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The main problem is to understand and to find periodic symmetric orbits in the n-body problem, in the sense of finding methods to prove or compute their existence, and more importantly to describe their qualitative and quantitative properties. In order to do so, and in order to classify such orbits and their symmetries, computers have been extensively used in many ways since decades. We will focus on some very special symmetric orbits, which occur as symmetric critical points (local minimizers) of the gravitational Lagrangean action functional. The exploration of the loop space of the n-point configuration space, raised some computational and mathematical questions that couldd be interesting. The aim of the article is to explain how such questions and issues were considered in the development of a software package that combined symbolic algebra, numerical and scientific libraries, human interaction and visualization.

The latest evidence suggests that multi-hazards and their interrelationships (e.g., triggering, compound, and consecutive hazards) are becoming more frequent across Europe, underlying a need for resilience building by moving from single-hazard-focused to multi-hazard risk assessment and management. Although significant advancements were made in our understanding of these events, mainstream practice is still focused on risks due to single hazards (e.g., flooding, earthquakes, droughts), with a limited understanding of the stakeholder needs on the ground. To overcome this limitation, this paper sets out to understand the challenges for moving towards multi-hazard risk management through the perspective of European stakeholders. Based on five workshops across different European pilots (Danube Region, Veneto Region, Scandinavia, North Sea, and Canary Islands) and an expert workshop, we identify five prime challenges: i) governance, ii) knowledge of multi-hazards and multi-risks, iii) existing approaches to disaster risk management, iv) translation of science to policy and practice, and v) lack of data. These challenges are inherently linked and cannot be tackled in isolation with path dependency posing a significant hurdle in transitioning from single- to multi-hazard risk management. Going forward, we identify promising approaches for overcoming some of the challenges, including emerging approaches for multi-hazard characterisation, a common understanding of terminology, and a comprehensive framework for guiding multi-hazard risk assessment and management. We argue for a need to think beyond natural hazards and include other threats in creating a comprehensive overview of multi-hazard risks, as well as promoting thinking of multi-hazard risk reduction in the context of larger development goals.

Central configurations play an important role in the dynamics of the n-body problem, and have been studied as relative equilibria, critical points, or projective fixed points of maps on configuration spaces. We describe some results on central configuration as fixed points of quotient maps, and then on the inverse problem in dimension 1, i.e. finding (positive or real) masses which make a given collinear configuration central. We study the inverse problem as a fixed point problem for multi-valued maps.

Central configurations play an important role in the dynamics of the nnn-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed points of self-maps defined on the shape space, and some results on the inverse problem in dimension 111, i.e. finding (positive or real) masses which make a given collinear configuration central. This survey article introduces readers to the recent results of the author, also unpublished, showing an application of the fixed point theory.

We compute the fixed point index of non-degenerate central configurations for the n-body problem in the euclidean space of dimension d, relating it to the Morse index of the gravitational potential function Ū induced on the manifold of all maximal O(d)-orbits. In order to do so, we analyze the geometry of maximal orbit type manifolds, and compute Morse indices with respect to the mass-metric bilinear form on configuration spaces.

Central configurations are solutions of the equations (formula presented), where U denotes the potential function and each qj is a point in the d-dimensional Euclidean space E ≅ Rd, for j = 1..., n. We show that the vector of the mutual differences (formula presented) satisfies the equation (formula presented), where Pm is the orthogonal projection over the spaces of 1-cocycles and (formula presented). It is shown that differences qij of central configurations are critical points of an analogue of U, defined on the space of 1-cochains in the Euclidean space E, and restricted to the subspace of 1-cocycles. Some generalizations of well known facts follow almost immediately from this approach.

Central configurations of n point particles in $${E \approx \mathbb{R}^d}$$E≈Rd with respect to a potential function U are shown to be the same as the fixed points of the normalized gradient map $${F = -\nabla_{M}U / ||\nabla_{M}U||_{M}}$$F=-∇MU/||∇MU||M , which is an SO(d)-equivariant self-map defined on the inertia ellipsoid. We show that the SO(d)-orbits of fixed points of F are all fixed points of the map induced on the quotient by SO(d), and we give a formula relating their indices (as fixed points) with their Morse indices (as critical points). At the end, we give an example of a nonplanar relative equilibrium which is not a central configuration.