Related papers: On the n-body problem on surfaces of revolution
Let $(\mathcal{M},g)$ be a Riemannian manifold and $\mathcal{N}$ a $\mathcal{C}^2$ submanifold without boundary. If we multiply the metric $g$ by the inverse of the squared distance to $\mathcal{N}$, we obtain a new metric structure on…
We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we…
We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree $-\beta$, $\beta\ge 2$. For $\beta>2$, the sets of initial conditions leading to…
The exact solution to the Einstein equations that represents a static axially symmetric source deformed by an internal quadrupole is considered. By using the Poincare section method we numerically study the geodesic motion of test…
This paper is a continuation and an extension of our recent work [3] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In [3], the analytic behaviour of the Laplacian…
We investigate the existence of stationary fronts in a coupled system of two sine-Gordon equations with a smooth, "hat-like" spatial inhomogeneity. The spatial inhomogeneity corresponds to a spatially dependent scaling of the sine-Gordon…
Invariant manifolds are the skeleton of the chaotic dynamics in Hamiltonian systems. In Celestial Mechanics, for instance, these geometrical structures are applied to a multitude of physical and practical problems, such as to the…
In this paper, generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge, more generally, generic $n$-type edge, which is invariant under a helicoidal motion in Euclidean $3$-space admits non-trivial isometric…
Based on a novel type of Sobolev-Poincar\'e inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of…
A strengthened canonical quantization scheme for the constrained motion on a curved hypersurface is proposed with introduction of the second category of fundamental commutation relations between Hamiltonian and positions/momenta, whereas…
A certain class of surface motions, including those of a relativistic membrane minimizing the 3-dimensional volume swept out in Minkowski-space, is shown to be equivalent to 3-dimensional steady-state irrotational inviscid isentropic…
We construct a class of global, dynamical solutions to the 3d Euler equations near the stationary state given by uniform "rigid body" rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter…
A novel method for calculation of the motion and radiation reaction for the two-body problem (body plus particle, the small parameter m/M being the ratio of the masses) is presented. In the background curvature given by the Schwarzschild…
Given the importance of quantum reference frames (QRFs) to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of QRFs, which…
The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies…
We develop a systematic approach for proving the existence of choreographic solutions in the gravitational $n$ body problem. Our main focus is on spatial torus knots: that is, periodic motions where the positions of all $n$ bodies follow a…
A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and…
We present the explicit solution to the geodesic equations in a warped de Sitter space-time proposed by Randall-Sundrum. We find that a test particle moves in the bulk and is not restricted on a 3-brane (to be taken as our universe). On the…
In the context of the Newtonian N-body problem, we prove the existence of a partially hyperbolic motion with prescribed positive energy and any initial collisionless configuration. Moreover, it is a free time minimizer of the respective…
A 3-parameter family of helical tubular surfaces obtained by screw revolving a circle provides a useful pedagogical example of how to study geodesics on a surface that admits a 1-parameter symmetry group, but is not as simple as a surface…