Radiation Reaction and Center Manifolds Markus Kunze
Mathematisches Institut der Universitat Koln Weyertal 86, D-50931 Koln, Germany email:
[email protected] Herbert Spohn
Zentrum Mathematik and Physik Department, TU Munchen D-80290 Munchen, Germany email:
[email protected] Abstract
We study the e ective dynamics of a mechanical particle coupled to a wave eld and subject to the slowly varying potential V ("q) with " small. To lowest order in " the motion of the particle is governed by an e ective Hamiltonian. In the next order one obtains \dissipative" terms which describe the radiation reaction. We establish that this dissipative dynamics has a center manifold which is repulsive in the normal direction and which is global, in the sense that for given data and suciently small " the solution stays on the center manifold forever. We prove that the solution of the full system is well approximated by the e ective dissipative dynamics on its center manifold.
1 Introduction At the beginning of this century, in the context of the Maxwell-Lorentz equations, radiation reaction was one of the most outstanding problems in theoretical physics. It was left sort of un nished when theoreticians turned to quantum electrodynamics. In this paper we study radiation reaction in the mathematically somewhat more accessible case of a scalar wave eld. We believe that our results provide good indications on the e ective dynamics for a charge coupled to the Maxwell eld [14]. To explain in more detail the physical context we have to set up the model rst. We consider a particle, position q(t) 2 IR3 and momentum p(t) 2 IR3 , with \charge" distribution of total charge e
Z
= d3x(x):
We require that is smooth, radial, and supported in a ball of radius R,
2 C01(IR3 ) ; (x) = r (jxj) ; (x) = 0 for jxj R :
(C )
The particle is coupled to the scalar wave eld (x; t) with the canonically conjugate momentum eld (x; t), x 2 IR3 . In addition the particle is subject to an external potential, V , whose properties will be listed below. We assume that the potential is slowly varying on the scale of the charge distribution, i.e., on the scale set by R . Formally we introduce the dimensionless parameter ", " 1, and consider the scale of potentials V ("q), " ! 0. The equations of motion for the coupled system are _ (x; t) = (x; t); _ (x; t) = (x; t) ? (x ? q(t));
Z (1.1) q_(t) = q p(t) 2 ; p_(t) = ?"rV ("q(t)) + d3x (x; t)r(x ? q(t)) : 1 + p(t)
The dynamics governed by (1.1) has three distinct time scales, wellseparated as " ! 0. On the microscopic time scale, t = O(1), the particle moves along an essentially straight line and the eld adjusts itself stationarily. On a time scale O("?1), that we call the macroscopic scale, the particle feels the potential and responds to it with an e ective kinetic energy which 1
incorporates the coupling to the eld. This scale was studied in [5]. The particle looses energy through radiation at a rate roughly proportional to q(t)2 . Thus on the macroscopic time scale, friction through radiation is of order ". To resolve such an e ect we have to go to even longer times or to look with higher precision. The friction time scale is the subject of our paper. The dynamics of (1.1) is of Hamiltonian form. We need a few facts in the case the external potential vanishes, V = 0. Then (1.1) has the energy Z Z H0 ( ; ; q; p) = (1+ p2)1=2 + 21 d3x (j(x)j2 + jr (x)j2)+ d3x (x)(x ? q) and the conserved total momentum Z P ( ; ; q; p) = p + d3x (x)r(x): The minimum of H0, at xed P , is attained at Sq;v = ( v (x ? q); v (x ? q); q; pv )
(1.2)
p where v 2 V = fv : jvj < 1g, pv = v= 1 ? v2, v = ?v r v , and ^v (k) = ?^(k)=[k2 ? (v k)2 ]; the hat denotes Fourier transform. We call Sq;v the soliton centered at q; v. It has the normalized energy Es(v) = H0(Sq;v ) ? H0 (Sq;0) 2 ? v2 1 log 1 + jvj = (1 ? v2)?1=2 ? 1 + 3me 2(1 ? ? v2) 2jvj 1 ? jvj and the total momentum Ps(v) = P (Sq;v ) 1 1 1 + j v j 2 ?1=2 = v(1 ? v ) + 3mev 2v2(1 ? v2) ? 4jvj3 log 1 ? jvj : (1.3) R
Here me = 31 d3k j^(k)j2 k?2 is the mass of the particle due to the coupling to the eld. We note that because of the Hamiltonian structure we have the identity v(dPs=dv) = (dEs=dv). Taking Sq;v as initial conditions for (1.1) with V = 0 we obtain a solution travelling at constant velocity v, Sq;v (t) = ( v (x ? q ? vt); v (x ? q ? vt); q + vt; pv ) ; v 2 V : 2
Let us call fSq;v : q 2 IR3 ; v 2 Vg the six{dimensional soliton manifold, S . Thus, for V = 0, if we start initially on S the solution remains on S and moves along the straight line t 7! q0 + v0t. In fact, if we start close to S , then S is approached asymptotically, [6]. When the particle is subject to a slowly varying external potential, then the rough picture is that the solution will remain close to S in the course of time. For simplicity we assume throughout that the initial datum for (1.1) lies exactly on S , i.e., (1.4) ( (0); (0); q(0); p(0)) = Sq0;v0 ; possible generalizations being discussed below. At this point it is instructive to transform (1.1) to the macroscopic spacetime scale in such a way that the eld energy remains constant. Then the macroscopic variables, denoted by a 0, are p t = "?1t0; x = "?1x0 ; q(t) = "?1q0(t0 ); and (x; t) = " 0(x0; t0): We also set "(x) = "?3("?1 x): R R In particular, "(x) = 0 for jxj "R and d3 x"(x) = d3 x(x). With this convention, omitting the primes and indicating explicitly only the "dependence of q0(t0 ), we arrive at p (1.5) (x; t) = (x; t) ? ""(x ? q"(t)); " " q_ (t) = v (t); Z p " " " m0 (v (t))v_ (t) = ?rV (q (t)) + " d3x (x; t)r"(x ? q"(t)): Here m0(v) is the p 3 2 3 matrix de ned through m0 (v)v_ = v_ + 3 (v v_ )v with (v) = 1= 1 ? v . Rather than momenta as in (1.1), we use velocities which turns out to be more convenient in our context. The initial soliton (1.4) transforms to Sq"0;v0 = ( "v0 (x ? q0); v"0 (x ? q0); q0; v0); (1.6) p )2 ] Rand v" = ?v r "v . Thus, on the where ^"v (k) = ? "^("k)=[k2 ? (v kp macroscopic scale, the total charge is " d3x(x), whereas Z 1 me = 3 " d3kj^"(k)j2 k?2 3
is independent of ". Eqs. (1.5) are again of Hamiltonian form. The energy Z 1 Hmac ( ; ; q; v) = (v) + V (q) + 2 d3 x (j(x)j2 + jr (x)j2) p Z + " d3x (x)"(x ? q) (1.7) is conserved under (1.5). It is bounded from below, as Hmac ( ; ; q; v) V (q) ? 3me independently of ". There is another, very instructive way to think about the initial value problem (1:5); (1:6). We prescribe initial data at t = ? , > 0, which have nite energy and some smoothness. We refer to [6] for the precise conditions. We solve (1.5) for V = 0 up to time t = 0. Then in the limit ! 1 the data at t = 0 are exactly of the form (1.6). For t > 0 the external forces are acting. Clearly this causes some mismatch, which is re ected by a non{smoothness of the elds ( ; ) at the light cone fx : jxj = t; t > 0g in the limit " ! 0. Under suitable assumptions on V and for jjL2 suciently small we proved in [5] that (1.8) jq_"(t)j v < 1 ; jq"(t)j C ; and j :::q " (t)j C uniformly in " and t 2 IR, and that the limit lim q"(t) = r(t)
"!0+
(1.9)
exists. Here r(t) is the solution of Hamilton's equations of motion with the e ective Hamiltonian E (p) + V (q), cf. the de nition of E (p) below (1.3), which in terms of velocities read
r_ = u ; m(u)u_ = ?rV (r); (1.10) with initial data r(0) = q0, u(0) = v0 . Here m(u) = m0 (u) + mf (u), where mf (u) is the additional \mass" due to the coupling to the eld de ned by mf (u)u_ = 3me '(juj)u_ + juj?1'0 (juj)(u u_ )u (1.11) as a 33 matrix, where '(jvj) is the function appearing in the square brackets of Eq. (1.3). Note that the energy
H (r; u) = Es(u) + V (r) 4
(1.12)
is conserved by the solutions to (1.10). With this background information let us return to the radiation reaction as discussed by Abraham, Lorentz, Schott, and Dirac, cf. [16] for an excellent account. Of course, these theoretical physicists were interested in the electrodynamics of moving charges. We take here the liberty to transcribe their arguments to the case of a scalar wave equation. For the sake of discussion we reintroduce the bare mass m0 and state the equations for small velocities only. In our proof below, however, we will handle all v 2 V . At the beginning of this century the hope was to de ne a structureless elementary charge through a point charge limit. For this program, one had to model the charge distribution phenomenologically with the understanding that ner details should become irrelevant in the limit. In (1.1) we adopted the Abraham model of a rigid charge distribution. The point charge limit then corresponds to taking in (1.1) a xed "-independent potential and to let the diameter of the charge distribution tend to zero. If this diameter is set proportional to " and ifpwe compare with (1.5), then in the point charge limit the charge distribution "" is to be replaced by ", which in particular shows that the adiabatic limit of a slowly varying external potential is distinct from R R 3 3 the point charge limit, where e = d x" (x) =R d x(x) is independent of " and the electromagnetic mass diverges as 31 d3kj^"(k)j2k?2 = "?1me. A formal Taylor expansion leads to the e ective equation of motion
m0r = ?rV (r) ? "?1mer + ae2 :::r; (1.13) valid for small velocities r_, with some constant a > 0. Eq. (1.13) is the nonrelativistic limit of the Lorentz-Dirac equation, [10]. The standard argument, reproduced in many textbooks, e.g. [3], (with the notable exception of Landau and Lifshitz [9]) is to lump m0 and "?1me together and to take the limits " ! 0 and m0 ! ?1 at constant m0 + "?1me = mexp, the experimentally observed mass of the particle. Then (1.13) reads as mexpr = ?rV (r) + ae2 :::r : (1.14) Since this equation is of third order, one needs besides q0; v0 also u_ (0) as initial condition which has to be extracted somehow from the initial data of the full system. Even worse, (1.14) has solutions which are exponentially unbounded in time, the famous run-away solutions. Thus one needs an additional criterion to single out the solutions of physical relevance. Dirac [1], and 5
later Haag [2], argued that physical solutions have to satisfy the asymptotic condition lim r(t) = 0; (1.15) t!1 as a substitute for the missing initial condition r(0). The validity of the asymptotic condition has been checked only in trivial cases; see [10]. For general V one should expect the solutions to (1.13) to be chaotic. Physical and unphysical solutions might be badly mixed up. On a more practical level, the physical solutions are unstable and therefore dicult to compute numerically. To put it in the words of W. Thirring [15]: \...(1.14) has not only crazy solutions and there are attempts to separate sense from nonsense through special initial conditions. But one hopes that the true solution to the problem will look di erently and that the nature of the equations of motion is not so highly unstable that the act of balance can be achieved only through a stroke of good fortune in the initial conditions." This is indeed the case, as we are going to show in this paper, and our resolution requires just a little twist. If instead of the point charge limit we consider a slowly varying external potential, then on the macroscopic time scale, according to (1.5), Equation (1.13) reads (m0 + me )r = ?rV (r) + "ae2 :::r
(1.16)
which re ects that radiation reaction is a small correction to the Hamiltonian motion. In (1.16) the highest derivative appears with a small prefactor. Such di erential equations are studied in geometric singular perturbation theory. From there we know that (1.16) has a six-dimensional invariant center manifold I", which is only O(") away from the Hamiltonian manifold I0 = f(q; q;_ q) : (m0 + me)q = ?rV (q)g. For initial conditions slightly o I" the solution moves away from I" exponentially fast. On I", q_ is bounded away from 1, q is bounded, and the motion is governed by an e ective second order equation, cf. Eq. (4.9) below, which gives precisely the physical solutions. To establish such a result we have to prove that the solution to (1.5) stays indeed close to I". In our paper we carry out this program, essentially under the same conditions as in [5], namely a suciently di erentiable V and jjL2 small. Our main additional estimate is
j :::v " (t)j C 6
(1.17)
uniformly in " and t 2 IR. Thereby we can bound one further order in the rigorous Taylor expansion and obtain, setting q_" = v",
m(v")v_ " = ?rV (q") + "a(v")v" + "b(v"; v_ ") + "2f "(t); t "t1;
(1.18)
with jf "(t)j C and coecient functions a; b that will be de ned below. Clearly (1.18) should be compared with
r_ = u ; m(u)u_ = ?rV (r) + "a(u)u + "b(u; u_ ):
(1.19)
Our crucial observation is that the condition ju(t)j const: < 1 for all t holds only on the center manifold I". Thus the a priori estimate jq_"(t)j v < 1, see (1.8), together with the initial conditions r(0) = q0, u(0) = v0, uniquely singles out that solution of (1.19) which is to be compared with the true solution. The coecient functions a; b are proportional to e2. If the total charge e = 0, then the friction term in (1.18) vanishes identically and radiation reaction appears at a higher level of approximation. Since on the error term f "(t) in (1.18) we only know that it is uniformly bounded, the di erence jq"(t) ? r(t)j, with r(t) having initial conditions on I", can be bounded at best as "ect . Thus on the time scale t = O(1) we seem to be back to the result (1.9) already proved in [5]. To distinguish, from this point of view, between (1.19) and (1.10) we would have to control the di erence with a precision of order "2. At present we do not know whether this is possible, but nevertheless we can prove the weaker statement
jH (q"(t); v"(t)) ? H (r(t); u(t))j const: "2;
(1.20)
where H is the energy from (1.12). Thus on a surface of constant energy the di erence jq"(t) ? r(t)j could be of order ", whereas along rH it must be of order "2. In addition to (1.20) it may also be shown that in fact jq"(t) ? r(t)j "3 on the short time scale t = O("), a result that is quite natural from the viewpoint of singularly perturbed ODEs. On the original time scale of (1.1) this amounts at least to an estimate with precision "2 over time intervals of length O(1), a result that could not have been obtained from the bounds in [5]. Taking a somewhat broader perspective, the problem discussed here may be viewed as an in nite dimensional Hamiltonian system which relaxes to a 7
stationary solution through the emission of radiation. This phenomenon is fairly common and has been studied in the context of linear and nonlinear wave equations. We refer to [12, 13] for recent work, also containing more references w.r. to prior studies. For such problems one typically has a stationary eigenmode which turns into a resonance by coupling to propagating modes. In comparison, a simplifying feature in the present paper is that the localized and propagating degrees of freedom are already well separated on the level of the equations of motion. On the other hand, we provide a quantitative estimate on the relaxation process and not only a power law decay in time.
2 Main results We give some more details and state our main results precisely. First we have to establish the bound (1.17). Lemma 2.1 For jjL2 suciently small we have sup j :::v " (t)j C t2IR
for every solution of (1.5) which starts on the soliton manifold S . Both the constant C and the bound for jjL2 depend only on the initial data. The bound of Lemma 2.1 may be used to Taylor expand the self-force p Z Fs"(t) = " d3x (x; t)r"(x ? q"(t)) (2.1) in (1.5) as Fs"(t) = ?mf (v"(t))v_ "(t)+ "a(v"(t))v"(t)+ "b(v"(t); v_ "(t))+ O("2) ; t "t1 ; (2.2) which together with the second equation in (1.5) yields (1.18). Here mf is de ned in (1.11), and t1 = 2R =(1 ? v) is the microscopic time the wave equation needs to forget its data because of the compact support of and the velocity bound, cf. assumption (C ) and (1.8). The coecient functions are given by a(v)v = (e2 =24)(v rv )rv 2 = (e2 =12)[ 4v + 4 6(v v)v] ; (2.3) b(v; v_ ) = (e2 =32)(v_ rv )2 rv 2 = (e2 =4)[2 6(v v_ )v_ + 6 v_ 2v + 6 8(v v_ )2v]; (2.4)
8
p
v;_ v 2 IR3, with = 1= 1 ? v2, jvj < 1. Next we explain the existence and the role of the center-like manifolds I" in greater detail. We refer to [11, 4] for further background on geometric singular perturbation theory. To rewrite (1.19) as a singular perturbation problem, let x = (r; u) 2 IR3 V ; y = u_ 2 IR3 ; f (x; y) = (x2 ; y) 2 V IR3 ; and g(x; y; ") = a(x2 )?1 [m(x2 )y + rV (x1) ? "b(x2 ; y)] : Then (1.19) reads as x_ = f (x; y) ; "y_ = g(x; y; ") : (2.5) We intend to apply the results from [11] to (2.5) in order to nd a center-like manifold for the perturbed problem near the corresponding manifold for the (" = 0)-problem. With h(x) = ?m(x2 )?1rV (x1 ), let I0 = f(x; y) : g(x; y; 0) = 0g = f(r; u; u_ ) : m(u)u_ = ?rV (r)g = f(x; h(x)) : x 2 IR3 Vg (2.6) be this invariant manifold for (2.5) with " = 0. The ow on I0 is governed by the equation x_ = f (x; h(x)), or stated di erently, m(r_ )r = ?rV (r), the familiar Hamiltonian ow. To see that I0 is perturbed to some I" with " small, we have to modify the functions a(u), m(u), and b(u; u_ ) for juj close to one due to the singularity at juj = 1. This will cause no problems later on, since we already have the a priori bound jv"(t)j v < 1 for the velocity of the true system. In (4.4) below, we will x a small = (v) > 0 satisfying some estimates; depends only on bounds for the initial data, since v does so. Let K1? = IR3 fu 2 IR3 : juj 1 ? g; We continue a(u), m(u), and b(u; u_ ) with their values at juj = 1 ? to the missing in nite strip 1 ? < juj < 1. Then the basic assumptions (I ), (II ) from [11, p. 45] are satis ed, since I0 is also what is called normally hyperbolic, i.e. repulsive in the direction normal to I0 at an "-independent rate, see Lemma 4.1 below. Hence we nd "0 = "0() > 0 and a C 1 -function h(x; ") = h"(x) : IR3 V]0; "0] ! IR3 such that for " "0, I" = f(x; h"(x)) : x 2 IR3 Vg 9
is forward invariant for the ow (1.19) with the modi ed functions a; m; b. Since the modi ed equation agrees with (1.19) in the interior of K1?, we conclude that I" is locally invariant for the ow (1.19), i.e. the solution of the modi ed equation is the solution to the original equation as long as it does not reach the boundary set f(x; h"(x)) = (r; u; h"(r; u)) : juj = 1 ? g. The ow for " = 0 is then perturbed to x_ = f (x; h"(x)) for " "0. We will show in Theorem 4.4 below that for " 2]0; "1], with "1 > 0 suciently small, all solutions of (1.19) starting at points (r; u; h"(r; u)) 2 I" with juj v, will indeed stay away from the boundary f(r; u; h"(r; u)) : juj = 1 ? g for all future times. In addition, rV (r(t)) ! 0 and r(t) ! 0 as t ! 1, which is just the asymptotic condition (1.15) postulated by Dirac and Haag. If the potential is suciently con ning, then the solution trajectory on I" not only approaches the set of critical points for V in the long-time limit, but it converges to some de nite critical point. Moreover, we will show that for all solutions on the center manifold, u_ (t) and u(t) are bounded, and u(t) is bounded away from 1, uniformly in " and t. Conversely, every such solution to (1.19) has to lie on I". Thus I" indeed characterizes the physical solutions. To summarize, we have established now the existence of a center manifold I" with a well-de ned (semi-) ow on it that gives a unique solution to (1.19) for initial velocities bounded by v. For the potential V 2 C 3(IR3) we assume that it is bounded in the sense sup3 jV (q)j + jrV (q)j + jrrV (q)j + jrrrV (q)j < 1 : (U ) q2IR
The method works equally well for V 2 C 3(IR3) which is con ning, i.e., V (q) ! 1 as jqj ! 1 ; (U 0 ) as will be made more precise in Section 4, cf. Theorem 4.8. Our main result is the following Theorem 2.2 Assume (U ) or (U 0 ) for the potential, and let the initial data ( 0(x); 0(x); q0 ; v0) for (1.5) be given by (1.6). Let jjL2 and " "1 be suciently small, and introduce the center manifolds I" for the comparison dynamics (1.19) as explained above. At time "t1 = "2R =(1 ? v) we match the initial values, r("t1 ) = q "("t1 ), u("t1 ) = v "("t1 ), for the motion on the center manifold, i.e., the initial data for the comparison dynamics are (q"("t1); v"("t1 ); h"(q"("t1); v"("t1))) 2 I" : 10
Then for every > 0 there exists c( ) > 0 such that for all t 2 ["t1 ; "t1 + ]
jq"(t) ? r(t)j c( )" ; jv"(t) ? u(t)j c( )" ; and jv_ "(t) ? u_ (t)j c( )":
(2.7)
In addition we have the bound
jH (q"(t); v"(t)) ? H (r(t); u(t))j c( )"2:
(2.8)
Remarks 2.3 (i) As already mentioned at the end of the introduction, we can also show
jq"(t) ? r(t)j c( )"3 and jv"(t) ? u(t)j c( )"2 for t 2 ["t1; "t1 + " ], i.e., t = O("), cf. Proposition 5.1.
(2.9)
(ii) The construction of the center manifolds and the upper bound for jjL2 rely only on bounds for the data, but not on properties of a particularly chosen solution. Our main technical assumption is a suciently small jjL2 which is presumably not necessary. (iii) In [5] we did not require the true solution to start on the soliton manifold, but instead to start close to it. We refer the criterion [5, Thm. 2.6] for an \adiabatic" family of solutions. The same generality could be achieved in the present context, using an appropriately modi ed version of [5, Thm. 2.6]. In Section 8 we derive the relevant estimates, in particular (8.8), in full generality containing a non-zero initial di erence Z (0). The corresponding generalization of Theorem 2.2 is then straightforward. However, since we did not want to obscure our main achievement through technicalities, we decided to elaborate here the more accessible case of a trajectory starting right on the soliton manifold. In the same spirit we do not consider arbitrary time intervals of length , but only the particular ["t1; "t1 + ]. (iv) The existence of solutions to (1.1) is discussed in [5, Lemma 2.2]. For every initial value Y 0 = ( 0(x); 0 (x); q0; p0) 2 E we nd a unique (weak) solution Y () 2 C (IR; E ) such that Y (0) = Y 0. Here the state space is E = D1;2(IR3 ) L2(IR3) IR3 IR3 [where D1;2(IR3) = f 2 L6 (IR3) : jr j 2 L2 (IR3 )g] with norm jY jE = jr jL2 + jjL2 + jqj + jpj. 11
Having such fairly precise information on the particle trajectory we can also determine the adiabatic limit " ! 0 of the elds ( ; ) in (1.5) through the solution of the inhomogeneous wave equation. We generate the initial data as explained in the introduction. On the level of the comparison dynamics this means to extend r(t) and u(t) to negative times t 0 by r(t) = q0 + tv0 resp. u(t) = v0. Let the retarded time tret , depending on x and t, be the unique solution of tret = t ? jx ? r(tret)j, and let n^(x; t) = (x ? r(tret))=jx ? r(tret)j. Theorem 2.4 Under the conditions of Theorem 2.2 and for the elds ( ; ) from (1.5) we have for x 6= r(t) the pointwise limits ?1 1 (x; t) = ? e p lim 1 ? n ^ ( x; t ) u ( t (2.10) ret ) "!0 " 4jx ? r(tret )j and, except for the light cone fx : jxj = t > 0g, 1 (x; t) p lim "!0
"
= ?
e
4jx ? r(tret)j
?3
?3
1 ? n^(x; t) u(tret)
? 4jx ?er(t )j2 1 ? n^ (x; t) u(tret) ret
n^(x; t) u_ (tret ) (^n(x; t) u(tret) ? u(tret)2): (2.11)
The paper is organized as follows. Since the proof of Lemma 2.1 is rather technical, we moved it to an appendix, Section 8. The derivation of the representation (2.2) of the self-force term is the contents of Section 3. In Section 4 we give supplementary remarks on the behaviour of solutions on the center manifold, whereas in Section 5 we carry out the proofs of Theorem 2.2 and Proposition 5.1. Section 6 contains the proof of Theorem 2.4, and nally in Section 7 we determine the amount of energy radiated to in nity.
3 Representation of the self-force In this section we show that the self-force Fs"(t) from (2.1) can be written in the form (2.2). We carry out this computation on the original fast time 12
scale corresponding to (1.1) since we will need some of the arguments from [5]. Thus we consider
Z Fs(t) = d3x (x; t)r(x ? q(t)):
Since (x; t) = r (x; t) + 0(x; t), where 0 = 0 with the initial values 0(x; 0) = 0(x) and 0 (x; 0) = 0(x), and since Z t ds Z 1 r (x; t) = ? 4 t ? s d2 y (y ? q(s)) 0 jy?xj=t?s is the retarded potential, we can decompose accordingly,
Fs(t) = F0(t) + Fr (t) = h 0(; t); r( ? q(t))i + h r (; t); r( ? q(t))i :
Lemma 3.1 The function F0 (t) vanishes for t t1 = 2R=(1 ? v). Proof :3 Let U (t)3denote the group generated by the free wave equation in D1;2(IR ) L2 (IR ). Then (1.4) and Fourier transformation implies ( (x); (x)) = ? 0
0
Z0
?1
ds [U (?s)( ? q0 ? v0 s)](x)
with (x) = (0; (x)). Thus Kirchho 's formula yields, as a consequence of jv0j < 1, that 0(x; t) = 0 for jx ? q0j t ? R . Since jq(t) ? q0j vt, the claim follows. 2 Hence to show (2.2) it is enough to prove
Lemma 3.2 For t t1 , Fr (t) = ?mf (v(t))v_ (t) + a(v(t))v(t) + b(v(t); v_ (t)) + O("3) ; cf. (1.11), (2.3), and (2.4).
Proof : We follow the proof of [5, Lemma 5.1], but expand q(s) = q(t) ? v(t)(t ? s) + 21 v_ (t)(t ? s)2 ? 61 v(t)(t ? s)3 + O("3) 13
up to third order, which is allowed by Lemma 2.1. Through Fourier transformation we arrive at Zt Z Fr (t) = (?i) ds d3k j^(k)j2 jkkj sin jkj(t ? s) e?i(kv)(t?s) 0 e?i[? 21 (kv_ )(t?s)2 + 61 (kv)(t?s)3 ] + O("3) ;
R
with v = v(t), etc.. As in [5, Lemma 5.1], here and in the following 0t ds(: : :) R can be changed forth and back to tt?T ds(: : :) for all t; T t1. Because
e?i[? 21 (kv_ )(t?s)2 + 61 (kv)(t?s)3 ] = 1 + 2i (k v_ )(t ? s)2 ? 6i (k v)(t ? s)3 ? 81 (k v_ )2(t ? s)4 + O("3)
for t ? s = O(1) by (8.1) below, we obtain, for t; T t1 , ZT Z Fr (t) = (?i) d d3k j^(k)j2 jkkj sin jkj e?i(kv) 0 i h 1 + 2i (k v_ ) 2 ? 6i (k v) 3 ? 81 (k v_ )2 4 + O("3) : Let
Ip =
Then
ZT 0
d sinjkjkj j e?i(kv) p ; p = 0; : : : ; 4 :
(v rv )rv I1 = ?k(k v)I3 and (v_ rv )2 rv I1 = ik(k v_ )2 I4 :
R
Our claim now follows from Lemma 3.3 below, d3k j^(k)j2kI0 ! 0, and R (1=2) d3k j^(k)j2k(k v_ )I2 ! ?mf (v(t))v_ (t) for T ! 1; see [5, Appendix A]. 2
Lemma 3.3 We have the identity Z1 0
Z dt t d3kj^(k)j2 sinjkjkj jt e?i(kv)t = (e2=4) 2 : 14
Proof : Since ^(k) = ^r (jkj) is radial, and by transformation to polar coordinates,
Z
Z1 sin j k j t 4 ? i (kv)t d kj^(k)j jkj e = tjvj dR j^r (R)j2 sin(Rt) sin(Rtjvj) : 0 Thus for xed T > 0, ZT Z dt t d3kj^(k)j2 sinjkjkj jt e?i(kv)t 0 Z 1 dR sin( R (1 + j v j ) T ) 2 2 sin(R(1 ? jv j)T ) : ? = jvj R j^r (R)j 1 ? jvj 1 + jvj 3
2
0
R
To complete the proof we only need to verify that d3kj^(k)j2 jkj?3 sin jkjT ! e2 =4 as T ! 1. To see this, let ^(k ) = jk j?3 sin(jk jT ). Then
Z
Z Z d3 kj^(k)j2 ^(k) = (2)?3=2 d3x(x) d3y (y) (x ? y) q and we are going to show (x) ! =2 as T ! 1. We have, by transformation to polar coordinates, Z Z 1 sin(s) sin(sjxj=T ) 3=2 3 ^ ? ik x ds s (2) (x) = d k (k)e = 4 ! 22 sjxj=T 0 for T ! 1. This completes the proof. 2
4 More about the center manifold In this section we explain the behaviour of solutions on the center manifold. First we show that the unperturbed manifold I0 from (2.6) is hyperbolic in normal direction.
Lemma 4.1 The eigenvalues of Dy g(x; y; 0) = a(x2 )?1 m(x2 ) are bounded below by a positive constant, uniformly in x = (r; u) with r 2 IR3 and juj 1 ? , for all prescribed 2]0; 1]. 15
Proof : By [8, Thm. 2, p. 185], a(u) and m(u) can be simultaneously trans-
formed to diagonal form through a single non-singular matrix B . In addition, denoting by bj 6= 0 the j th column of B and by j the j th eigenvalue of a(u)?1m(u), one has j a(u)bj = m(u)bj , j = 1; 2; 3. Multiplication by bj leads to j (e2 =12) 3[ b2j + 4 3 (v bj )2 ] b2j + 3 (v bj )2 , and thus j (3=e2) ?3 . 2 Since a(u), m(u) are modi ed to be constant outside juj 1 ? , their corresponding eigenvalues are uniformly bounded below for juj < 1. As a consequence of Lemma 4.1 the manifolds I" are unstable at some exponential rate et for solutions in the normal direction. We note that, by [11, Thm. 2.1], supfjh"(r; u)j : (r; u) 2 IR3 V ; " 2]0; "0 ]g c = c(): (4.1) Our next aim is to prove global existence of solutions to (1.19) forward in time which start over Kv = IR3 fu 2 IR3 : juj vg on the center manifold, provided " "1 with "1 > 0 suciently small. For this purpose we introduce a suitable Lyapunov function. Lemma 4.2 Let G"(r; u; u_ ) = H (r; u) ? "(a(u)u_ ) u = Es(u) + V (r) ? "(a(u)u_ ) u : Then along solutions (r(t); u(t); u_ (t)) of (1.19) we have d G (r; u; u_ ) = ?"(e2 =12) [6 8(u u_ )2 + 6 u_ 2]: (4.2) dt " Proof : Observing that (a(u)u_ ) u = (e2 =12) 6 (1 + 3u2)(u u_ ); this is a straightforward calculation. 2 Through the Lyapunov function G" we can control the long time behaviour. Theorem 4.3 Let (U ) or (U 0 ) hold and let any global solution (r(t); u(t)) of (1.19) be given such that supt0 ju(t)j u(") < 1 and supt0 ju_ (t)j c("), for possibly "-dependent constants u(") and c("). Then u_ (t) ! 0; u(t) ! 0; and rV (r(t)) ! 0 as t ! 1: 16
Proof : Denoting by c(") or C (") general "-dependent constants, by Lemma 4.2 we have along a trajectory
c(")
ZT 0
ZT u_ 2 dt ? dtd G" dt 0 = ?Es(u(T )) ? V (r(T )) + "(a(u(T ))u_ (T )) u(T ) +Es(u0) + V (r0) ? "(a(u0)u_ 0) u0 C ("; data):
For theR last estimate observe inf r2IR3 V (r) > ?1 in both cases (U ) and (U 0 ). Thus 01 u_ 2 dt C ("; data) and, by (1.19), also supt0 ju(t)j C ("; data). Hence we::: conclude u_ (t) ! 0 as t ! 1. Next, di erentiation of (1.19) yields supt0 j u (t)j C ("; data), and thus from u_ (t) ! 0 we nd u(t) ! 0. Therefore rV (r(t)) ! 0 follows from the equation (1.19). 2 In the demonstration of the following theorem we use the sublevel sets
fG" cg = f(r; u; u_ ) : G"(r; u; u_ ) cg and fH cg = f(r; u) : H (r; u) cg for c 2 IR. However, before proceeding, we rst have to introduce an
appropriate = (v) > 0 small to modify the functions a(u), m(u), and b(u; u_ ) outside juj 1 ? , cf. Section 2. To do this, we assume (U ) from now on. The case (U 0 ) is discussed in the remarks below. Since V is bounded and v < 1, we can nd c0 2 IR such that Kv fH c0g. Then as a consequence of Es(u) ! 1 for juj ! 1, we have
o n s0 = sup juj : (r; u) 2 fH c0 + 1g for some r 2 IR3 < 1 :
Let us de ne
= minf(1 ? v)=2; (1 ? s0)=2g > 0:
(4.3) (4.4)
Theorem 4.4 Assume the potential V to satisfy the condition (U ). Then there exists "1 > 0 depending only upon v such that for " 2]0; "1 ] all solutions of (1.19) starting at points (r; u; h"(r; u)) 2 I", juj v, stay away from the boundary f(r; u; h"(r; u)) : juj = 1 ? g for all future times. In particular, solutions exist globally.
Proof : Let us denote the bound c() from (4.1) by c1 and let us x ca > 0 such that ja(u)j ca for all juj < 1. We recall that a(u) was modi ed to be constant outside juj 1 ? . We de ne "1 = minf"0; (2cac1 )?1g > 0. 17
Let (r; u) 2 Kv. Then G"(r; u; h"(r; u)) = H (r; u) ? "(a(u)h"(r; u)) u c0 +ca c1". Because of Lemma 4.2 the set fG" c0 +ca c1"g is forward invariant and the solution remains in this set for all future times. On the other hand, since v 1 ? 2 < 1 ? , the solution of the modi ed problem is a solution to (1.19) and stays on I", at least for a short times. For the xed time span where this holds the solution is of the form (r1 ; u1; h"(r1; u1)) and we have H (r1; u1) = G"(r1 ; u1; h"(r1; u1))+ "(a(u1)h"(r1; u1)) u1 c0 + ca c1 " + cac1 " = c0 +2cac1 " c0 +1 for " "1. Therefore by (4.3), ju1j s0 1 ? 2 < 1 ? . This argument shows that in fact the solution is con ned to f(r; u; u_ ) : juj 1 ? 2g. Hence the solution of the modi ed problem exists, is a solution to (1.19), and stays on I" for all future times. 2
Corollary 4.5 In the setting of Theorem 4.4, for solutions of (1.19) starting on I", supfju(t)j : t 2 IR; " 2]0; "1]g 1 ? 2 < 1 ; and supfju_ (t)j : t 2 IR; " 2]0; "1]g + supfju(t)j : t 2 IR; " 2]0; "1 ]g c():
(4.5)
In particular by Theorem 4.3 u_ (t) ! 0; u(t) ! 0; and rV (r(t)) ! 0 as t ! 1: Proof : The rst estimate was mentioned already in the preceding proof. For the second we note that (4.1) applies, since the trajectory stays on the center manifold, u_ (t) = h"(r(t); u(t)). Concerning the last bound, we may write h"(r; u) = ?m(u)?1rV (r) + h1;"(r; u) with jh1;"(r; u)j c()" (4.6) for (r; u) 2 IR3 V , see [11, Thm. 2.9]. By (1.19), j"uj ja(u)?1jjm(u)h1;"(r; u) ? "b(u; u_ )j c()"; so we are done. 2
Solutions on I" are uniformly bounded, in the sense of the corollary; in general a bound on r(t) cannot be expected, e.g. in a scattering situation. Conversely, as to be shown next, solutions with uniformly bounded u(t), u_ (t), and u(t) are con ned to the center manifolds. 18
Proposition 4.6 Suppose we have a family (r"(t); u"(t)), " 2]0; "2], of solutions to (1.19) such that
supfju"(t)j : t 2 IR; " 2]0; "2]g u < 1 ; and supfju_ "(t)j : t 2 IR; " 2]0; "2]g + supfju"(t)j : t 2 IR; " 2]0; "2]g c2: Then for suciently small " the solutions have to lie on I".
Proof : Note that we can construct I" here by modifying a(u); m(u), and b(u; u_ ) to be constant outside, say, fu : juj (1 + u)=2g. According to
[11, Thm. 2.1 (ii)] there exists > 0 such that for all " small and solutions (x(t); y(t)) to (2.5) the condition supt2IR jy(t) ? h(x(t))j implies that the solution has to lie on I". With x(t) = (r"(t); u"(t)) and y(t) = u_ "(t), this condition is veri ed since we obtain from (1.19) and the assumed bounds ju_ "(t) + m(u"(t))?1rV (r"(t))j c" , the latter for " small. 2 The asymptotic condition, r(t) ! 0, of Dirac and Haag is also sucient for a solution to lie on I", in the following sense.
Proposition 4.7 Suppose a family (r"(t); u"(t)), " 2]0; "2], of solutions to
(1.19) is given such that
supfju"(t)j : t 2 IR; " 2]0; "2]g u < 1 and r"(t) = u_ "(t) ! 0 as t ! 1 for each " 2]0; "2 ]. Then for sucient small " the solutions have to lie on I".
Proof : Fix > 0. Since Theorem 4.3 applies, we nd in the notation of
Proposition 4.6 jy(t) ? h(x(t))j = ju_ "(t) + m(u"(t))?1rV (r"(t))j =2 for t t("), with some t("). Thus the solution remains (=2)-close to I0 after time t("), and hence by (4.6) also -close to I" for " small. Since I" is normally hyperbolic (repulsive) at an "-independent rate and since > 0 was arbitrary, this can only happen if the solution was already contained in I". 2 Corollary 4.5 provides a partial information on the long time behavior of the solutions to (2.5) on the center manifold. Roughly one can distinguish two 19
classes. (i) (scattering): The particle enters a domain where ?rV = 0 at r1 with velocity u1. If the straight line trajectory r1 + u1t, t 0, is contained in this domain, then the particle travels freely to in nity. Physically this is a scattering trajectory. In this case limt!1 u(t) = u1 6= 0, whereas the position has no limit. (ii) (bounded motion): We assume that jr(t)j const: and that within this ball the critical points of V form a discrete set. Then by Corollary 4.5 and by continuity we have limt!1 u(t) = 0 and limt!1 r(t) = r1, where r1 is one of the critical points of V . If r1 is a stable critical point, then the relaxation is exponentially fast, as can be seen from linearization around the xed point. Clearly (i) and (ii) do not exhaust all possibilities. The critical points of V could lie on a sphere. If V is con ning, one would still expect convergence to a de nite r1. Moreover, V could vanish inside a ball. If ?rV is pointing towards the ball, then close to each turning point the particle looses energy. Thus limt!1 u(t) = 0, whereas the position has no limit. The potential could decrease so slowly at in nity that no de nite velocity is approached. All these cases have to be studied separately. Up to now we discussed bounded potentials satisfying (U ). In the introduction we claimed that our results remain valid also for con ning potentials satisfying (U 0 ). In this case, since V is unbounded, we have no longer Kv fH c0g for some c0 2 IR as in Theorem 4.4 above. However, by energy conservation, one can derive the a priori bound supt2IR jq"(t)j M for solutions to the true system (1.5) on the macroscopic time scale. Thus the motion is bounded also in the q-direction and it suces to build the center manifold for the e ective equation (1.19) over the bounded domain KM; v = fr 2 IR3 : jrj M g fu 2 IR3 : juj vg ; enlarged to a suitable KM +1;1? such that solutions starting over KM; v stay away from the boundary of KM +1;1? for " > 0 suciently small. In this manner we obtain
Theorem 4.8 Assume (U 0 ) holds for the potential, and let KM; be de ned as above. Then there exists "1 > 0 depending only on the initial data such that for " 2]0; "1 ] all solutions of (1.19) starting at points (r; u; h"(r; u)) 2 I", (r; u) 2 KM; , exist globally. Moreover, these solutions are uniformly bounded, supfjr(t)j : t 2 IR; " 2]0; "1]g c();
20
supfju(t)j : t 2 IR; " 2]0; "1]g 1 ? 2 < 1; and supfju_ (t)j : t 2 IR; " 2]0; "1]g + supfju(t)j : t 2 IR; " 2]0; "1 ]g c(): (4.7) In addition,
u_ (t) ! 0; u(t) ! 0; and rV (r(t)) ! 0 as t ! 1:
Proof : The proof is similar to the one of Theorem 4.4. Concerning the boundedness, note that again for some c0 2 IR and " > 0 small, KM; fH c0 g f(r; u) : G"(r; u; h"(r; u)) c0 + ca c1"g fH c0 + 1g: Thus all solutions starting over KM; will remain on the manifolds over fH c0 + 1g. Since this set is independent of " and compact by (U 0 ), the solutions must be uniformly bounded, because h" is uniformly bounded.
2
On the center manifold the motion is governed by the (second order) equation x_ = f (x; h"(x)): (4.8) Since the existence of h" is established only abstractly, Eq. (4.8) is somewhat implicit. From [11, (2.9-1) & Thm. 2.9] we know that h" depends smoothly on ". Thus (4.8) can be expanded in ". Including the rst Taylor term we pick up an error of order "2 , which is of the same order as the error between the true and the comparison dynamics on the center manifold. For consistency we should stop then at this order. We make the ansatz h"(r; u) = h0 (r; u) + "h1(r; u) + h2;"(r; u) ; jh2;"(r; u)j c()"2 for (r; u) 2 IR3 V . Then
m(u)h0(r; u) = ?rV (r); and h1(r; u) is determined through
Dxh0(r; u)f (r; u; h0(r; u)) = Dy g(r; u; h0(r; u); 0)h1(r; u)+D"g(r; u; h(r; u); 0); 21
see [11, (2.9-1) & Thm. 2.9]. Computing the respective derivatives one arrives at m(u)h1 (r; u) = a(u) ? m(u)?1r2V (r)u d ?1 + m(u) r V (r); m(u)?1rV (r) du ? 1 + b(u; m(u) rV (r)) and the e ective second order equation r_ = u ; m(u)u_ = ?rV (r) + "m(u)h1(r; u) (4.9) of the particle motion on the center manifold.
5 Comparison of the true and the e ective system In this section we prove Theorem 2.2. Since we have ju(t1)j = jv"(t1)j v by (1.8), Theorem 4.4, resp. Theorem 4.8, implies that the solution trajectory of the system with the modi ed functions a(u), m(u), and b(u; u_ ) is indeed a solution trajectory to (1.19). Recall that, by (1.18) and (1.19), m(v")v_ " = ?rV (q") + "a(v")v" + "b(v"; v_ ") + "2f "(t); t "t1; (5.1) m(u)u_ = ?rV (r) + "a(u)u + "b(u; u_ ); (5.2) with jf "(t)j C . Using the bounds (1.8) and (4.5), resp. (4.7), we infer the weaker estimate m(v")v_ " = ?rV (q") + O("); t t1; m(u)u_ = ?rV (r) + O("); which has been proved already in [5]. Hence (2.7) follows by the argument there. To show (2.8) we compute as in Lemma 4.2, using (5.1), d G (q"(t); v"(t); v_ "(t)) dt " = "2f "(t)v"(t)h i ?"(e2 =12) (v_ "(t))6v_ "(t)2 + 6 (v_ "(t))8 (v"(t) v_ "(t))2 ; t "t1 : 22
Since r("t1) = q"("t1 ) and u("t1) = v"("t1 ), using the uniform bounds, we have for t "t1 jH (q"(t ); v"(t)) ? H (r(t); u(t))j " (a(v"(t))v_ "(t)) v"(t) ? (a(u(t))u_ (t)) u(t)
ds "2jf "(s)v"(s)j + "(e2=12) (v_ "(s))6v_ "(s)2 ? (u_ (s))6u_ "(s)2 "t1 " 8 " 2 2 " 8 +6 (v_ (s)) (v (s) v_ (s)) ? (u_ (s)) (u(s) u_ (s)) h " i C" jv (t) ? u(t)j + jv_ "(t) ? u_ (t)j + C"2t Zt h i +C" ds jv"(s) ? u(s)j + jv_ "(s) ? u_ (s)j "t1 2 C" (1 + t) C"2; by (2.7) for t = O(1). This concludes the proof of Theorem 2.2. 2 Finally we show that on a microscopic time scale our results track the true trajectory with a higher precision, cf. (2.9), Remark 2.3(i). +
Zt
Proposition 5.1 We have jq"(t) ? r(t)j c"3 and jv"(t) ? u(t)j c"2; t = O(");
i.e., (2.9) holds.
Proof : De ne (s) = "?1q"("s) ? "?1r("s); v"("s) ?0u("s); "v_ "(1"s) ? "u_ ("s)
0 1 0 _ s) = A (s) + (s), where A = B for s t1 . Then ( @ 0 0 1 CA and (s) = 0 0 0 2 " " (0; 0; v ("s) ? u("s)), whence j(s)j c" jq"("s) ? r("s)j + jv"("s) ? u("s)j + jv_ "("s) ? u_ ("s)j + "2 c(j (s)j + "3); s t1 ; by (5.1), (5.2), and the uniform bounds. Therefore by the variation of constants formula and Gronwall's inequality for s 2 [t1; t1 + ], j (s)j c( )(j (s1)j + "3). Consequently, q"("t1) = r("t1) and v"("t1) = u("t1) yields "?1jq"(t) ? r(t)j + jv"(t) ? u(t)j c( ) "jv_ "("t1) ? u_ ("t1)j + "3 23
for t 2 ["t1; "t1 + " ]. By (5.1) and (5.2), jv_ "("t1) ? u_ ("t1)j c", so that (2.9) follows. 2
6 Adiabatic limit of the elds We prove Theorem 2.4. Let U (t) again denote the fundamental solution of the wave equation in D1;2(IR3) L2 (IR3 ). We set Z (x; t) = ( (x; t); (x; t)) as well as " = (0; "). Then the inhomogeneous wave equation in (1.5) is solved as
Zt p Z (x; t) = [U (t)Z (; 0)](x) ? " ds [U (t ? s) ( ? q"(s))](x): 0
"
Since
p Z0 Z (x; 0) = ? " ds [U (?s)"( ? q0 ? v0s)](x); ?1 cf. Lemma 3.1, we have for t > 0 p Z t ds [U (t ? s) ( ? q"(s))](x); Z (x; t) = ? " " ?1
where we extended the position to negative times t 0 by q"(t) = q0 + v0 t. Thus by the solution formula for the wave equation Z 3 p1" (x; t) = ? 4jdx ?y yj "(y ? q"(t ? jx ? yj)) (6.1) and (x; t) = _ (x; t). For " ! 0, q"(t) ! r(t), cf. (1.9), with r(t) extended to negative times by r(t) = q0 + v0 t. Moreover, "(x) = "?3("?1x) ! e0 in the sense of distributions. Hence the transformation z = y ? q"(t ? jx ? yj), det(dy=dz) = [1 ? v"(t ? jx ? yj) (x ? y)=jx ? yj]?1, in (6.1) yields the pointwise convergence (2.10), except on the worldline of the particle, since the integrand in (6.1) is singular at y = x, i.e. for x = r(t) which corresponds to tret = t. The analogous argument works for (x; t). In the limit " ! 0, is discontinuous at the light cone fx : jxj = tg, which we avoided due to our assumption. 2 24
7 Radiated energy Let ER;q" (t) (t + R) be the energy, particle plus eld, at time t + R in a ball of radius R centered at q"(t). For R > "R this energy changes as
d E " (t + R) dt R;q (t) i h d H (t = 0) ? 1 Z 3 2 2 = dt d x j ( x; t + R ) j + jr ( x; t + R ) j mac 2 fjx?q"(t)j>Rg Z = R2 d2! (q"(t) + R!; t + R) ! r (q"(t) + R!; t + R) j!j=1 2 Z h + R2 d! (! v"(t)) j(q"(t) + R!; t + R)j2 j!j=1 i (7.1) +jr (q"(t) + R!; t + R)j2 ; where we used that the total energy is conserved. ER changes because there is energy owing back and forth between particle and eld, and because energy is lost irreversibly to in nity. To separate both contributions we take the limit R ! 1. Using (6.1) and the relation t + R ? jq"(t) + R! ? yj = t + ! (y ? q"(t)) + O(1=R) for bounded jyj, we arrive at d E " (t + R) I "(t) = Rlim !1 dt R;q (t) Z Z = ?"(4)?2 d2! (1 ? ! v"(t)) d3y "(y ? q"(t + ! [y ? q"(t)])) j!j=1 2 " " (1 ?!! v_ v("t(+t +!! [y[y??q q("t()])t)]))2 cf. [7, Sec. 3] for details on a similar calculation. In fact, there the ball of radius R was centered at the origin and the second summand in (7.1) is absent. To let " ! 0, we again transform to z = y ? q"(t + ! [y ? q"(t)]), det(dy=dz) = [1 ? ! v"(t + ! [y ? q"(t)])]?1, use "(x) ! e0 in the sense of distributions, and insert the identity y = q"(t) for z = 0 to obtain Z ? 1 " 2 ? 2 lim " I (t) = ?e (4) d2! (1 ? ! u(t))?5 (! u_ (t))2 "!0 j!j=1 = ?(e2 =12) [6 8(u(t) u_ (t))2 + 6 u_ (t)2 ]; 25
in agreement with (4.2). Alternatively, we could rst take the limit " ! 0 in (7.1). Using Theorem 2.4 we nd, with ( ; ) denoting the limit elds from (2.10), (2.11), ?1 d E " (t + R) IR (t) = "lim " !0Z dt R;q (t) = R2 d2 ! (r(t) + R!; t + R) ! r (r(t) + R!; t + R) j!j=1 2 Z h + R2 d2! (! u(t)) j(r(t) + R!; t + R)j2 j!j=1 i +jr (r(t) + R!; t + R)j2 : Since both and r have one term proportional to R?1 and other contributions of order R?2, in the limit R ! 1 only the product of the two leading terms survives and it follows that Z 2 ? 2 lim I (t) = ?e (4) d2! (1 ? ! u(t))?5 (! u_ (t))2; R!1 R j!j=1
as before. We note that the radiated energy is of order " and it therefore suces to use the e ective dynamics to order one, i.e., ignoring the radiation reaction.
8 Appendix: Proof of Lemma 2.1 In this appendix we prove Lemma 2.1. Since we need to use some identities from [5], we switch back to the original time scale of (1.1). Hence we have to show Lemma 2.1 For solutions of (1.1) with initial values satisfying (1.4), i.e., starting on the soliton manifold, and for jjL2 suciently small we have sup j :::v (t)j C"3 : t2IR
The constant C and the bound on jjL2 depend only on the data.
Proof : From [5, Lemma 2.2 and Prop. 4.1] we already know the bounds supt2IR jv(t)j v < 1 ; supt2IR jv_ (t)j + supt2IR jp_(t)j C" and supt2IR jv(t)j + supt2IR jp(t)j C"2; (8.1) 26
for jjL2 suciently small. The constants v and C appearing in (8.1) do not depend on the particular solution, but only on bounds for the initial values. Denote ! ! ' ( x; t ) ( x; t ) ? v (t) (x ? q (t)) Z (x; t) = (x; t) = (x; t) ? (x ? q(t)) : v(t) Then, cf. [5], with L(t) = r v(t) + _ ,
Z
p(t) = ?" r V ("q(t)) v(t) + d3x (L(t)')(x + q(t); t)r(x) =: ?"2 r2V ("q(t)) v(t) + M (t): Therefore ::: p (t) = ?"2r2 V ("q(t)) v_ (t) ? "3r3 V ("q(t))(v(t); v(t)) + M_ (t) : (8.2) Below we will show 2
2
Lemma 8.1 The estimate
jM_ (t)j C "3 + jjL2
Z t j :::v (s)j ds 0 1 + (t ? s)2
holds. Then according to (8.2), (8.1), and assumption (U ) on the potential,
j :::p (t)j C "3 + jjL2
Z t j :::v (s)j ds : 0 1 + (t ? s)2
(8.3)
Since
j :::v j
d p ::: d2 p 3 p d p p p p = dp 1 + p2 +3 dp2 1 + p2 (p;_ p) + dp3 1 + p2 (p;_ p;_ p_) C (j :::p j + "3); the claim of Lemma 2.1 obtains from (8.3) by taking jjL2 small enough. 2 Thus it remains to give the 27
Proof of Lemma Z D 8.1 : First note E M_ (t) = d3x (L(t)Z (; t))(x); r (x ? q(t)) IR2 ; (x) = ((x); 0); where L(t)Z = rZ v_ (t) + (r2Z )(v(t); v(t)) + 2rZ_ v(t) + Z. Because Z_ = AZ ? B , with ! r v v (t) (x ? q (t)) v_ (t) A( ; ) = (; ) and B (x; t) = r (x ? q(t)) v_ (t) ; (8.4) v v(t) we obtain d (L(t)Z ) = A(L(t)Z ) ? L(t)B + 2 [(r2 Z )(v; v_ ) + rZ_ v_ ] + rZ v : dt Let U (t) again denote the group generated by the free wave equation on D1;2(IR3) L2 (IR3 ). Then D E M_ (t) = U (t)[L(0)Z (; 0)]; r ( ? q(t)) L2 (IR2 )
Zt D E + ds ? U (t ? s)[L(s)B (; s)]; r( ? q(t)) L2 (IR2 ) 0 D + 2 U (t ? s)[(r2Z (; s))(v(s); v_ (s)) + rZ_ (; s) v_ (s)]; E r ( ? q(t)) L2(IR2) D E + U (t ? s)[rZ (; s) v(s)]; r ( ? q(t)) L2 (IR2 ) =: T0 + T1 + T2 + T3 : We estimate each term Tj separately, keeping all parts which do contain only initial values. Note that here according to (1.4) we have Z (x; 0) = 0, so all these terms vanish. Nevertheless, we wanted to derive the general form of the estimate; see Remark 2.3(iii). Estimate of T3 : Since Z_ = AZ ? B , we nd
Z (t) = U (t)Z (0) ?
Zt 0
ds U (t ? s)B (; s);
and hence Z E t D ds U (t)[rZ (; 0) v(s)]; r ( ? q(t)) L2 (IR2 ) T3 =
(8.5)
Zs D E ? 0 ds 0 d U (t ? )[rB (; ) v(s)]; r ( ? q(t)) L2 (IR2 ) =: T3;0 + T3;1 : 0
Zt
28
Then Lemma 8.2 below and (8.1) imply through integration by parts in the d3x-integral, Zt Zs T3;1 C"2 ds d 1 + (t"? )3 C"3 : 0 0 Estimate of T0 : This term is determined solely through the data. Estimate of T1 : If we calculate the form of L(t)B = rB v_ +(r2B )(v; v)+ 2rB_ v + B explicitly ! from (8.4), fortunately many terms cancel, and we nd with v = v , v
L(t)B = rv v (x ? q) :::v +3r2v v (x ? q)(v;_ v) + r3v v (x ? q)(v;_ v;_ v_ ) : Now we may argue analogously to the estimate of T3 and Lemma 8.2 to obtain with ! ~(x) = [U (t ? s)L(s)B (; s)](x) ; ~(x) the estimate
jr ~(x + q(t))j 1 + (Ct ? s)2 ("3 + j :::v (s)j) ; jxj R ; t s : (8.6)
Here we have used (8.1) and some of the estimates
jrrv v (x)j + jrr2v v (x)j + jrr3v v (x)j jr2rv v (x)j + jr2r2v v (x)j + jr2r3v v (x)j jr3rv v (x)j + jr3r2v v (x)j + jr3r3v v (x)j jr4rv v (x)j + jr4r2v v (x)j + jr4r3v v (x)j jrrv v (x)j + jrr2v v (x)j + jrr3v v (x)j jr2rv v (x)j + jr2r2v v (x)j + jr2r3v v (x)j jr3rv v (x)j + jr3r2v v (x)j + jr3r3v v (x)j for x 2 IR3 and jvj v. From (8.6) we conclude T1 = ?
Zt 0
ds
Z
jxjR
d x r ~(x+q(t))(x) C "3+jjL2 3
29
C (1 + jxj)?2 ; C (1 + jxj)?3 ; C (1 + jxj)?4 ; C (1 + jxj)?5 ; C (1 + jxj)?3 ; C (1 + jxj)?4 ; C (1 + jxj)?5 ; (8.7)
Z t j :::v (s)j ds : 0 1 + (t ? s)2
Estimate of T2 : Let P (t)Z = r2Z (; t)v(t) + rZ_ (; t). Then d (P (t)Z ) = P (t)Z_ + (r2Z )v_ = A(P (t)Z ) ? P (t)B + (r2 Z )v_ : dt Therefore by de nition of T2,
Zt D E T2 = 2 ds U (t)[(P (0)Z (; 0)) v_ (s)]; r ( ? q(t)) L2(IR2) 0 Zt Zs D h i +2 ds d U (t ? ) ? P ( )B (; ) + (r2Z (; ))v_ ( ) v_ (s); 0 0 E r( ? q(t)) L2 (IR2 ) =: T2;0 + T2;1 + T2;2 : To estimate T2;1 , observe P (t)B = rrv v (x ? q) v + rr2v v (x ? q)(v;_ v_ ) : Hence we may argue as before to nd jT2;1 j C"3. In order to bound T2;2 , similarly to the estimate of T3 we again use (8.5) to get T2;2 Zt Zs D E = 2 ds d U (t)[r2 Z (0)(v_ ( ); v_ (s))]; r ( ? q(t)) L2(IR2) 0 Z t 0Z s Z D ?2 0 ds 0 d 0 d U (t ? )[r2 B (; )(v_ ( ); v_ (s))]; E r ( ? q(t)) L2(IR2) =: T2;2;0 + T2;2;1 : By (8.7) and the argument of Lemma 8.2 then Zt Zs Z 3 3 T2;2;1 ds d d 1 + (C" t ? )4 C" : 0 0 0 Summarizing all above estimates for T0 {T3, we hence arrive at Z t j :::v (s)j 3 _ jM (t)j C " + jjL2 0 1 + (t ? s)2 ds 30
D
+ U (t)[L(0)Z (; 0)]; r ( ? q(t))
E
2
2
L (IR ) Zt D E + 2 ds U (t)[(P (0)Z (; 0)) v_ (s)]; r( ? q(t)) L2 (IR2 ) Z0t Z s D E + 2 ds d U (t)[r2 Z (0)(v_ ( ); v_ (s))]; r ( ? q(t)) L2 (IR2 ) Z t0 D 0 E + ds U (t)[rZ (; 0) v(s)]; r ( ? q(t)) L2 (IR2 ) : (8.8) 0
Concerning the terms that contain data, these vanish here since Z (x; 0) = 0 as a consequence of (1.4). This completes the proof of Lemma 8.1. 2 In case of solutions starting not on, but close, to the soliton manifold as discussed in Remark 2.3(iii), conditions on the data have to be imposed to ensure the last four terms in (8.8) can also be estimated by C"3. In [5, Thm. 2.6] and Section 4 of that paper details are carried out for derivatives of one order less. Above we used the following lemma. Lemma 8.2 The estimate kr[U (t ? )rB (; )]( + q(t))kR C 1 + (t"? )3 ; t ; (8.9)
holds.
Proof : Such estimates have already been used in [5], but we nevertheless
include some details of the argument. Let ! ~(x) = [U (t ? )rB (; )](x) ~(x) for xed t; . By Kirchho 's formula for the solution to the wave equation and by (8.4), r ~(x + q(t)) Z h 1 2 = d y (t ? )r2 rv v( ) (y ? q( )) v_ ( ) 4(t ? )2 jy?x?q(t)j=(t? ) +r2 rv v( ) (y ? q( )) v_ ( ) i +r3 rv v( ) (y ? q( ))(v_ ( ); y ? x ? q(t)) : (8.10) 31
Now jxj R and jy ? x ? q(t)j = (t ? ) yields jy ? q( )j (t ? ) ? v(t ? ) ? R = (1 ? v)(t ? ) ? R by (8.1). As a consequence of (8.7), hence (8.9) follows from (8.10). 2
Acknowledgement: We thank A. Komech for useful discussions.
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[10] F. Rohrlich: Classical charged particles. 2nd edition, Addison-Wesley, Reading, MA, 1990. [11] K. Sakamoto: Invariant manifolds in singular perturbation problems for ordinary di erential equations. Proc. Roy. Soc. Edinburgh Sect. A 116, (1990) 45-78. [12] A. Soffer & M. Weinstein: Time dependent resonance theory. Geom. Funct. Anal. 8, (1998) 1086-1128. [13] A. Soffer & M. Weinstein: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, (1999) 9-74. [14] H. Spohn: Runaway charged particles and center manifolds. Preprint 1998. [15] W. Thirring: A course in mathematical physics, vol. 2: classical eld theory. Springer, New York-Wien 1978. [16] A.D. Yaghjian: Relativistic dynamics of a charged sphere. Lecture Notes in Physics m 11, Springer, Berlin-New York 1992.
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