In this paper we develop symbolic computation algorithms to investigate finiteness of central configurations for the planar $n$-body problem. Our approach is based on Albouy-Kaloshin's work on finiteness of central configurations for the 5-body problems. In their paper, bicolored graphs called $zw$-diagrams were introduced for possible scenarios when the finiteness conjecture fails, and proving finiteness amounts to exclusions of central configurations associated to these diagrams. Following their method, the amount of computations becomes enormous when there are more than five bodies. Here we introduce matrix algebra for determination of both diagrams and asymptotic orders, devise several criteria to reduce computational complexity, and verify finiteness mostly through automated deductions. For the planar six-body problem, our first algorithm effectively narrows the proof for finiteness down to 117 $zw$-diagrams, the second algorithm eliminates 31 of them, the last algorithm eliminates 62 other diagrams except for masses in some co-dimension 2 variety in the mass space, and leaving 24 cases unsolved.
In an earlier paper \cite{yu2021Finiteness}, we showed that there are finitely many stationary configurations (consisting of equilibria, rigidly translating configurations, relative equilibria and collapse configurations) in the planar four-vortex problem. However, we only established finiteness of collapse configurations in the sense of prescribing a collapse constant. In this paper, by developing ideas of Albouy-Kaloshin and Hampton-Moeckel to do an analysis of the singularities, we further show that there really are finitely many collapse configurations in the four-vortex problem. This is a…
A symmetric planar central configuration of the Newtonian six-body problem $x$ is called cross central configuration if there are precisely four bodies on a symmetry line of $x$. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses $m_1,m_2,m_3,m_4,m_5=m_6$ there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dim…
Consider a planar polynomial vector field $X$, and assume it admits a symbolic first integral $\mathcal{F}$, i.e. of the $4$ classes, in growing complexity: Rational, Darbouxian, Liouvillian and Riccati. If $\mathcal{F}$ is not rational, it is sometimes possible to reduce it to a simpler class first integral. We will present algorithms to reduce symbolic first integral to a lower complexity class. These algorithms allow to find the minimal class first integral and in particular to test the existence of a rational first integral except in the case where $\mathcal{F}$ is a $k$-Darbouxian first i…
Recently, the theory concerning piecewise smooth vector fields (PSVFs for short) have been undergoing important improvements. In fact, many results obtained do not have an analogous for smooth vector fields. For example, the chaoticity of planar PSVFs, which is impossible for the smooth ones. These differences are generated by the non-uniqueness of trajectory passing through a point. Inspired by the classical fact that one-dimensional discrete dynamic systems can produce chaotic behavior, we construct a conjugation between the shift map and PSVFs. By means of the results obtained and the techn…
An Efficient Symbolic Algorithm for Computing Lyapunov Constants in Planar Switching Systems
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；College of Mathematics and Physics, Beijing University of Chemical Technology（北京化工大学理学院）
Central configurations and relative equilibria are an important facet of the study of the $N$-body problem, but become very difficult to rigorously analyze for $N>3$. In this paper we focus on a particular but interesting class of configurations of the 5-body problem: the equilateral pentagonal configurations, which have a cycle of five equal edges. We prove a variety of results concerning central configurations with this property, including a computer-assisted proof of the finiteness of such configurations for any positive five masses with a range of rational-exponent homogeneous potentials (…
The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-α_j$: \[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional. Compared with the classical $N$-centre problem w…
In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in $\tilde{\mathcal{O}}(N^{ω+1})$, where $N$ is the bound on the degree of a representation of the first integral and $ω\in [2;3]$ is the exponent of linear algebr…
