Adiabatic invariance III: The equation x = −Vx (x, ω)

ANNALS

OF

PHYSICS:

1-12

29,

Adiabatic

(1964)

Invariance

III: The

Equation

x = -V,

(x,

W)

J. E. LITTLEROOD Trinity

College,

Cambridge,

England

In the equation of the title w = w(t) is a slowly --m < t < a. If w is replaced by a constant equal w, at t = 7, the modified equation has an integral y&i” and for periodic

+

V(z,

w,)

varying to the

function momentary

= c,

appropriate I’ (see Fig. 1 below), and CI > 0, there motion between z = X’ and S, of period p=&

Associated

with

this

J(T)

=

x (V(X,w,) s X’

there

-

is an “Adiabatic

zi dx = 242

of t for value

I’(.$, w,)ln2

is a “momentary”

dl.

Invariant”

x (V(S,

w,)

-

V(&

w,) 1-1’2 d[.

s X’

+

It is shown that J(t) is constant in - CC < t < a measure of the slowness of variation of w.

cc to error

O(&),

where

c is

1. Our subject is the well-known equation 2 = -Vz(x, where V, = dV/ax, and w varies slowly with wr = W(T) , we have the modified equation ii = -Vz(t,

wr),

w), t

( 1:I

in - 00 <

E(T) = +I>

i(T)

t

< a. If w is fixed at = j.(T),

(2)

which has the integral E, = %I” + V(E, w,) = c,

E, being the “energy.” With appropriate V, and assuming positive energy E, at t

= 7, there are values [ = X,‘, X, (see Fig. 1) for which $ vanishes; we have f$‘g = V(X,‘,

cd,) - V(& co,) = V(X,

and there is a periodic motion, of period

) w,) -

I’((, w,);

i 3 1)

2

LITTLEWOOD

0

x:

X1-

X,

X

FIG. 1 in which ,$moves forwards and backwards between X,’ and X, , each such move occupying a time $5~ Associated with this (‘momentary periodic” motion at time 7 there is an “Adiabatic Invariant”

J(T)= I$gdt = 2~9 j-1: (V(X7,~7)- V(t,w>}l” dt.

(5)

We wish to prove that J(t) is approximately constant in - 00 < t < 00. We are aware that the problem is complicated by the possibility of (‘escape over a hump” (e.g., over ill, in Fig. 1)) and to deal with this we must cover much preliminary work. Figure 1. is concerned with the state of things at an arbitrary timet = r, and so varies slowly with r. The abscissais x and the ordinate E, . x, , i:, are the values at t = Tforthesolution (x, i) of (1) withx = x0, 5 = koat t = O;E,isthe energy (6) where .$ris the t(~) of (2). The curve I? is the graph of V(x, w,) (it will be apparent that E, = V(X,‘, w,) = V(X, , w,). We make hypotheses (HI), (H,), (H3). In these c’s are positive constants, in general not the same from one occurrence to the next. (For example, we might write c < [ V, 1 < c. Distinguishing c’s would involve a confusing number of suffixes, and our convention leaves us free to use suffixes for other purposes.)

(HI), Vz., Vu, Vzz, Vu,, V,, are, for all x, t, continuous and bounded by c’s. After (HI) we have V,, = V,, . Finally, we shall be concerned only with a range of x of length c at most, and we may then revise (H,) to hold for this range of x only (and all t) .

ADIABATIC

INVARIASCE

:3

III

The appropriate hypotheses about the slow variation of w are, in terms of a small 6: (H,), (i) ti (and &) is continuous for ~1.1.t. (ii) ( 6~1 < CC( - m < t < w). (iii) JY, 1w 1dt < CE. (v) J”cc / 6 1 dt < y, where y is a su.iently small constant depending on the c’s in (H,), (H,), ( H, ). It is condition (iv), with y a small enough constant depending only on the c’s occurring in our hypotheses, that is needed to secure “no escape over a hump.” It will appear (as the reader will easily believe) that it has the consequence that r varies, over - 3~ < 7 < CC, only to an extent measured by y. Before going on, we note a consequence of ( HR) ( iii) and ( iv j, which we record as

Let ~12,~= J:” ( ti j dt. Since 6~is continuous 1ti(fZjj for some tl, tz of (71, 7~ + 1). Hence

are have M, =

j ~j(t~)l , M,, =

and the result of the Lemma follows. The next set of hypotheses concerns the horizontal lines I1 , or AH, . . , l4 , ot DC; let them have ordinates hl , hz , h3 , h4 . We suppose now: (H3), (i) h1 - hz = h, - h0 = c (this may be thought of as small) ; h, - h4 = c. (ii) In the jigure for 7 = 0 the point (x0 , I?,) lies half-way’ between 1, and I4 vertically, and lies between the arcs FK and HG of J? in the wide sense. The heights h, ... , hq are accordingly constant, and 1, , . . . , 1, are jxed horizontal lines in the (z, E) space. We suppose, further, in the figure for 7 = 0: (iii) The area ABCD contains no point of r other than those of fhe arcs =1D, (‘B; (iv) - V,(x, wo ) lies between c’s on arc AD; Vz(x, woj lies between c’s on arc C’B; (v) c < length DC < c. ’ We might

equally

well

take

it to be between

Ii0

and

!)/io of the way.

4

LITTLEWOOD

2. We now need LEMMA 2. Subject to (HI), (Hz), ( H3), and provided that y of (Hz) is a au& ciently small c, depending on the c’s so far mentioned, we have the following state of things in jig. 1 for an arbitrary time r of - ~0 < r < 00 : (i) The area FGHK contains no point of r other than the boundary arcs FD, HG. (ii) The point (z, , E7) lies in the region @ consisting of the closed area FGHK less FG and HK (so hq < h, = E, < hz). (iii) There are new constants c, depending on the c’s sofar mentioned (now including -y), such that c < -Vz(x, w,) on arc FK, c < V%(x, CO,) < c on arc HG; and c < length KH < length FG < c. It should be noted that Fig. 1 may become modified for some r to the extent that, e.g., Aa2 may sink below II , but not below 12; Ml may rise above l3 , but not above l4 . It follows

from

(H1,&

that, for a fixed x,

From this it is intuitive that for small enough y the results (i) and (iii) of the lemma will hold. The proofs would be easy in detail, but awkward and tedious to read. We will omit them, observing only that for a fixed V of the range hh 5 V 6 hz there will be an inverse function x(V) satisfying &x/aV = l/V2 , which likewise (since 1 V, I > c) differs from its value for 7 = 0 by an amount small with y. 3. In what follows c’s will first of all depend only on c’s already mentioned, then on the new ones; generally, each new c depends only on previous ones. We use suffixes to identify particular c’s in an argument. This use is temporary, and we restart suffixes at 1 in each new context. The constants of O’s will be of type c. We have still to prove Lemma 2 (ii). Consider now the solution (xt , &) of (1) with x = 20 , li = 20 at time t = 0. At t = 0, (x0 , E,) is in a, and in particular hh < E, < ht . Let T be the greatest value of T (possibly and in actuality co) for which E, is in the open range h, < E, < hz for the open range -T < t < T, or R, say. For 7 of R the horizontal line 1, through (z, , E,) meets I? in (X,‘, E,), (X, , E,), say (X,’ < X,) , and we have g%” + V(x, ) w,) = VW’ Abbreviate Vw(x, CO)to W(x, w).

) a,) = V(X,

, a,).

(8)

ADIABATIC

INVARIANCE

.)

III

Let 2, + 6x,, etc., be values at t = 7 + 87. In using the 6 notation we will ignore squares of 6’s; equations L = R between 6’s are to be interpreted as “lim L/ST = lim R/ST as 6~ + 0.” For small enough 87, 7 + 6~ is in R, since R is open, and similarly E, + 6E, lies strictly between h, and hz . From (8 j, ,?k(C + 6a$ = V(X,’

+ 6X,‘, wr + ScLb)- T’(x, + 8x, ) (3, + f&d,)

= B(X,

+ 6X, ) Wr + sbJ,>- V(x:, + 6x:, ) Wr + 6w,).

(9’)

Also 68, = ~‘,ch = -Vz(xr

) cd,)87 )

6x, = .i$r.

(10)

K-on1 (9), (lo), and (l), - irVs(.L’, ) w,)67 = 3&&i&= +i(&

+ 6i,)” - ,&”

+ bW(X,

= 6X,V,(x:,

) w,)

) w,j - 8Z,TT,(X, ) w,) - 6w,W(x, ) cd,‘).

The term on the extreme left cancels with the third term on the extreme right, by virtue of (10) ; hence gx,vz(x,

) w,) = -Gw,{W(X, x7 = -k(W(X,

) w,j - W(z, ) cd,)}, ) 0,) - W(x, ) W,)}/VJXT,

c&j,

(11)

and similarly with X,’ for X, . 4. Equation ( 11) is a key result (when, as it will be apparent, it is shown to be valid for all r). We use it first, however, to show that, for ‘Tof R, 2, satisfies X,’ I zr 5 X, (recall that X,‘, X, are defined by the intersections of 1, with I’, and that x, lies between them throughout R needs to be proved). Since / F’, j > c in (11) we have X, = O(t) , and, differentiating again, we see easily that (since ( V, ( > c) 2, = O(E). Similarly for X,‘. So for 7 of R 2, ) 2, ) ,“ir,‘, .zT’ = O( E). Hence (for 7 of

(12j

R)

6(& - 2,) = 67(% - 8,) = hi-V,(X,)

co,) + O(E)},

by (1). The curly bracket is negative, by Lemma 1 (iii), when x, is near X, , and it follows that for increasing 7 (of R), say, 2, , being initially 5 X, at 7 = 9, can never cross X, . Similarly for X,’ and for decreasing 7. We infer that X,’ 5 X, 5 XT(7 E Rj. Next, since Xl, = O(ti,) by (ll),

X, can vary, for 7 of

(13) R, by at most

6

LITTLEWOOD

and so, since 1 V, 1 < c on FK and HG, h, = E, can vary at most by a cy. Since h, starts at 7 = 0 half-way between 1, and 1, , it can never reach either of these for 7 of R if y is a suitably small c. This fact combined with (13) is equivalent to (x, , E,) belonging to CR. Now no condition on T has occurred in this argument: it follows that T = ~0, and that for all 7, (2, , E,) belongs to CR. This completes the proof of Lemma 1 (ii) and so of the whole lemma. We have further established ( 11) for all 7. 6. We have now completed the rather vicious circles and establish Lemma 2.

elaborate

argument

needed to avoid

We have from (5)

Since the integrand tends to 0 with 6’s when 6X,’ and 6X, in the limits of integration. So t VW7

+ 87) = L;:

[{V(X,

, 0,) -

t = X,’ or X, we may suppress

vet, @,I } + 6X,Vz(X,,

+ b{W(X, or, substituting a &UT

UT)

, w,> - Wk a,>)11’2dC;,

from (ll),

+ 6s) = J-1: [(V(X)

UT) -

V(E, %I)

+ S%(W(X,)

a,> - WC.5 ~1 111” dE,

so that

((VPL, ~7) - V(f, w>I-““{W(z, ,w,) - Wkw)} dt(14) = U(x7,w>= U(T), say. (Note that (for a given function V) X, , X,‘, and therefore of z, and W, only, and so ultimately of 7, x0 , and & .) 6. Our aim is now to prove that U(t)

li, are functions

satisfies

+P,

s7

U(t) dt = O(:‘*)

for all 7. From this we shall find it straightforward 0( 2”) for all 1.

to deduce that J(t)

= C+

ADIABhTIC

INVARIhNCE

III

We have, from (4) and (14), 1;(T) = - UIiT) + U,(T), UIiT) = lx;: (ViX,

UT) - Vi& w,) pm,

%I 4,

( 1.5i

U?(T) = (J 5 +1 p,)W(zG ) 0,). We need two further lemmas. LmfnfA4 3. Let

Th,en

c < p, < c. LIGMMS 4. Let r1 = r + p, . In accordancewith fh.erlotation (2), let $ = Er be the solution 14

g = -Vz(&

w,),

with initial conditions lr = x, , g7 = k, at t = 7. Then jar 7 j t 5 q we have (i) lt - .rt = O(C); (ii) If’(sl, Ot) = TV(&) w,) + O(t) = W(t;, , w,) + O(E) = lV(‘$t, w,) + W,); (iii) Cl(t) - t1(7) = O(E1!2); (iv) p, - p, = O(2”). '7. PROOF OF LEMMA 3. Since 1V,, 1 < c, the arcs FK, GH of I’ can be extended to FK, and FK2 , where K,H, is a horizontal line a small enough distance c below KH, and such that I’, lies between values -c on FK, , and between values c on GH, . If we denote the abscissaeof AT,‘, F, . . . for short by S,‘, F, . . . we have

Kl - K > c,

H - HI > c,

HI - r;, > c,

FIG. 2

G-F<<.

8

LITTLEWOOD

We now distinguish the following cases: (i) [ lies between X,’ and K, or between H1 and Xt ; (ii) 5 lies between K, and H1 . In the second alternative of (i) , say, we have c(Xt - r> s D s 4x-t

- 0,

N = (Xt - Wdf’,

4,

where .$ lies between HI and Xt , so that c < V, < c. Then c < Q < c as desired, and the first alternative of (i) is similar. In case (ii) ,$- Xt’, Xt - E, and N each lie between c’s, and so therefore does Q. This is the first part of Lemma 3, and for the second we have from the first

c< pt/s x-;((t- Xt'>(Xt - ‘$1 p 4 < c, and so c < p, < c. 8. PROOF OF LEMMA

Et = -vz(tt,

4. For (i) we have 2 = -V2(xt,wt

071,

since V,, = O(1) and mt - w, = O(l)&(t’) 22 = -Vz(Zt

) = -Vz(zt

) w,> + O(e),

= O(E). If u = .$ - zt we have

+ u, w,) + Vz(zt , w,) + O(E) = O(u) + O(E)

since V,, = O(1). Hence Iti1

~clvu~

+czc)

with initial conditions ~(7) = ti( T) = 0. It follows by a familiar majorant argument that ] u 1 5 v, where ti = c12(y+ c~c), ti( 7) = v( 7) = 0. Since v = c&osh (q(t - T)) - l] for t of (7, TV), this gives j u ] < CEand hence (i) of Lemma 3. Next, since W, and W, are 0 ( 1) , W(xt,

CL?>- W(xt,

0,) = O(w, - W,)‘Ww(xt,

WY = O(E)O(l)

= O(e),

and similarly WC&, 4

- W(b)

w> = O(e)

Further W(xt , Wt) - W(ft,

at) = O(xt - ‘$t)W&‘,

a> = O(E).

These results establish part (ii). Next, Ul(t) is given by the second formula (15) for U1(T), with t written

ADIABATIC

INVARIANCE

formula we may replace W( f, at) by W( 4, wT j,

throughout for T. In the resulting with an error which, by (ii), is O(E) /xy

, at) - V(c;, w)}-~‘*

{V(Xt

We can now treat (iii) and (iv) together; u*w

= Ly:

9

III

tvc&

d.$ = O(e)pt

each of Ul(t)

, w> -

Vk

4

= O(E).

and pt is of the form

l-“2FW

df

where F([j is continuous and 0( 1) in X,’ 5 t 5 Xt , and the function F (though not the range of its [) is independent of t. Thus r.l( t) is U*(t) with F( 5) = W({, w7), and p, is U*(t) with F(t) = d2 (and no error term O(t)). To prove (iii) and (iv) it is sufficient to prove that U;*(i)

-

u*(T)

= O(T2).

Let k be a c large enough to satisfy a certain reyuirement also to satisfy k >

1Xt - XT I ,I x,’

which

is possible by (6). A little consideration

U*(t)

-

U*(T)

below, and, initially,

- x7’ I ) now shows that

= T1 + T2 + Ta ,

T1 = s,“:l”,: -- k,

I( V(X,

, at) - V(& at) )-1’2 -

w 1 IV’(E)

T, = l,

(V(Xt

Ta = -

I-R. iV(XT,

1V(Xv , %I

dE,

, wt 1 w,) -

(16)

VLC, w> I-“‘F(E)

d.5

V(& w,) I-““F(t)

d$,

where R, , R2 are each a pair of intervals of length O(k). Since Q = O(1) and F(t) = O(1) we have Tz = O(1) 1

RI

{ (6 - X,‘)(X,

The worst case is clearly that in which and we have

- 0 )-“’

the intervals

d&

of RI abut on Xt and X1’,

T, = 0( 1). [( Xt - ~)1’2]~Z~:-ckE + (a similar term).

10

LITTLEWOOD

SO

Tz = 0((k~)1’2},

T3 = O( (k~)~‘~},

(17)

T3 being similar to T, . The integrand in T, is of the form [IV(XT

I w)

-

V(L w,) + 8clp2

- { V(X,

for some ~1 and 1 d 1 5 1. Now namely, (X,’ + kc, X, - Ice), V(XT,

a)

-

vcr,

) w,) - V(& wpz]

in the

range

a (6)

of integration

w> 2 c([ - X,‘)(X,

(18) of Tl ;

- I$),

by Lemma 3, 1 c2kc. We choose k to be the greater of 2cI/cz and the value needed for the earlier requirement. Then I 8ClE

I/iV(XT,

w>

-

vet,

a>}

6

35,

and so (18) is of the form O(E){V(XT)

w,) -

V(S, w,)}-3’z

= O(e>l([

- X,‘>(X,

- ‘$))-3’2,

and finally TI =

O(E) S,“=;:

{ ([ - X,‘>(X,

-

.$) )-”

ds$ = O(:‘2).

This, combined with (li), completes the proof that U*(t) and hence the proof of Lemma 4. 9. From 1” T

U(t) =-

(15) and Lemma 4 (ii), dt = - 1” U,(t) 7

1” T

(U,(T)

- p, Ul(7)

= O(:“)

+ PA- Ul(7)

Now

in the last integral 3&‘/z

(iv),

dt + /” Uz(t) dt T

+ O(t1’2)}

= O(?)

(iii),

- U*( 7) = O( 2’2),

dt + 1” (;442p, T

+ %‘2pr + !iv’2

1” {W(Et, 7 s,” W(&,

Et is periodic,

+ O(c”2)JW(x~, ~7) + O(c)) w)

dt

4.

(19)

and we have

/-I WC&, w,> dt = 5542 T

jk W(tt,

w) dt

w> Wit

ADIABATIC

INVARIANCE

31

III

ill which

so that

Thus the square bracket

in ( 19) is 0, and

f i ‘I U(t)dt = O(tl’Y 10. By (14) and (20) we have, on integrating J(T1) - ~(7)

= 42 j” U(t)& 7

dt = 42471)

(200)

by parts, jr1 U(t)

dt (21)

-

,,‘a ;’

r

,-{ti

j’

T

U(T)

d+ dt;

also J(t)

- J(T)

= 0( 1) jT’ 161 dt = O(E)(T T

(22)

5 t 5 TV).

Now by Lemma 4, U(t)

= O(1) R’

((5 - X,%X,

-

E))-“2dt

= O(1).

Hence, by (%I),

J(n)

- J(T) = O( P2GJ) } + O( 1) j”

T

/ ij 1 dt.

c,23 j

Let T$ = 71 + p(~~), etc., and let T-~ , T-Z , . . . he such that T = T O = T-I + (these will certainly exist since pi lies between p(T-11, T-1 = 7-2 + p(7-21, ... C’S and is continuous). Given t, let it lie in (~~-1 , T,,). By (22) and (23) J(t)

- J(T)

= O(E”~)

5 l&(~n) j +0(l) n=-cc

= O(2”)

2 13(Tn) / + O(t), n=--m

jm /o/ dt + O(E) --m (24)

by (H?) (iii). Now since T,+~ -

7n > e the number of rn lying in a given interval

m, 111+ 1

12

LITTLEWOOD

is at most c. It follows that iJ

I4Gz)l

s c&w

by Lemma 1. Then from (24)) J(t)

- J(7)

= O(F)

for all t and 7, and so finally J(t) where C is a constant. RECEIVED:

March 6, 1964

= c + O(P2),

< c,