A note on a family of non-gravitational central force potentials in dimension one

Accepted Manuscript A note on a family of non-gravitational central force potentials in dimension one M. Alvarez-Ram´ ırez, M. Corbera, Josep M. Cors, A. Garc´ ıa PII: S0893-9659(17)30133-7 DOI: http://dx.doi.org/10.1016/j.aml.2017.04.020 Reference: AML 5241 To appear in: Applied Mathematics Letters Received date : 12 March 2017 Accepted date : 19 April 2017 Please cite this article as: M. Alvarez-Ram´ ırez, et al., A note on a family of non-gravitational central force potentials in dimension one, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.04.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A note on a family of non-gravitational central force potentials in dimension one M. Alvarez-Ram´ıreza, M. Corberab, Josep M. Corsc, A. Garc´ıaa,∗ aDepartamento de Matem´aticas, UAM–Iztapalapa, 09340 Iztapalapa, M´exico City, M´exico bFacultat de Ci`encies i Tecnologia. Universitat de Vic-Universitat Central de Catalunya (UVic-UCC), C. de la Laura, 13, 08500 Vic, Spain cDepartament de Matem`atiques , Universitat Polit`ecnica de Catalunya, 08242 Manresa (Barcelona), Spain. Abstract In this work we study a one-parameter family of differential equations and the different scenarios that arise with the change of parameter. We remark that these are not bifurcations in the usual sense but a wider phenomenon related with changes of continuity or differentiability. We offer an alternative point of view for the study for the motion of a system of two particles which will always move in some fixed line, we take Rfor the position space. If we fix the center of mass at the origin, so the system reduces to that of a single particle of unit mass in a central force field. We take the potential energy function U(x) = |x|β, where xis the position of the single particle and βsome positive real number. Keywords: Singularities; collisions; non-gravitational interactions. 1. Introduction1 In 1981, R. McGehee [1] investigated geometrically the regularization of2 binary collisions of classical particle systems with non-gravitational interactions.3 R. McGehee considered the motion near a collision of a particle in the vector4 field given by the homogeneous potential U(x) = −|x|−α, where x∈R2is the5 position of a single particle and αis a positive real number. McGehee showed6 by appropriate coordinate transformations that the singularity corresponding7 to a double collision (x= 0) is blown up to a collision manifold, after that the8 time variable is rescaled appropriately, and finally the vector field is extended9 smoothly to this manifold. He noted that there exits a bifurcation at α= 2.10 More recently, Xia and Jard´on-Kojakhmetov [2] investigated the topological11 structure of the same system as αvaries along the entire real line R. This study12 ∗Corresponding author Email addresses: [email protected] (M. Alvarez-Ram´ırez), [email protected] (M. Corbera), [email protected] (Josep M. Cors), [email protected] (A. Garc´ıa) Preprint submitted to Elsevier May 22, 2017 β Uβ(x)U0 β(x) 0< β < 1 continuous, not Lipschitz not continuous β= 1 continuous, Lipschitz not continuous 1< β < 2 differentiable continuous, not Lipschitz 2≤βdifferentiable differentiable Table 1: The properties of the potential Uβ(x) and U0 β(x) at x= 0. recovers the results of the previous reference and tries to extend the analysis to13 α≤0. They use the McGehee techniques without considering that for α≤014 the potential U(x) is defined at x= 0 and claimed the occurrence of bifurcations15 at α= 0 and α= 2. The aim of this paper is to show that the study of the flow16 near the origin is relevant in the global flow. This fact was ignored by Xia and17 Jardon-Kojakhmetov [2].18 In order to fix the ideas we define the concept of bifurcation and remark that19 the different phase spaces that arise with the change of parameter α < 0 do not20 fulfill this definition. Let Xα(q) be a family of vector fields where q∈D⊂Rn 21 and α∈R, a bifurcation is a change in the topological or analytical behavior22 of the flow when the parameter αpasses a value α0∈R. We note that D, the23 domain of the vector field is the same for all the family.24 We are going to study the differentiability and continuity properties of the25 family of central force Hamiltonians and potentials of [1, 2] only for the case26 n= 1. We omit proofs because they are straightforward and adds nothing to27 the paper.28 2. Equations of motion29 Let us consider the Hamiltonian: H(x, y) = y2 2−Uβ(x), where x,y∈R,β > 0 and Uβ(x) = |x|β. The associated Hamiltonian equations are: ˙x=y, ˙y=U0 β(x) = βx|x|β−2.(1) Let us remark that βcorresponds to −αin [1, 2]. The next proposition gives30 the continuity and differentiability properties of Uβ(x) and U0 β(x) depending on31 the value of β.32 Proposition 1. The properties of the potentials Uβ(x) = |x|βand Hamiltonian33 vector fields X(x, y)=(X1(y), X2(x)) = y, U0 β(x)where β > 0are:34 1. If x6= 0 the potentials Uβ(x)and their associated Hamiltonian vector fields35 X(x, y)are differentiable.36 2. X1(−y) = −X1(y),X2(−x) = −X2(x)and Uβ(x) = Uβ(−x).37 2 β U0 β(x)U00 β(x) 0< β < 1x > 0: U0 β(x)>0, lim x→0+U0 β(x) = ∞x6= 0: U00 β(x)<0 x < 0: U0 β(x)<0, lim x→0− U0 β(x) = −∞ lim x→0U00 β(x) = −∞ β= 1 x > 0: U0 β(x)=1 x6= 0: U00 β(x)=0 x < 0: U0 β(x) = −1 1< β < 2U0 β(0) = 0, x > 0: U0 β(x)>0x6= 0: U00 β(x)>0 x < 0: U0 β(x)<0 lim x→0U00 β(x)=+∞ 2≤β U0 β(0) = 0, x > 0: U0 β(x)>0x6= 0: U00 β(x)>0 x < 0: U0 β(x)<0U00 β(0) = 0 Table 2: The properties of U0 β(x) near x= 0. 3. The properties of Uβ(x)and U0 β(x)at x= 0 are given in the Table 1.38 4. The properties of U0 β(x)near x= 0 are described in the Table 2.39 5. There is only one minimum of Uβ(x)at x= 0:U(0) = 0.40 Let ϕ(t)=(x(t), y(t)) be the solution of the Hamiltonian vector field X(x, y)41 with initial conditions (x0, y0), H(x0, y0) = hand x06= 0.42 The next proposition describes the flow if x6= 0.43 Proposition 2. The solution ϕ(t)satisfies44 1. If y(t)>0then the component x(t)is increasing, and if y(t)<0then the45 component x(t)is decreasing.46 2. If x(t)>0then the component y(t)is increasing, and if x(t)<0then the47 component y(t)is decreasing.48 3. Analysis of the different scenarios49 In this section we analyse the behaviour of ϕ(t) as the parameter βchanges.50 3.1. 0< β ≤151 In this case the vector field is not defined in x= 0.52 Proposition 3. If x0<0and h≥0, then there exists t1=t1(x0,|y0|)>053 and one of the following statements hold.54 1. h≥0and y0>0then lim t→t− 1 ϕ(t) = (0,√2h).55 2. h≥0and y0<0then lim t→ −t+ 1 ϕ(t) = (0,−√2h).56 If h < 0the solution is defined for all t∈R, see Figure 1.57 3 !1!2!3 1 2 3 x 1 2 !1 !2 y h < 0 h= 0 h > 0 Figure 1: Distinct solutions of (1) with 0 < β ≤1 depending on the value of h. There is a similar proposition mutatis mutandis for the right half plane.58 We observed that although the solutions in the left half plane and y > 059 with energy h > 0 has maximal interval (−∞, t1), they can be identified with60 the orbits in the right half plane with the same energy and y > 0. With this61 identification all solutions will be defined for all time. Moreover, they preserve62 the continuous dependence with respect to initial conditions. The curves just63 found are not differentiable in the crossing with the y-axis. In a similar way we64 can work with the orbits with energy h > 0 and y < 0.65 In the case of zero energy there are two possible choices for the continuation66 of the curves. In one hand, neither one will preserve the continuous dependence67 with respect to initial conditions, in the other hand one of the resulting curves68 will be differentiable.69 3.2. 1< β < 270 In this case the vector field is defined in x= 0, but is not Lipschitz. Now,71 when h > 0 the solutions that start in x0<0 and y0>0 pass the line x= 072 and enter into the first quadrant in a differential way. In a similar way we can73 identify the orbits with y0<0. For h= 0 we have the following result:74 Proposition 4. If x0<0and h= 0, then there exists t1=t1(x0,|y0|)such75 that76 1. if y0>0then x(t1) = 0,y(t1)=0.77 2. if y0<0then x(−t1)=0,y(−t1)=0.78 3. x(t) = y(t) = 0 is a solution for t∈R.79 There is an analogous proposition for the right half plane.80 Let us observe that we can obtain solutions passing through the origin with81 the following procedure: we start for instance with a solution of the form 1 of82 Proposition 4 (i.e. a solution with x0<0 and y0>0), we follow it until it83 reaches the origin, we stay at the origin as many time as we wish and leave the84 origin either following a solution of the form 2 of Proposition 4 (i.e. a solution85 with x0<0 and y0<0), or following a solution of the analogous of Proposition86 4 with x0>0 and y0>0. Depending on the different choices we have different87 solutions. See Figure 3.88 4 !1!2!3 1 2 3 x 1 2 !1 !2 y h < 0 h= 0 h > 0 Figure 2: Distinct solution of (1) with 1 < β < 2 depending on the value of h. Proposition 5. The initial value problem (1) with ϕ(0) = (0,0) has infinitely89 many solutions, see Figure 3.90 The Proposition 5 does not contradict the Existence and Uniqueness Theo-91 rem of Ordinary Differential Equations since the vector field is not Lipschitz in92 (0,0), see [3].93 3.3. 2≤β94 This case is differentiable, so the usual results and techniques of Hamiltonian95 dynamical systems can be used to describe the flow. Clearly the phase space96 is similar to the one of a hyperbolic fixed point. All solution are unique and97 defined for all time. In particular the solution with zero energy tends to the98 origin back or forward but in a infinity time.99 4. Conclusions100 In a differential equation with parameters defined in a domain it loosely said101 that a bifurcation occurs in a specific parameter if the behavior of its solutions102 changes. However, in the Hamiltonian system (1) associated to the potential103 Uβ(x) = |x|βwhere β∈Rthere are several troubles. First, there are different104 domains of definition when the parameter βvaries. Second, as it is shown the105 Proposition 1 there are different possibilities in the behavior of the solutions.106 For example, the solutions with energy h= 0 and that start in the second107 quadrant and if 0 < β ≤1 reach the origin in a finite time and they can not108 be continued. If 1 < β < 2 the solutions reach the origin in a finite time and109 they can be continued in a not unique form. If 2 ≤βthe solutions never reach110 the origin. However the change of behavior of the flow does not correspond to111 bifurcations.112 Acknowledgments113 M. Corbera and Josep M. Cors wish to thank the Department of Mathe-114 matics, UAM-Iztapalapa, Mexico, where some of this work was carried out. M.115 5 t1t2 t x t1t2 t x a) b) t1t2 t x t1t2 t x c) d) Figure 3: Examples of solutions of (1) with ϕ(t) = (0,0) for some t∈R. a) Solution with x(t)>0, y(t)<0 for t < t1,x(t) = 0, y(t) = 0 for t∈[t1, t2], and x(t)>0, y(t)>0 for t > t2. b) Solution with x(t)<0, y(t)>0 for t < t1,x(t) = 0, y(t) = 0 for t∈[t1, t2], and x(t)<0, y(t)<0 for t>t2. c) Solution with x(t)<0, y(t)>0 for t<t1,x(t) = 0, y(t)=0 for t∈[t1, t2], and x(t)>0, y(t)>0 for t>t2. d) Solution with x(t)>0, y(t)<0 for t<t1, x(t) = 0, y(t) = 0 for t∈[t1, t2], and x(t)<0, y(t)<0 for t>t2. 6 Alvarez-Ram´ırez and A. Garc´ıa were partially supported by the grant: Red de116 cuerpos acad´emicos Ecuaciones Diferenciales. Proyecto sistemas din´amicos y117 estabilizaci´on. PRODEP 2011-SEP, Mexico. M. Corbera and Josep M. Cors118 are partially supported by MINECO Grant Number MTM2013-40998-P.119 5. References120 [1] R. McGehee, Double collisions for a classical particle system with non-121 gravitational interactions, Comment. Math. Helv. 56 (1981) 524–557. doi:122 10.5169/seals-43257.123 [2] L. Xia, H. Jard´on-Kojakhmetov, Bifurcations of a non-gravitational inter-124 action problem, Appl. Math. Comput. 251 (2015) 253–257. doi:110.1016/125 j.amc.2014.11.066.126 [3] S. W. Hirsch, S. Smale, Differential equations, dynamical systems and linear127 algebra., Academic Press.128 7