45 research outputs found
Projective dynamics and classical gravitation
Given a real vector space V of finite dimension, together with a particular
homogeneous field of bivectors that we call a "field of projective forces", we
define a law of dynamics such that the position of the particle is a "ray" i.e.
a half-line drawn from the origin of V. The impulsion is a bivector whose
support is a 2-plane containing the ray. Throwing the particle with a given
initial impulsion defines a projective trajectory. It is a curve in the space
of rays S(V), together with an impulsion attached to each ray. In the simplest
example where the force is identically zero, the curve is a straight line and
the impulsion a constant bivector. A striking feature of projective dynamics
appears: the trajectories are not parameterized.
Among the projective force fields corresponding to a central force, the one
defining the Kepler problem is simpler than those corresponding to other
homogeneities. Here the thrown ray describes a quadratic cone whose section by
a hyperplane corresponds to a Keplerian conic. An original point of view on the
hidden symmetries of the Kepler problem emerges, and clarifies some remarks due
to Halphen and Appell. We also get the unexpected conclusion that there exists
a notion of divergence-free field of projective forces if and only if dim V=4.
No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure
Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups
Given a closed, orientable surface of constant negative curvature and genus
, we study the topological entropy and measure-theoretic entropy (with
respect to a smooth invariant measure) of generalized Bowen--Series boundary
maps. Each such map is defined for a particular fundamental polygon for the
surface and a particular multi-parameter.
We present and sketch the proofs of two strikingly different results:
topological entropy is constant in this entire family ("rigidity"), while
measure-theoretic entropy varies within Teichm\"uller space, taking all values
("flexibility") between zero and a maximum, which is achieved on the surface
that admits a regular fundamental -gon. We obtain explicit formulas for
both entropies. The rigidity proof uses conjugation to maps of constant slope,
while the flexibility proof -- valid only for certain multi-parameters -- uses
the realization of geodesic flow as a special flow over the natural extension
of the boundary map.Comment: 22 pages, 11 figure
Symbolic dynamics: from the -centre to the -body problem, a preliminary study
We consider a restricted -body problem, with and
homogeneous potentials of degree -\a<0, \a \in [1,2). We prove the
existence of infinitely many collision-free periodic solutions with negative
and small Jacobi constant and small values of the angular velocity, for any
initial configuration of the centres. We will introduce a Maupertuis' type
variational principle in order to apply the broken geodesics technique
developed in the paper "N. Soave and S. Terracini. Symbolic dynamics for the
-centre problem at negative energies. Discrete and Cont. Dynamical Systems
A, 32 (2012)". Major difficulties arise from the fact that, contrary to the
classical Jacobi length, the related functional does not come from a Riemaniann
structure but from a Finslerian one. Our existence result allows us to
characterize the associated dynamical system with a symbolic dynamics, where
the symbols are given partitions of the centres in two non-empty sets.Comment: Revised version, to appear on NoDEA Nonlinear Differential Equations
and Application
Symbolic dynamics for the -centre problem at negative energies
We consider the planar -centre problem, with homogeneous potentials of
degree -\a<0, \a \in [1,2). We prove the existence of infinitely many
collisions-free periodic solutions with negative and small energy, for any
distribution of the centres inside a compact set. The proof is based upon
topological, variational and geometric arguments. The existence result allows
to characterize the associated dynamical system with a symbolic dynamics, where
the symbols are the partitions of the centres in two non-empty sets