Ratbay Myrzakulov

As is well known, for our dynamical system, the Euler-Lagrange equations read as
( )
d ∂L37
− ∂L 37
= 0, (4.9)
dt ∂Ṙ ∂R
( )
d ∂L37
dt ∂T
˙ − ∂L 37
= 0, (4.10)
∂T
( )
d ∂L37
− ∂L 37
= 0. (4.11)
dt ∂ȧ ∂a
Hence, using the expressions
we get
∂L 37
∂Ṙ = −6ɛ 1F RR a 2 ȧ, (4.12)
∂L 37
∂T
˙
∂L 37
∂ȧ
respectively. Here
= −6ɛ 1 F RT a 2 ȧ, (4.13)
= −12(ɛ 1 F R − ɛ 2 F T )aȧ − 6ɛ 1 (F RR Ṙ + F RT ˙ T + F Rψ ˙ψ)a 2 + a 3 F T vȧ + a 3 F R uȧ, (4.14)
a 3 F TT
(
T − v − 6ɛ 2
ȧ 2
a 2 )
(
)
a 3 F RR R − u − 6ɛ 1 (ä
a + ȧ2
a 2 )
= 0, (4.15)
= 0, (4.16)
U + B 2 F TT + B 1 F T + C 2 F RRT + C 1 F RTT + C 0 F RT + MF + 6ɛ 2 a 2 p = 0, (4.17)
U = A 3 F RRR + A 2 F RR + A 1 F R , (4.18)
A 3 = −6ɛ 1 Ṙ 2 a 2 , (4.19)
A 2 = −12ɛ 1 Ṙaȧ − 6ɛ 1 ¨Ra 2 + a 3 Ṙuȧ, (4.20)
A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 ˙uȧ − a 3 u a , (4.21)
B 2 = 12ɛ 2 Taȧ ˙ + a 3 Tvȧ, ˙
(4.22)
B 1 = 24ɛ 2 ȧ 2 + 12ɛ 2 aä + 3a 2 ȧvȧ + a 3 ˙vȧ − a 3 v a , (4.23)
C 2 = −12ɛ 1 a 2 ṘT, ˙
(4.24)
C 1 = −6ɛ 1 a 2 T˙
2 , (4.25)
C 0 = −12ɛ 1 Taȧ ˙ + 12ɛ 2 Ṙaȧ − 6ɛ 1 a 2 ¨T + a 3 Ṙvȧ + a 3 Tuȧ, ˙
(4.26)
M = −3a 2 . (4.27)
If F RR ≠ 0,
F TT ≠ 0, from Eqs. (4.17) and (4.17), it is easy to find that
R = u + 6ɛ 1 (Ḣ + 2H2 ), T = v + 6ɛ 2 H 2 , (4.28)
so that the relations (4.1) are recovered. Generally, these equations are the Euler constraints of
the dynamics. Here a,R,T are the generalized coordinates of the configuration space. On the
other hand, it is also well known that the total energy (Hamiltonian) corresponding to Lagrangian
L 37 is given by
H 37 = ∂L 37
∂ȧ ȧ + ∂L 37
Ṙ + ∂L 37
∂Ṙ ∂T
˙ T˙
− L 37 . (4.29)
Hence using (4.12)-(4.14) we obtain
H 37 = [−12(ɛ 1 F R − ɛ 2 F T )aȧ − 6ɛ 1 (F RR Ṙ + F RT ˙ T + F Rψ ˙ψ)a 2 + a 3 F T vȧ + a 3 F R uȧ]ȧ
−6ɛ 1 F RR a 2 ȧṘ − 6ɛ 1F RT a 2 ȧ ˙ T − [a 3 (F − TF T − RF R + vF T + uF R + L m )−
6(ɛ 1 F R − ɛ 2 F T )aȧ 2 − 6ɛ 1 (F RR Ṙ + F RT ˙ T + F Rψ ˙ψ)a 2 ȧ]. (4.30)
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