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Register a new userOn the Number of Equilibria Balancing Newtonian Point Masses with a Central Force
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We consider the critical points (equilibria) of a planar potential generated by $n$ Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular restricted $n$-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by $n+1$, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in $n$. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as ``ring configurations, consisting of $n-1$ equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as $5n-5$ equilibria. We verify analytically that the ring configuration has at least $5n-5$ equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwells problem from electrostatics and the image counting problem from gravitational lensing.
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We consider the spatial central force problem with a real analytic potential. We prove that for all analytic potentials, but the Keplerian and the Harmonic ones, the Hamiltonian fulfills a nondegeneracy property needed for the applicability of Nekhoroshevs theorem. We deduce stability of the actions over exponentially long times when the system is subject to arbitrary analytic perturbation. The case where the central system is put in interaction with a slow system is also studied and stability over exponentially long time is proved.
The goal of this paper is to investigate the normal and tangential forces acting at the point of contact between a horizontal surface and a rolling ball actuated by internal point masses moving in the balls frame of reference. The normal force and static friction are derived from the equations of motion for a rolling ball actuated by internal point masses that move inside the balls frame of reference, and, as a special case, a rolling disk actuated by internal point masses. The masses may move along one-dimensional trajectories fixed in the balls and disks frame. The dynamics of a ball and disk actuated by masses moving along one-dimensional trajectories are simulated numerically and the minimum coefficients of static friction required to prevent slippage are computed.
We study the first Hilbert coefficient (after the multiplicity) $e_1$ of a local ring $(A,m). $ Under various circumstances, it is also called the {bf Chern number} of the local ring $A.$ Starting from the work of D.G. Northcott in the 60s, several results have been proved which give some relationships between the Hilbert coefficients, but always assuming the Cohen-Macaulayness of the basic ring. Recent papers of S. Goto, K. Nishida, A. Corso and W. Vasconcelos pushed the interest toward a more general setting. In this paper we extend an upper bound on $e_1$ proved by S. Huckaba and T. Marley. Thus we get the Cohen-Macaulayness of the ring $A$ as a consequence of the extremal behavior of the integer $e_1.$ The result can be considered a confirm of the general philosophy of the paper of W. Vasconcelos where the Chern number is conjectured to be a measure of the distance from the Cohen-Macaulyness of $A.$ This main result of the paper is a consequence of a nice and perhaps unexpected property of superficial elements. It is essentially a kind of Sally machine for local rings. In the last section we describe an application of these results, concerning an upper bound on the multiplicity of the Sally module of a good filtration of a module which is not necessarily Cohen-Macaulay. It is an extension to the non Cohen-Macaulay case of a result of Vaz Pinto.
The shaking force balancing is a well-known problem in the design of high-speed robotic systems because the variable dynamic loads cause noises, wear and fatigue of mechanical structures. Different solutions, for full or partial shaking force balancing, via internal mass redistribution or by adding auxiliary links were developed. The paper deals with the shaking force balancing of the Orthoglide. The suggested solution based on the optimal acceleration control of the manipulators common center of mass allows a significant reduction of the shaking force. Compared with the balancing method via adding counterweights or auxiliary substructures, the proposed method can avoid some drawbacks: the increase of the total mass, the overall size and the complexity of the mechanism, which become especially challenging for special parallel manipulators. Using the proposed motion control method, the maximal value of the total mass center acceleration is reduced, as a consequence, the shaking force of the manipulator decreases. The efficiency of the suggested method via numerical simulations carried out with ADAMS is demonstrated.
Recent tokamak experiments employing off-axis, non-inductive current drive have found that a large central current hole can be produced. The current density is measured to be approximately zero in this region, though in principle there was sufficient current drive power for the central current density to have gone significantly negative. Recent papers have used a large aspect-ratio expansion to show that normal MHD equilibria (with axisymmetric nested flux surfaces, non-singular fields, and monotonic peaked pressure profiles) can not exist with negative central current. We extend that proof here to arbitrary aspect ratio, using a variant of the virial theorem to derive a relatively simple integral constraint on the equilibrium. However, this constraint does not, by itself, exclude equilibria with non-nested flux surfaces, or equilibria with singular fields and/or hollow pressure profiles that may be spontaneously generated.
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