0111}} {g_{1101}}+8 {g_{0102}} {g_{1110}} +8 {g_{0012}} {g_{1200}}-24 {g_{0120}} {g_{2001}}+51 {g_{1020}} {g_{2010}} \\ &{} +24 {g_{0021}} {g_{2100}}+51 {g_{0030}} {g_{3000}}+4 i \omega _{2} {g_{1111}} )-\omega _{1}^{4} \omega _{2}^{3} (31 {g_{0210}} {g_{1002}}+34 {g_{0201}} {g_{1011}}-17 {g_{0111}} {g_{1101}}+34 {g_{0102}} {g_{1110}} \\ &{}+31 {g_{0012}} {g_{1200}}-6 {g_{0120}} {g_{2001}}+12 {g_{1020}} {g_{2010}}+6 {g_{0021}} {g_{2100}}+12 {g_{0030}} {g_{3000}}+17 i \omega _{2} {g_{1111}} )+\omega _{1}^{3} \omega _{2}^{4} \bigl(12 {g_{0102}} {g_{0201}} \\ &{}+12 {g_{0003}} {g_{0300}}-6 {g_{0210}} {g_{1002}}+34 {g_{1020}} {g_{1101}} -17 {g_{1011}} {g_{1110}}+6 {g_{0012}} {g_{1200}}+31 {g_{0120}} {g_{2001}} +34 {g_{0111}} {g_{2010}} \\ &{}+31 {g_{0021}} {g_{2100}}+i\omega _{2} (4 {g_{0202}}-17 {g_{2020}} ) \bigr)-\omega _{1}^{5} \omega _{2}^{2} \bigl(51 {g_{0102}} {g_{0201}}+51 {g_{0003}} {g_{0300}}-24 {g_{0210}} {g_{1002}}+8 {g_{1020}} {g_{1101}} \\ &{} -4 {g_{1011}} {g_{1110}}+24 {g_{0012}} {g_{1200}}+8 {g_{0120}} {g_{2001}}+8 {g_{0111}} {g_{2010}}+8 {g_{0021}} {g_{2100}}+i (17 {g_{0202}}-4 {g_{2020}} ) \omega _{2} \bigr) \\ &{}+4 \omega _{1} \omega _{2}^{6} (-2 {g_{1020}} {g_{1101}}+ {g_{1011}} {g_{1110}}+ {g_{0120}} {g_{2001}}-2 {g_{0111}} {g_{2010}} + {g_{0021}} {g_{2100}}+i \omega_{2} {g_{2020}} ) \bigr], \end{aligned}$$

(90)

$$\begin{aligned} D_{21} =&\frac{-i}{4 \omega _{1}^{5} \omega _{2}-17 \omega _{1}^{3} \omega _{2}^{3}+4 \omega _{1} \omega _{2}^{5}} \bigl[-12 ( {g_{1020e1}} {g_{2010}}+ {g_{1020}} {g_{2010e1}}+ {g_{0030e1}} {g_{3000}}+ {g_{0030}} {g_{3000e1}} ) \omega _{2}^{7} \\ &{}+4 \omega _{1}^{7} \bigl(3 ( {g_{0102e1}} {g_{0201}}+ {g_{0102}} {g_{0201e1}}+ {g_{0003e1}} {g_{0300}}+ {g_{0003}} {g_{0300e1}} ) +i \omega _{2} {g_{0202e1}} \bigr)- 4 \omega _{1}^{6} \omega _{2} ( {g_{0210e1}} {g_{1002}} \\ &{}+ {g_{0210}} {g_{1002e1}}-2 {g_{0201e1}} {g_{1011}}-2 {g_{0201}} {g_{1011e1}} + {g_{0111e1}} {g_{1101}}+ {g_{0111}} {g_{1101e1}}-2 {g_{0102e1}} {g_{1110}}-2 {g_{0102}} {g_{1110e1}} \\ &{}+ {g_{0012e1}} {g_{1200}} + {g_{0012}} {g_{1200e1}}-i \omega _{2}{g_{1111e1}} ) +\omega _{1}^{2} \omega _{2}^{5} (8 {g_{0210e1}} {g_{1002}}+8 {g_{0210}} {g_{1002e1}}+8 {g_{0201e1}} {g_{1011}} \\ &{}+8 {g_{0201}} {g_{1011e1}}- 4 {g_{0111e1}} {g_{1101}}-4 {g_{0111}} {g_{1101e1}}+8 {g_{0102e1}} {g_{1110}}+8 {g_{0102}} {g_{1110e1}} + 8 {g_{0012e1}} {g_{1200}}+8 {g_{0012}} {g_{1200e1}} \\ &{}-24 {g_{0120e1}} {g_{2001}}-24 {g_{0120}} {g_{2001e1}}+ 51 {g_{1020e1}} {g_{2010}}+51 {g_{1020}} {g_{2010e1}}+24 {g_{0021e1}} {g_{2100}}+24 {g_{0021}} {g_{2100e1}} \\ &{}+51 {g_{0030e1}} {g_{3000}}+51 {g_{0030}} {g_{3000e1}}+4 i {g_{1111e1}} \omega _{2} )- \omega _{1}^{4} \omega _{2}^{3} (31 {g_{0210e1}} {g_{1002}}+31 {g_{0210}} {g_{1002e1}}+34 {g_{0201e1}} {g_{1011}} \\ &{}+34 {g_{0201}} {g_{1011e1}}- 17 {g_{0111e1}} {g_{1101}}-17 {g_{0111}} {g_{1101e1}}+34 {g_{0102e1}} {g_{1110}}+34 {g_{0102}} {g_{1110e1}}+ 31 {g_{0012e1}} {g_{1200}} \\ &{}+31 {g_{0012}} {g_{1200e1}}-6 {g_{0120e1}} {g_{2001}}-6 {g_{0120}} {g_{2001e1}}+ 12 {g_{1020e1}} {g_{2010}}+12 {g_{1020}} {g_{2010e1}}+6 {g_{0021e1}} {g_{2100}} \\ &{}+6 {g_{0021}} {g_{2100e1}}+ 12 {g_{0030e1}} {g_{3000}}+12 {g_{0030}} {g_{3000e1}}+17 i \omega _{2}{g_{1111e1}} )+ \omega _{1}^{3} \omega _{2}^{4} \bigl(12 {g_{0102e1}} {g_{0201}}+12 {g_{0102}} {g_{0201e1}} \\ &{}+12 {g_{0003e1}} {g_{0300}}+12 {g_{0003}} {g_{0300e1}}- 6 {g_{0210e1}} {g_{1002}}-6 {g_{0210}} {g_{1002e1}}+34 {g_{1020e1}} {g_{1101}}+34 {g_{1020}} {g_{1101e1}} \\ &{}- 17 {g_{1011e1}} {g_{1110}}-17 {g_{1011}} {g_{1110e1}}+6 {g_{0012e1}} {g_{1200}}+6 {g_{0012}} {g_{1200e1}} + 31 {g_{0120e1}} {g_{2001}}+31 {g_{0120}} {g_{2001e1}} \\ &{}+34 {g_{0111e1}} {g_{2010}}+34 {g_{0111}} {g_{2010e1}}+ 31 {g_{0021e1}} {g_{2100}}+31 {g_{0021}} {g_{2100e1}}+i \omega _{2} (4 {g_{0202e1}}-17 {g_{2020e1}} ) \bigr) \\ &{}- \omega _{1}^{5} \omega _{2}^{2} \bigl(51 {g_{0102e1}} {g_{0201}}+51 {g_{0102}} {g_{0201e1}}+51 {g_{0003e1}} {g_{0300}}+51 {g_{0003}} {g_{0300e1}}- 24 {g_{0210e1}} {g_{1002}}-24 {g_{0210}} {g_{1002e1}} \\ &{}+8 {g_{1020e1}} {g_{1101}}+8 {g_{1020}} {g_{1101e1}}- 4 {g_{1011e1}} {g_{1110}}-4 {g_{1011}} {g_{1110e1}} +24 {g_{0012e1}} {g_{1200}}+24 {g_{0012}} {g_{1200e1}} \\ &{}+ 8 {g_{0120e1}} {g_{2001}}+8 {g_{0120}} {g_{2001e1}}+8 {g_{0111e1}} {g_{2010}}+8 {g_{0111}} {g_{2010e1}}+ 8 {g_{0021e1}} {g_{2100}}+8 {g_{0021}} {g_{2100e1}} \\ &{}+i \omega _{2} (17 {g_{0202e1}}-4 {g_{2020e1}} ) \bigr)+ 4 \omega _{1} \omega _{2}^{6} (-2 {g_{1020e1}} {g_{1101}}-2 {g_{1020}} {g_{1101e1}}+ {g_{1011e1}} {g_{1110}}+ {g_{1011}} {g_{1110e1}} \\ &{}+ {g_{0120e1}} {g_{2001}}+ {g_{0120}} {g_{2001e1}}-2 {g_{0111e1}} {g_{2010}}-2 {g_{0111}} {g_{2010e1}}+ {g_{0021e1}} {g_{2100}}+ {g_{0021}} {g_{2100e1}} +i {g_{2020e1}} \omega _{2} ) \bigr], \end{aligned}$$

(91)

$$\begin{aligned} D_{22} =&\frac{-i}{4 \omega _{1}^{5} \omega _{2}-17 \omega _{1}^{3} \omega _{2}^{3}+4 \omega _{1} \omega _{2}^{5}} \bigl[-12 ( {g_{1020A}} {g_{2010}}+ {g_{1020}} {g_{2010A}}+ {g_{0030A}} {g_{3000}}+ {g_{0030}} {g_{3000A}} ) \omega _{2}^{7} \\ &{}+ 4 \omega _{1}^{7} \bigl(3 ( {g_{0102A}} {g_{0201}}+ {g_{0102}} {g_{0201A}}+ {g_{0003A}} {g_{0300}}+ {g_{0003}} {g_{0300A}} )+i \omega _{2}{g_{0202A}} \bigr)-4 \omega _{1}^{6} \omega _{2} ( {g_{0210A}} {g_{1002}} \\ &{}+ {g_{0210}} {g_{1002A}}-2 {g_{0201A}} {g_{1011}}-2 {g_{0201}} {g_{1011A}} +{g_{0111A}} {g_{1101}}+ {g_{0111}} {g_{1101A}}-2 {g_{0102A}} {g_{1110}}-2 {g_{0102}} {g_{1110A}} \\ &{}+ {g_{0012A}} {g_{1200}} + {g_{0012}} {g_{1200A}}-i \omega _{2} {g_{1111A}} )+\omega _{1}^{2} \omega _{2}^{5} (8 {g_{0210A}} {g_{1002}}+8 {g_{0210}} {g_{1002A}}+8 {g_{0201A}} {g_{1011}} +8 {g_{0201}} {g_{1011A}} \\ &{}-4 {g_{0111A}} {g_{1101}}-4 {g_{0111}} {g_{1101A}}+8 {g_{0102A}} {g_{1110}}+8 {g_{0102}} {g_{1110A}} +8 {g_{0012A}} {g_{1200}}+8 {g_{0012}} {g_{1200A}}-24 {g_{0120A}} {g_{2001}} \\ &{}-24 {g_{0120}} {g_{2001A}}+51 {g_{1020A}} {g_{2010}} +51 {g_{1020}} {g_{2010A}}+24 {g_{0021A}} {g_{2100}}+24 {g_{0021}} {g_{2100A}} +51 {g_{0030A}} {g_{3000}} \\ &{}+51 {g_{0030}} {g_{3000A}}+4 i \omega _{2} {g_{1111A}} )-\omega _{1}^{4} \omega _{2}^{3} (31 {g_{0210A}} {g_{1002}}+31 {g_{0210}} {g_{1002A}}+34 {g_{0201A}} {g_{1011}}+34 {g_{0201}} {g_{1011A}} \\ &{} -17 {g_{0111A}} {g_{1101}}-17 {g_{0111}} {g_{1101A}}+34 {g_{0102A}} {g_{1110}}+34 {g_{0102}} {g_{1110A}}+31 {g_{0012A}} {g_{1200}}+31 {g_{0012}} {g_{1200A}} \\ &{}-6 {g_{0120A}} {g_{2001}}-6 {g_{0120}} {g_{2001A}}+ 12 {g_{1020A}} {g_{2010}}+12 {g_{1020}} {g_{2010A}}+6 {g_{0021A}} {g_{2100}}+6 {g_{0021}} {g_{2100A}}+ 12 {g_{0030A}} {g_{3000}} \\ &{}+12 {g_{0030}} {g_{3000A}}+17 i \omega _{2} {g_{1111A}} ) +\omega _{1}^{3} \omega _{2}^{4} \bigl(12 {g_{0102A}} {g_{0201}}+12 {g_{0102}} {g_{0201A}}+12 {g_{0003A}} {g_{0300}}+12 {g_{0003}} {g_{0300A}} \\ &{}-6 {g_{0210A}} {g_{1002}}-6 {g_{0210}} {g_{1002A}}+34 {g_{1020A}} {g_{1101}}+34 {g_{1020}} {g_{1101A}} -17 {g_{1011A}} {g_{1110}}-17 {g_{1011}} {g_{1110A}} \\ &{}+6 {g_{0012A}} {g_{1200}}+6 {g_{0012}} {g_{1200A}}+ 31 {g_{0120A}} {g_{2001}}+31 {g_{0120}} {g_{2001A}}+34 {g_{0111A}} {g_{2010}}+34 {g_{0111}} {g_{2010A}} \\ &{}+31 {g_{0021A}} {g_{2100}}+31 {g_{0021}} {g_{2100A}}+i \omega _{2} (4 {g_{0202A}}-17 {g_{2020A}} ) \bigr)- \omega _{1}^{5} \omega _{2}^{2} \bigl(51 {g_{0102A}} {g_{0201}}+51 {g_{0102}} {g_{0201A}} \\ &{}+51 {g_{0003A}} {g_{0300}}+51 {g_{0003}} {g_{0300A}}- 24 {g_{0210A}} {g_{1002}}-24 {g_{0210}} {g_{1002A}}+8 {g_{1020A}} {g_{1101}}+8 {g_{1020}} {g_{1101A}} \\ &{}- 4 {g_{1011A}} {g_{1110}}-4 {g_{1011}} {g_{1110A}}+24 {g_{0012A}} {g_{1200}}+24 {g_{0012}} {g_{1200A}} +8 {g_{0120A}} {g_{2001}} +8 {g_{0120}} {g_{2001A}}+8 {g_{0111A}} {g_{2010}} \\ &{}+8 {g_{0111}} {g_{2010A}}+8 {g_{0021A}} {g_{2100}}+8 {g_{0021}} {g_{2100A}} +i \omega _{2} (17 {g_{0202A}}-4 {g_{2020A}} ) \bigr)+4 \omega _{1} \omega _{2}^{6} (-2 {g_{1020A}} {g_{1101}} \\ &{}-2 {g_{1020}} {g_{1101A}}+ {g_{1011A}} {g_{1110}}+ {g_{1011}} {g_{1110A}}+{g_{0120A}} {g_{2001}}+ {g_{0120}} {g_{2001A}}-2 {g_{0111A}} {g_{2010}}-2 {g_{0111}} {g_{2010A}} \\ &{} +{g_{0021A}} {g_{2100}}+ {g_{0021}} {g_{2100A}}+i \omega _{2} {g_{2020A}} ) \bigr], \end{aligned}$$

(92)

$$\begin{aligned} D_{23} =&\frac{-i}{4 \omega _{1}^{5} \omega _{2}-17 \omega _{1}^{3} \omega _{2}^{3}+4 \omega _{1} \omega _{2}^{5}} \bigl[-12 ( {g_{1020e2}} {g_{2010}}+ {g_{1020}} {g_{2010e2}}+ {g_{0030e2}} {g_{3000}}+ {g_{0030}} {g_{3000e2}} ) \omega _{2}^{7} \\ &{}+4 \omega _{1}^{7} \bigl(3 ( {g_{0102e2}} {g_{0201}}+ {g_{0102}} {g_{0201e2}}+ {g_{0003e2}} {g_{0300}}+ {g_{0003}} {g_{0300e2}} )+ i \omega _{2}{g_{0202e2}} \bigr)-4 \omega _{1}^{6} \omega _{2} ( {g_{0210e2}} {g_{1002}} \\ &{}+ {g_{0210}} {g_{1002e2}}-2 {g_{0201e2}} {g_{1011}} -2 {g_{0201}} {g_{1011e2}}+ {g_{0111e2}} {g_{1101}}+ {g_{0111}} {g_{1101e2}}-2 {g_{0102e2}} {g_{1110}}-2 {g_{0102}} {g_{1110e2}} \\ &{} + {g_{0012e2}} {g_{1200}}+ {g_{0012}} {g_{1200e2}}-i \omega _{2} {g_{1111e2}} )+\omega _{1}^{2} \omega _{2}^{5} (8 {g_{0210e2}} {g_{1002}}+8 {g_{0210}} {g_{1002e2}} +8 {g_{0201e2}} {g_{1011}} \\ &{}+8 {g_{0201}} {g_{1011e2}}- 4 {g_{0111e2}} {g_{1101}}-4 {g_{0111}} {g_{1101e2}}+8 {g_{0102e2}} {g_{1110}} +8 {g_{0102}} {g_{1110e2}}+8 {g_{0012e2}} {g_{1200}}+8 {g_{0012}} {g_{1200e2}} \\ &{}-24 {g_{0120e2}} {g_{2001}}-24 {g_{0120}} {g_{2001e2}} + 51 {g_{1020e2}} {g_{2010}}+51 {g_{1020}} {g_{2010e2}}+24 {g_{0021e2}} {g_{2100}}+24 {g_{0021}} {g_{2100e2}} \\ &{}+51 {g_{0030e2}} {g_{3000}}+51 {g_{0030}} {g_{3000e2}}+4 i \omega _{2}{g_{1111e2}} )- \omega _{1}^{4} \omega _{2}^{3} (31 {g_{0210e2}} {g_{1002}}+31 {g_{0210}} {g_{1002e2}}+34 {g_{0201e2}} {g_{1011}} \\ &{}+34 {g_{0201}} {g_{1011e2}}- 17 {g_{0111e2}} {g_{1101}}-17 {g_{0111}} {g_{1101e2}}+34 {g_{0102e2}} {g_{1110}}+34 {g_{0102}} {g_{1110e2}}+ 31 {g_{0012e2}} {g_{1200}} \\ &{}+31 {g_{0012}} {g_{1200e2}}-6 {g_{0120e2}} {g_{2001}}-6 {g_{0120}} {g_{2001e2}}+ 12 {g_{1020e2}} {g_{2010}}+12 {g_{1020}} {g_{2010e2}}+6 {g_{0021e2}} {g_{2100}} \\ &{}+6 {g_{0021}} {g_{2100e2}}+ 12 {g_{0030e2}} {g_{3000}}+12 {g_{0030}} {g_{3000e2}}+17 i\omega _{2} {g_{1111e2}} )+\omega _{1}^{3} \omega _{2}^{4} \bigl(12 {g_{0102e2}} {g_{0201}}+12 {g_{0102}} {g_{0201e2}} \\ &{}+12 {g_{0003e2}} {g_{0300}}+12 {g_{0003}} {g_{0300e2}}- 6 {g_{0210e2}} {g_{1002}}-6 {g_{0210}} {g_{1002e2}}+34 {g_{1020e2}} {g_{1101}}+34 {g_{1020}} {g_{1101e2}} \\ &{}- 17 {g_{1011e2}} {g_{1110}} -17 {g_{1011}} {g_{1110e2}}+6 {g_{0012e2}} {g_{1200}}+6 {g_{0012}} {g_{1200e2}}+ 31 {g_{0120e2}} {g_{2001}}+31 {g_{0120}} {g_{2001e2}} \\ &{} +34 {g_{0111e2}} {g_{2010}}+34 {g_{0111}} {g_{2010e2}}+ 31 {g_{0021e2}} {g_{2100}}+31 {g_{0021}} {g_{2100e2}}+i \omega _{2} (4 {g_{0202e2}}-17 {g_{2020e2}} ) \bigr) \\ &{}- \omega _{1}^{5} \omega _{2}^{2} \bigl(51 {g_{0102e2}} {g_{0201}}+51 {g_{0102}} {g_{0201e2}}+51 {g_{0003e2}} {g_{0300}}+51 {g_{0003}} {g_{0300e2}} 24 {g_{0210e2}} {g_{1002}}-24 {g_{0210}} {g_{1002e2}} \\ &{}+8 {g_{1020e2}} {g_{1101}}+8 {g_{1020}} {g_{1101e2}}- 4 {g_{1011e2}} {g_{1110}}-4 {g_{1011}} {g_{1110e2}}+24 {g_{0012e2}} {g_{1200}}+24 {g_{0012}} {g_{1200e2}} \\ &{}+ 8 {g_{0120e2}} {g_{2001}}+8 {g_{0120}} {g_{2001e2}}+8 {g_{0111e2}} {g_{2010}}+8 {g_{0111}} {g_{2010e2}}+ 8 {g_{0021e2}} {g_{2100}}+8 {g_{0021}} {g_{2100e2}} \\ &{}+i \omega _{2} (17 {g_{0202e2}}-4 {g_{2020e2}} ) \bigr)+ 4 \omega _{1} \omega _{2}^{6} (-2 {g_{1020e2}} {g_{1101}}-2 {g_{1020}} {g_{1101e2}}+ {g_{1011e2}} {g_{1110}}+ {g_{1011}} {g_{1110e2}} \\ &{}+ {g_{0120e2}} {g_{2001}}+ {g_{0120}} {g_{2001e2}}-2 {g_{0111e2}} {g_{2010}}-2 {g_{0111}} {g_{2010e2}}+ {g_{0021e2}} {g_{2100}}+ {g_{0021}} {g_{2100e2}}+i \omega _{2} {g_{2020e2}} ) \bigr]. \end{aligned}$$

(93)