1.1.1 Explicit solution of the Zhukovsky–Volterra gyrostat
The first objective is the explicit integration and a qualitative study of behavior of the classical generalization of the Euler top: the system describing the free motion of a gyrostat: a rigid body carrying inside a symmetric rotator whose axis is fixed in the body. As was shown by N.E.Zhukovsky [64] and, independently, by V. Volterra [58], the system can be reduced to the following equations describing the evolution of the total angular momentum vector M ∈ R3:
M = M˙ × (aM − g), M = (M1, M2, M3)T, (1.1) where g∈ R3is a constant vector characterizing the axial angular momentum of the rotator. In the case g = 0, these equation reduce to the classical Euler top problem.
The motion of the gyrostat in space is then described by solutions of the Poisson equations
˙γ = γ× ω(t), where ω = aM (t)− g, ω being the angular velocity vector.
Like the Euler top, the system (1.1) has two quadratic integrals, which, however, are not all homogeneous in Mi,
f1(M ) = M12+ M22+ M32 = k, (1.2) f2(M ) = a1M12+ a2M22+ a3M32− 2M1g1− 2M2g2− 2M3g3 = l k, l = const.
(1.3) Then, according to a theorem of the algebraic geometry (see e.g., [23]), for generic values of the constants k, l, the complex invariant variety of the Zhukovsky–Volterra (ZV) system (1.2) is an elliptic curveE. However, in contrast to the classical Euler top, explicit integration of the ZV system and, especially, the explicit description of the motion of the gyrostat in space given by solutions of the Poisson equations in practice appears to be a much more complicated problem.
In [58] Volterra presented expressions for the components of the momenta M and of the rotation matrix of the gyrostat in terms of sigma-functions and exponents, however these expressions include several undetermined parameters and only provide the structure of the solution, but not explicit formulas.
We bridge these gaps, namely,
1) To find expressions for the components of momenta Mi as elliptic functions on the curveE by using a new rational parametrization for Mi in terms of some canonical coordinates onE, whose dependence on time is known;
2) To derive all explicit solutions of the Poisson equations and obtain trigonometric functions of the Euler angles as functions of time t.
It is expected that the obtained explicit formulas provide an effective description of the motion of the gyrostat in space and be useful in practice.
Zhukovski-Voltera system described above has the property of being
bi-Hamiltonian. By using this property and applying the new scheme for topo-logical analysis of bi-Hamiltonian system [12] we construct bifurcation diagram of momentum mapping, given by integrals f1, f2:
Φ(f1, f2) :R3(M1, M2, M3)→ R2(f1, f2).
We describe the set of critical points, verify the non-degeneracy condition and de-termine the stability of equilibrium points. By using the parametric description of critical points, we obtained the bifurcation diagram of momentum mapping Φ and analyze the topological type of common level of integrals f1, f2.
Notice, that the standard scheme for describing set of critical point and analysis of their stability consists of the Jacobi matrix and Hessian of the restriction f1 onto simplectic leaf, whereas this new technics has allowed us to answer all this questions without difficult computations.
1.1.2 Separation of variables, explicit integration and bifurcation analysis of the Steklov–Lyapunov systems
The classical Steklov integrable case of the Kirchhoff equations M = M˙ × ∂H
∂M + p× ∂H
∂p, p = p˙ × ∂H
∂p, (1.4)
where M, p ∈ R3 are the vectors of the impulsive momentum and the impulsive force, and H = H(M, p) is the Hamiltonian, which is quadratic in M, p. given by
H = 1 2
∑3 α=1
(
bαMα2+ 2νbβbγMαpα+ ν2bα(bβ− bγ)2p2α )
, (1.5)
bi and ν being arbitrary constants, was first integrated in terms of theta-functions of 2 arguments by F. K¨otter [38] in 1900. However, the method of integration was not indicated in that paper and, moreover, the solutions presented contain several undetermined parameters, which make impossible to apply them in practice.
The second group of objectives of the thesis are
1) To revise the separation of variables and explicit integration of the classical Steklov–Lyapunov systems. Namely, we give a geometric interpretation of the separating variables;
2) then, applying the Weierstrass root functions, obtain an explicit theta-function solution to the problem.
3) construct and analyze the bifurcation diagram for the Steclov-Lyapunov system by using the bi-Hamiltonian properties of the system [12] and then, indicate on the plane (h1, h2) the domains of real motion, describe the type of the special motion for each segment of the bifurcation curves and do stability analysis for critical periodic solutions.
1.1.3 Bifurcation analysis of Rubanovskii sistem
Apart from the classical Steklov integrable case of the Kirchhoff equations, in [50]
V. Rubanovsky found its gyroscopic generalization describing the motion of a gy-rostat in an ideal fluid and also the rigid body in presence of non-zero circulation.
In contrast to the Steklov Hamiltonian (1.5), the Hamiltonian of the gyroscopic generalization contains linear terms in M, p and has the form
H1 = 1 2
∑3 α=1
(
bα(Mα− 2gα)2+ 2νbβbγMαpα
+ν2bα(bβ− bγ)2p2α+ 4ν(bβ+ bγ)gαpα
)
, (1.6)
ν, g1, g2, g3 = const.
Like in the case of the Zhykovsky–Volterra gyrostat, here the vector g characterizes the axial angular momentum of the rotator inside the body.
The Kirchhoff equations with the Hamiltonian (1.6) possess a second integral quadratic in M, p.
It can be observed that under the change of variables M → z
2zα = Mα− (bβ+ bγ)pα, α = 1, 2, 3 , (α, β, γ) = (1, 2, 3) these equations take the form
˙
z = z× (Bz − g) − Bp × (Bz − g), p = p˙ × (Bz − g).
and, as was shown in [27], the latter equations admit the following Lax pair with skew-symmetric matrices and an elliptic parameter s
L(s) = [ L(s), A(s) ] ,˙ L(s), A(s)∈ so(3), s ∈ C , (1.7) L(s)αβ = εαβγ(√
s− bγ(zγ+ spγ) + gγ/√ s− bγ
) , A(s)αβ = εαβγ1
s
√
(s− bα)(s− bβ) (bγzγ− gγ) . (1.8) Writing out the characteristic equation for L(s) we arrive at the following quadratic integrals
J1 =⟨p, p⟩, J2 = 2⟨z, p⟩ − ⟨Bp, p⟩, H1 = 1
2⟨z, Bz⟩ − ⟨z, g⟩, H2 = 1
2⟨z, z⟩ − ⟨Bz, p⟩ + ⟨p, g⟩, B = diag(b1, b2, b3).
Lax-pair to the Rubanovskii case [27] allows to describe a bi-Hamiltoian struc-ture corresponding to this system. Using the fact that Rubanovskii system is bi-Hamiltonian and applying new techniques [12] we solve the following problems:
1) description of the singularities of the momentum mapping defined by four inte-grals
Φ :R6(z, p)→ R4(J1, J2, H1, H2);
2) stability analysis for closed trajectories;
3) non-degeneracy and stability analysis for equilibria;
4) some property of bifurcation diagram of Rubanovskii system.