Heinz Hanßmann, Philip Holmes
On the global dynamics of Kirchhoff's equations :
Rigid body models for underwater vehicles
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ABSTRACT.  We study the Kirchhoff model for the motion of a
rigid body submerged in an incompressible, irrotational,
inviscid fluid in the absence of gravitational forces and
torques.  Symmetries allow reduction to a two
degree-of-freedom Hamiltonian system.  In [7] the existence
and stability of pure and mixed mode equilibria was studied
and, in [7] \S 5.2, the system was averaged, allowing further
reduction to one degree of freedom.  We give an
interpretation of the averaged Hamiltonian function as a
normal form of order one.  Iterating the process we obtain
the normal form of order two, thus resolving a degeneracy
noted in [7], and allowing us to prove that the (integrable)
normal form of order two has heteroclinic orbits between
`pure 2' and between the `pure 3' modes in a range of
parameter values, and, at a critical (bifurcation) value,
heteroclinic cycles linking pure 2 and pure 3 modes.  We
discuss the implications for the original system and the
full rigid body motions.