We consider periodic orbits in the circular restricted 3-body problem, where the third (small) body is a solar sail. In particular, we consider orbits about equilibrium points in the Earth-Sun rotating frame which are high above the ecliptic plane, in contrast to the classical 'halo' orbits about the collinear equilibria. It is found that due to coupling in the equations of motion, periodic orbits about equilibria are naturally present at linear order. Using the method of Lindstedt-Poincaré, we construct nth order approximations to periodic solutions of the nonlinear equations of motion. It is found that there is much freedom in specifying the position and period/amplitude of the orbit of the sail, high above the ecliptic and looking down on the Earth. A particular use of such solutions is presented, namely the year-round constant imaging of, and communication with, the poles. We find that these orbits present a significant improvement on the position of the sail when viewed from the Earth, compared to a sail placed at equilibrium.
The stability of the Cauchy horizon in spherically symmetric self-similar collapse is studied by determining the flux of scalar radiation impinging on the horizon. This flux is found to be finite.
Structures are traditionally designed to be stable. However, unstable configurations (such as buckling) can in principle be controlled in smart structures using embedded sensors and actuators. In this paper we explore a new means of reconfiguring smart structures by connecting multiple unstable configurations. Methods from dynamical systems theory are used firstly to identify sets of unstable configurations in a simple smart structure model, and then to connect them through so-called heteroclinic connections in the phase space of the problem. The instability inherent in the structure is then actively utilised to provide an effective new way of transitioning between configurations of the structure in a controlled manner.
The conjugate locus of a point $p$ in a surface $\mathcal{S}$ will have a
certain number of cusps. As the point $p$ is moved in the surface the conjugate
locus may spontaneously gain or lose cusps. In this paper we explain this
`bifurcation' in terms of the vanishing of higher derivatives of the
exponential map; we derive simple equations for these higher derivatives in
terms of scalar invariants; we classify the bifurcations of cusps in terms of
the local structure of the conjugate locus; and we describe an intuitive
picture of the bifurcation as the intersection between certain contours in the
tangent plane.Comment: Accepted in Journal of Geometry and Physics April 201
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