On the dynamics of the rigid body with a fixed point: periodic orbits and integrability

Abstract

The aim of the present paper is to study the periodic orbits of a rigid body with a fixed point and quasi-spherical shape under the effect of a Newtonian force field given by different small potentials. For studying these periodic orbits, we shall use averaging theory. Moreover, we provide information on the $\mathcal{C}^{1}$-integrability of these motions.

Appendix

Now we shall present the basic results from averaging theory that we need for proving the results of this paper.

The next theorem provides a first- and second-order approximation for the periodic solutions of a periodic differential system; for the proof, see Theorems 11.5 and 11.6 of Verhulst [4, 11], and [9].

Consider the differential equation

$$ \dot{x}=\varepsilon F_{1}(t,x)+\varepsilon^{2}F_{2}(t,x)+ \varepsilon ^{3}R(t,x,\varepsilon),\quad x(0)=x_{0} $$

(7)

with $x\in D\subset\mathbb{R}^{n}, t\geq0$. Moreover, we assume that both F ₁(t,x) and F ₂(t,x) are T periodic in t. Separately, we consider in D the averaged differential equation

$$ \dot{y}=\varepsilon f_{1}(y)+\varepsilon^{2}f_{2}(y), \quad y(0)=x_{0} , $$

(8)

where

Under certain conditions, equilibrium solutions of the averaged equation turn out to correspond with T-periodic solutions of Eq. (8).

Theorem 5

Consider the two initial value problems (7) and (8). Suppose:

• (i) F ₁, its Jacobian ∂F ₁/∂x, its Hessian ∂ ² F ₁/∂x ², F ₂ and its Jacobian ∂F ₂/∂x are defined, continuous, and bounded by an independent constant ε in [0,∞)×D and ε∈(0,ε ₀].

• (ii) F ₁ and F ₂ are T-periodic in t (T independent of ε).

• (iii) y(t) belongs to Ω on the interval of time [0,1/ε].

Then the following statements hold.

• (a) For t∈[1,ε] we have that x(t)−y(t)=O(ε), as ε→0.

• (b) If p is a singular point of the averaged equation (8) and

  $$\det \biggl( \frac{\partial(f_{1}+\varepsilon f_{2})}{\partial y} \biggr) \bigg \vert _{\mathrm{y}=p}\neq0, $$

  then there exists a T-periodic solution φ(t,ε) of Eq. (7), which is close to p such that φ(0,ε)→p as ε→0.

• (c) The stability or instability of the limit cycle φ(t,ε) is given by the stability or instability of the singular point p of the averaged system (8). In fact, the singular point p has the stability behavior of the Poincaré map associated to the limit cycle φ(t,ε).