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The cotangent bundle of a 4-sphere is a symplectic manifold.  Suppose $M$ is a submanifold.  Is the seven-dimensional restriction of the cotangent bundle to $M$ a contact manifold?  The submanifold is obviously not compact, and included within it the cotangent bundle of $M$.  If the answer is yes, it would be extremely helpful to proceed to check whether Hamiltonian systems on M have closed periodic orbits.
A very simpleminded check is to consider a local chart on $T^*S^4$ with coordinates $(x_1, \dots, x_4, y_1, \dots, y_4$ and the symplectic form $\omega = \sum_i dx_i\wedge dy_i$ with coordinates at least locally chosen so $x_4=0$ on the submanifold.  The restriction of $\omega$ simply gets rid of $dy_4\wedge dx_4$ and so the one-form $\eta = x_1 dy_1+x_2dy_2+x_3dy_3$ would be a contact form.  Note that this form does not depend on $x_4$ at all.  If this is right, we should be able to treat Hamiltonian systems using the machinery built up by various mathematicians.