1. Introduction
Research Contribution
Received: 10 March 2024. Accepted: 10 September 2024
Key words and phrases. billiards, gravity, foci, paraboloid mirror, confined trajectories
MSC Classification Mathematics Subject Classification. 35A01, 65L10, 65L12, 65L20, 65L70
In 2015, the first study of the dynamics in a two-dimensional cone were performed by [LM15] , [Lan15] showing that certain quantities of the one-dimensional framework map one-to-one to the embedded surface in
In general, the motion of the particle is highly non-trivial and a neat expression for the trajectory at each time is not accessible. For this reason, following [Mas14] , [Mas20] , the confined domains for a point like particle bouncing in a parabolic, one-dimensional cavity under the influence of a homogeneous gravitational force were derived through a geometric-analytic approach. In particular the author showed in [Mas20] that the geometric interpretation of the integrable system is that the foci of consecutive flight parabolae all lie on a common circle of fixed radius with center being the focus of the parabolic mirror.
Recently associated foci curves and confined domains for a particle ideally bouncing inside general one-dimensional boundaries were obtained in [Jau23a] . As a generalization Galavotti and Jauslin [GJ20] considered the so called Boltzman system (a particle under the influence of a Keplerian potential being reflected along a straight line within
In the work of Borisov et al. [ABM19] the frictionless motion of a point mass on an elliptic (or hyperbolic) paraboloid
In the following the confined domains for a particle bouncing inside a rotational symmetric paraboloid under the influence of a constant gravitational force parallel to the axis of symmetry is studied. Our analysis will show that some one-dimensional features obtained e.g. in [Mas20] , [Jau23a, Jau23b] will carry over (in some cases) to the two-dimensional scenario. Due to the additional rotational movement associated to conserved angular momentum along the
The structure of this work presents as follows: in Section 2 we will briefly introduce all necessary assumptions and general ideas that we will benefit from in our later analysis. Before diving into a general analysis, Section 3 will show that under certain restrictions the two-dimensional particle motion can be reduced to the one-dimensional force free case within a circle. For this case, corresponding to maximal angular momentum, we give a neat expression for the particles position within the cavity at any time
We are considering the movement for a particle of mass
The focus of the paraboloid is centered at the origin of the coordinate system and
For a general point
As usual, the trajectory of one specific flight parabola can be written as a function of time
where
Note that for a given flight parabola
In analogy to the one-dimensional case (see [Mas20] , [Jau23a] )
As a last component we state the law of reflection in vector form whenever the particle hits the boundary of the mirror at a point
Here we used in the last step the fact, that
From our setup it is clear that the allowed values for
Due to rotational symmetry it thus is sufficient to determine the restrictions on
Both conditions are fulfilled if
i.e. the
Demanding for the reflection points all to lie on the same circle of radius
As mentioned before in the introduction, this result is consistent with former studies, e.g. [Fed01] , [Ves88] , where the trajectories inside a general quadric are tangent to other confocal quadrics.
Considering the flight parabolas dividing the rotational flight surface
As a remark for
The trajectory of the particle in our szenario now is completely determined by the parameters
where the particle at time
corresponds to the propagation time between two consecutive point of reflections.
Applying the law of reflection in
i.e. both foci lie on a common sphere of radius
The vertex
Thereby we used that for the focal length of Section 2 holds
where
restricts the allowed values for the orientation of
Due to rotational symmetry it is sufficient to consider the case
with the restriction on the radius
In order to obtain expressions for the associated envelope curves we define a new quantity
The envelope curves restricting the confined domains then are obtained eliminating
In this section four limiting cases in terms of the reduced angular momentum
Solving the system ( 26 ) yields the envelope curves denoted by
This reproduces the results obtained geometrically in [Mas20] and analytically in [Jau23a] . Since the motion lies in a common plane containing the
For
This envelope curve is reminiscent of the Higgs-potential in particle physics, in which in the cases
where
In these cases, the second square root for the height function of Theorem 3 vanishes, resulting in a single height function for
with
For future works it would be interesting to generalize our results to other rotational symmetric domains. Also, the motion in a non-constant, e.g. Keplerian-field, would be of interest, generalizing e.g. the results obtained in [Fed01] , [JZ24] .
We would like to thank Dan Reznik for the inspiring conversation leading to this work. Also we thank the anonymous reviewer for careful reading of the manuscript and the many insightful comments and suggestions.
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