On a linearly damped 2 body problem

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center.


Introduction
The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. In the previous paper [3], the author raised the hypothesis that some phenomena such as carboniferous gigantism of arthropods and vegetals (cf. [2,5] ), the appearent expansion of the universe (cf. [4])or slight deformation of electrical devices over decades might be explained by an atomic contraction phenomenon (with respect to the size of the proton) coming from a very weak environmental dissipation produced by what we consider to be vacuum. When a dissipation is added to the basic equation modelling Coulomb's central force (with q the elementary charge, m the mass of the electron and ε 0 the vacuum permittivity) or its equivalent Newton's law for planets (where G is the gravitational constant and M S the mass of the sun) written in complex form in the orbital plane with a suitable set of axes and length unit, new interesting phenomena appear. This leads us to consider, as the simplest possible dissipative perturbation of the conservative problem including both equations, the modified ODE In [3], considering the difficulty of exhibiting decaying solutions for the equation (2), the author investigated a slightly more complicated case where we allow the charge q to decay exponentially in time. This led to the equation for which explicit exponentially decaying solutions depending on two real parameters can be exhibited. More precisely for α = δ we found the fast rotating spiraling solutions whereas for α = 3 2 δ , we found the uniformly rotating spiraling solutions Although it would probably be interesting to do, at the level of [3] we did not try to prove wellposedness (global existence) for general initial data not leading to collision with the center, nor did we try to elucidate global behavior of trajectories other than the previous special solutions. Before deciding to do that, one must first try to see whether or not the simpler model (2) is sufficient to identify a contraction phenomenon, and possibly determine the extent of its stability.
In the present paper, we derive some partial results on (2) together with some heuristic discussion. More precisely, in Section 2, we prove that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up in finite time either, even in the classical context where arbitrarily large velocities are allowed. In Section 3, we prove that u(t) remains bounded under a smallness condition involving both initial radius and velocity. In Section 4, we prove that all bounded non-vanishing solutions converge to 0 at infinity in t. Finally, in Section 5, some formal calculations suggest the existence of special orbits with an asymptotically spiraling convergence to the center. We conclude by a few heuristic remarks.

A global existence result
For convenience we recall our basic equation (2) u where we simply wrote c instead of c 0 , since for mathematicians c is not always the velocity of light. In order to solve this equation with u = u(t) ∈ R 2 = C we introduce the amplitude and the phase and we shall drop the variable t when it does not lead to confusion. Here all derivatives are with respect to t. From the formulas we conclude that (4) is equivalent to the system of two real equations We observe first that when trying to solve (4) for t ≥ 0, the initial value u(0) = 0 is excluded by the singularity, while whenever u(0) = 0 we shall obtain at least a local solution for any initial velocity u ′ (0). Moreover, multiplying (6) by r we reduce it to with This relation expresses the variation of the momentum of the solution, which was constant in the conservative case δ = 0 and decays exponentially when δ > 0. By plugging in (5) the value of θ ′ given by (7), we are left with an equation involving r(t) only: We now state our first result Theorem 2.1. Assuming M = 0, the unique local non-vanishing solution r = r(t) of (8) with any initial data r(0) = r 0 = 0; r ′ (0) = r ′ 0 ∈ R is global and we have for some constants Proof. The existence and uniqueness of local non-vanishing solutions is obvious. Let us introduce the scalar function F , defined as long as the solution r exists and does not vanish by the formula: We have immediately In particular for all t ∈ [0, T max) we have which can be rearranged in the more convenient form providing at once the two inequalities The conclusions follow immediately with

Bounded trajectories
The next result does not require the condition M = 0.
Theorem 3.1. Let u 0 = 0 and assume the initial smallness condition Then the local solution u of (4) on [0, T ) with initial conditions u(0) = u 0 ; u ′ (0) = u ′ 0 satisfies the inequality In particular if u does not vanish in finite time, u is a global bounded non-vanishing solution .
Proof. We start from the inequality and we observe that if F 0 < 0, the inequality . Now we compute F 0 in terms of the initial data. In fact we have the formula so that for t = 0 we find The previous observation now gives that under condition (14), we have

Convergence to 0 of non-vanishing bounded solutions
As in the case of the conservative equation (1) which is well known to have elliptic and parabolic trajectories, we suspect that (4) may have some unbounded solutions. But the next result shows that in sharp constrast with (1), (4) does not have any periodic trajectory at all. Proof. We introduce the total energy Since u never vanishes, it is clear that u ∈ C 2 (R + ) and we have In particular, E(t) is non-increasing. Then we have two possibilities Then In particular u ′ ∈ L 2 (R + , C). We have assumed that u(t) is bounded, hence precompact in R + with values in C. Therefore if (16) is not satisfied we may assume that for some sequence t n tending to +∞ lim As an immediate consequence of the previous theorem and the results of Sections 2 and 3, we obtain Corollary 4.2. Assume Im(u 0 u ′ 0 ) = 0 and |u 0 ||u ′ 0 | 2 < 2c. Then the local solution of (4) with initial conditions u(0) = u 0 ; u ′ (0) = u ′ 0 tends to 0 as t tends to infinity.

A formal calculation for spiraling solutions 6 Concluding remarks
The previous calculations suggest the possible existence of special orbits with an asymptotically (exponentially fast) spiraling convergence to the center. More precisely if we succeed in finding, by whichever method, a solution r of (8)with for a certain C = C(M ) > 0, from this we can build a solution u = re iθ with θ ′2 ∼ Ke 6δt so that |u ′ | 2 = r ′2 + θ ′2 r 2 ≥ ke 2δt for some positive k and t large. The kinetic energy blows-up exponentially in such a case, which is not absurd since the potential energy also.
Remark 6.1. This construction seems interesting, but even in case it works this is not entirely satisfactory for the following reason: as in the case of (3), the family of spiraling orbits built in this way would depend on only two real parameters (the moment and the initial phase) whereas the phase space has four dimensions. Remark 6.3. In the case M = 0, it should be possible to study if, as in the conservative case, solutions with initial velocity directed towards the center vanish in finite time, if necessary under some additional conditions. We did not study this question here since it is not our main concern.
Remark 6.4. After the simple model of Rutherford-Bohr explained in [1,6] and after the introduction of undulatory mechanics by L. De Broglie, the purely probabilistic model of E. Schrodinger (cf. [7]) has been considered the best atomic model since 1926. It does not seem obvious at all to introduce a damping mechanism in that model, and it might very well happen that in order to do that, an effective coupling between deterministic corpuscular and probabilistic wave conceptions of particles has to be devised, as was always advocated by L. De Broglie and even suggested by E. Schrodinger himself in [7].