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Adiabatic invariance IV: Note on a new method for Lorentz's pendulum problem
ANNALS
OF
PHYSICS:
Adiabatic
29,
13-18
(1964)
Invariance Lorentz’s
IV: Note Pendulum
on a New Problem
Method
for
J. E. LITTLEWOOD Trinity
College,
England
Cambridge,
The Note sketches an alternative method for the problem, which was discussed in (1). The new method seems likely to be better suited than the old for a discussion of the problem of multiple periods, and it has some formal interest.
1. The problem concerns the differential equation z + w2x = 0,
y = 2,
where w varies slowly over - 00 < t < co. The exact assumptions about W, in terms of a ‘?neasure of slowness” e, and positive constants b, are:
w > bo,
dnw W(n) = -40 dt”
as
1wCn)1 < b, en(n h l),
t-+*co(n
2 1);
m j w(*) 1 dt < !I;-1 en-*.
s-cc
Since then j U( 7’) - w( T>\ 5 1.f:’ I cj ( dt 1-+ 0 as 7, 71--+ 00 or - ~0, we have limw = I as T+ *a. Lorente’s conjecture was that the “Adiabatic Invariant”
J = J(t)
= E/w = x(ci? + w2x2)/w
(where E is the “energy”) is approximately constant over - 00 < t < ot . In my paper (1) ,l referred to as I, it was shown that (i) J = C + O(c) ; (ii) j = C + O(E~); (iii) th ere is no improvement over (i) or (ii); (iv) J(m) - J(-a) = O(cn) f or an arbitrary 12.In (ii) 7 is the average of J over the local period 2,/ w. The constants of O’s depend only on the b’s concerned. 2. There are two proofs in I. The first depends on Chandrasekhar’s transformation CC) 1 Some corrections
dtl = w dt,
21 = w1’2x,
to this are given at the end of the Note. 13
14
LITTLEWOOD
which leads to d2q
2 Wl'Xl
=
0,
fF+ 1
w1*
=
1+
(w-l'*).
511,
Now though we do not have J(h) = w1x12 + wl’y,2 = J(t), we do have lim J(t) = lim J( tl) as t + f 00 (when tl + f 00). Further, (C) can be iterated, and Q,, = O(e2”). The result (i) comes from a formula J(t)
- J(a)
= -kc-“sy
+ R~(w)
+ R;(w),
where R2 contains terms all of weight 2 in differentiations k2 + wti ; and R2’ = 0( 1) J?Wl R,(w) j dt, whereR3 hastermsall Now for t = - cc) (1) becomes
J(- O”) - J( co> = Jn( - ~0> - J,( 00> =
(1) of w, for example, of weight 3 at least.
0( 1) 11 1R3(Wn) j dt,
(2)
which is easily seen to be 0 ( c2nf2) . Thus (iv) follows from (i) by iteration of (C). It is desirable for further developments to have a proof independent of (C), which is then not available. In I a step by step process was given of finding successively better approximations to J(t) - J( m ), of the form
J(t) - J(m 1 = $ R,(w) +
O(1) /-I
1R,+I(w)
1 dt,
where R, contains terms all of weight m in differentiations and R,+l terms all of weight n + 1 at least; and in principle this could be carried out for any given numerical value of n, e.g., n = 8. The general formula (3), however, depends on the existence of an identity in the first n differential coefficients of an arbitrary indefinitely differentiable f( t) ; and it seems very difficult to give a direct proof of this identity, The difficulty is met in I by the use of Liouville’s theorem of constant area. 3. The object of the present Note is to sketch a method of successive approximation based on a new set of differential equations; the new method is a good deal simpler than that of I. In any case, it is reasonable in view of further problems to explore the different possible approaches, and in part,icular the new method seems likely to be more suited than the old to the problem of multiple periods. It has also, perhaps, some formal interest. Let R* = 2E = w2x2 + g2,
J = E/w
= xR2/w.
(4)
ADIABATIC
We suppose (naturally)
INVARIANCE
13
IV
that, at t = 0, R. > 0. The equations
z = c?R cos T,
?J=
-R
T = wt + 8,
sin 7,
define a cont,inuous 6 = 0(t) . The new method is to work variables. We have from (4) 2wch? + 2ylj + 2w”x.i 2R”
2RA 2R2
ri R
WOX2 K”
(5)
with R, 0 as dependent
ti 2 w cos r1
and so
J-zz J E’urther,
;
(
212 - 5 = cj cos &-. log R_” = R w w w>
from (5))
-Rsinr
= y = $ = --w?Rsin~(~
+ tit + 8) + cos~(w-~R
(bt + 4) sin 7 = ( E - 4 ) = - 5 sin” 7 cos 7, W
W
by (61, so that we have, writing’ cjt
Equations
+
4
=
-
i = A, i = A = ~(1 - .1&-‘Lj sin 7).
sin 27,
$$w-lch
18)
(7) and (8) are the formal basis of the ensuing argument.
4. As a corollary
of (6) we have
Hence
R = O(Ro),
1 !/ I = OiR”)
j x I = O(Ro),
Further log J(t>
J(O)
=
t iJ cos 27 (17
Io i
Since J”“, 1G/W 1dt exists, we have - W. - cos 27 d7 + 0 st w 2 To
avoid
denowtinntors
(G)“.
as
t+
m,
i all t 1.
(!J)
16
LITTLEWOOD
so that lim J(t)
= J( 00 ) exists as t -+ 00, and similarly
lim J(t)
= J( - co ) as
- w. J(t) cos 27 dr. log Jo = - st -w 6. Integrating
Straightforward
(10) by parts we have (recalling
calculation
1bsinA2t J(t) logJ(m) = zw
(10)
i = A)
shows this to be of the form
I 1-7, (11)
P = (91*’ sin 27 + fir’ sin 47) + (fd3’ + f$’ sin 27 + Lti3’ cos 27 + ~2:~’cos 47 + Qi3’ cos 67),
where Q2(pmi is a sum of terms each of weight m in differentiation (e.g., O&O + h3 + ~“a), divided by some power of w. The precise forms of the Q’s need not concern us. It follows easily from the hypotheses about w that
- ] d”’ 1dT = 0( em-‘) f-co
(m 2 1).
We have at once S”, 1P/A* ) dr = O(e) , and (11) consequently gives the result (i) . We will ignore (ii) and (iii), which depend on further steps (2 for (iii) ) in the approximation process below. The proofs would be simpler than the corresponding ones in I. Equation (11) gives in particular
J(-m)
log J(m)
= O(E)
andifweiterate(C),andrecallJ(fm) = J,(fa)forJ,= becomes0( :n+2), and this is equivalent to
J(a)
- J(-m)
J(w,),theO(t)
= O(R,$)
for arbitrary 12,which is (iv). 6. If we do not use (C) we must carry out successive approximations, of which (11) is the first, and finally apply Liouville’s theorem of constant area. We may sketch the work as follows, in which a(m) are not in general the same from one occurrence to the next.
ADIABATIC
Since A-’ = c
INVARIANCE
IV
17
fiCrn) sin”’ 7 is of the form (12)
the expression
is also of the form (12). Let us denote by 1 “integrated terms,” such as the first term on the right hand side of (ll), which vanish at t = f ~0. Since J?, / Q2’“’ / dt = 0( Ed) me have from (11)
mQ(P) cosVT dr
st
sin
d r st m sin cm
= x!z
The first term on the right is an I, and the second is easily seen to be of the form (12)) but with all m 2 P + 1. By successive stages (giving increasing m) it is clear that we shall arrive at a result of the form log $$
00
= c
I + c [- QCrn)dt + O(;). m t
(13)
This is closely parallel to the situation at the top of p. 242 of ref. 1, and we find by a parallel application of Liouville’s theorem that in (13) we have JY, QCn’ dt = 0 for each term of cm . Since the I vanish at t = - 00 we accordingly have log
~J(-~1 J(a)
= O(E%),
and so J( 00) -
J( - 00) = 0(2&e”),
as in Section V. I take this opportunity of correcting some mistakes in I. p. 234, line 8, for a, read 8, ; LEMMA 1, line 3 should read 1A j = . . . 5 BH 1~j (; line 4, for j CGdt read 1 ~j 1dt. p. 235, Eq. (3), for E read +; Eq. (4), for & read --h ; Eq. (5), for Q = read
18 -
LITTLEWOOD q1 = ; LEMMA
p. read p. p.
2, the proof should read: t xy dt = -x2 s to 2
[11t to
= O(1).
238, line 11, for ,$I read -& ; line 13, for q1 read -ql ; line 14, for d/dt * * * = -d/dt ..* = ; line 19, for @ read CL. 240, Section 9, line 5, for z = read XI = . 242, line 7 up, for Y’ read VI ; line 8 up, for iken read k,e2”.
RECEIVED:
March
6, 1964 REFERENCE
I. J. E. LITTLEWOOD,
Lorentz’s
pendulum
problem.
Ann.
Phys.
(N.Y.)
21, 233-242
(1963).
OF
PHYSICS:
Adiabatic
29,
13-18
(1964)
Invariance Lorentz’s
IV: Note Pendulum
on a New Problem
Method
for
J. E. LITTLEWOOD Trinity
College,
England
Cambridge,
The Note sketches an alternative method for the problem, which was discussed in (1). The new method seems likely to be better suited than the old for a discussion of the problem of multiple periods, and it has some formal interest.
1. The problem concerns the differential equation z + w2x = 0,
y = 2,
where w varies slowly over - 00 < t < co. The exact assumptions about W, in terms of a ‘?neasure of slowness” e, and positive constants b, are:
w > bo,
dnw W(n) = -40 dt”
as
1wCn)1 < b, en(n h l),
t-+*co(n
2 1);
m j w(*) 1 dt < !I;-1 en-*.
s-cc
Since then j U( 7’) - w( T>\ 5 1.f:’ I cj ( dt 1-+ 0 as 7, 71--+ 00 or - ~0, we have limw = I as T+ *a. Lorente’s conjecture was that the “Adiabatic Invariant”
J = J(t)
= E/w = x(ci? + w2x2)/w
(where E is the “energy”) is approximately constant over - 00 < t < ot . In my paper (1) ,l referred to as I, it was shown that (i) J = C + O(c) ; (ii) j = C + O(E~); (iii) th ere is no improvement over (i) or (ii); (iv) J(m) - J(-a) = O(cn) f or an arbitrary 12.In (ii) 7 is the average of J over the local period 2,/ w. The constants of O’s depend only on the b’s concerned. 2. There are two proofs in I. The first depends on Chandrasekhar’s transformation CC) 1 Some corrections
dtl = w dt,
21 = w1’2x,
to this are given at the end of the Note. 13
14
LITTLEWOOD
which leads to d2q
2 Wl'Xl
=
0,
fF+ 1
w1*
=
1+
(w-l'*).
511,
Now though we do not have J(h) = w1x12 + wl’y,2 = J(t), we do have lim J(t) = lim J( tl) as t + f 00 (when tl + f 00). Further, (C) can be iterated, and Q,, = O(e2”). The result (i) comes from a formula J(t)
- J(a)
= -kc-“sy
+ R~(w)
+ R;(w),
where R2 contains terms all of weight 2 in differentiations k2 + wti ; and R2’ = 0( 1) J?Wl R,(w) j dt, whereR3 hastermsall Now for t = - cc) (1) becomes
J(- O”) - J( co> = Jn( - ~0> - J,( 00> =
(1) of w, for example, of weight 3 at least.
0( 1) 11 1R3(Wn) j dt,
(2)
which is easily seen to be 0 ( c2nf2) . Thus (iv) follows from (i) by iteration of (C). It is desirable for further developments to have a proof independent of (C), which is then not available. In I a step by step process was given of finding successively better approximations to J(t) - J( m ), of the form
J(t) - J(m 1 = $ R,(w) +
O(1) /-I
1R,+I(w)
1 dt,
where R, contains terms all of weight m in differentiations and R,+l terms all of weight n + 1 at least; and in principle this could be carried out for any given numerical value of n, e.g., n = 8. The general formula (3), however, depends on the existence of an identity in the first n differential coefficients of an arbitrary indefinitely differentiable f( t) ; and it seems very difficult to give a direct proof of this identity, The difficulty is met in I by the use of Liouville’s theorem of constant area. 3. The object of the present Note is to sketch a method of successive approximation based on a new set of differential equations; the new method is a good deal simpler than that of I. In any case, it is reasonable in view of further problems to explore the different possible approaches, and in part,icular the new method seems likely to be more suited than the old to the problem of multiple periods. It has also, perhaps, some formal interest. Let R* = 2E = w2x2 + g2,
J = E/w
= xR2/w.
(4)
ADIABATIC
We suppose (naturally)
INVARIANCE
13
IV
that, at t = 0, R. > 0. The equations
z = c?R cos T,
?J=
-R
T = wt + 8,
sin 7,
define a cont,inuous 6 = 0(t) . The new method is to work variables. We have from (4) 2wch? + 2ylj + 2w”x.i 2R”
2RA 2R2
ri R
WOX2 K”
(5)
with R, 0 as dependent
ti 2 w cos r1
and so
J-zz J E’urther,
;
(
212 - 5 = cj cos &-. log R_” = R w w w>
from (5))
-Rsinr
= y = $ = --w?Rsin~(~
+ tit + 8) + cos~(w-~R
(bt + 4) sin 7 = ( E - 4 ) = - 5 sin” 7 cos 7, W
W
by (61, so that we have, writing’ cjt
Equations
+
4
=
-
i = A, i = A = ~(1 - .1&-‘Lj sin 7).
sin 27,
$$w-lch
18)
(7) and (8) are the formal basis of the ensuing argument.
4. As a corollary
of (6) we have
Hence
R = O(Ro),
1 !/ I = OiR”)
j x I = O(Ro),
Further log J(t>
J(O)
=
t iJ cos 27 (17
Io i
Since J”“, 1G/W 1dt exists, we have - W. - cos 27 d7 + 0 st w 2 To
avoid
denowtinntors
(G)“.
as
t+
m,
i all t 1.
(!J)
16
LITTLEWOOD
so that lim J(t)
= J( 00 ) exists as t -+ 00, and similarly
lim J(t)
= J( - co ) as
- w. J(t) cos 27 dr. log Jo = - st -w 6. Integrating
Straightforward
(10) by parts we have (recalling
calculation
1bsinA2t J(t) logJ(m) = zw
(10)
i = A)
shows this to be of the form
I 1-7, (11)
P = (91*’ sin 27 + fir’ sin 47) + (fd3’ + f$’ sin 27 + Lti3’ cos 27 + ~2:~’cos 47 + Qi3’ cos 67),
where Q2(pmi is a sum of terms each of weight m in differentiation (e.g., O&O + h3 + ~“a), divided by some power of w. The precise forms of the Q’s need not concern us. It follows easily from the hypotheses about w that
- ] d”’ 1dT = 0( em-‘) f-co
(m 2 1).
We have at once S”, 1P/A* ) dr = O(e) , and (11) consequently gives the result (i) . We will ignore (ii) and (iii), which depend on further steps (2 for (iii) ) in the approximation process below. The proofs would be simpler than the corresponding ones in I. Equation (11) gives in particular
J(-m)
log J(m)
= O(E)
andifweiterate(C),andrecallJ(fm) = J,(fa)forJ,= becomes0( :n+2), and this is equivalent to
J(a)
- J(-m)
J(w,),theO(t)
= O(R,$)
for arbitrary 12,which is (iv). 6. If we do not use (C) we must carry out successive approximations, of which (11) is the first, and finally apply Liouville’s theorem of constant area. We may sketch the work as follows, in which a(m) are not in general the same from one occurrence to the next.
ADIABATIC
Since A-’ = c
INVARIANCE
IV
17
fiCrn) sin”’ 7 is of the form (12)
the expression
is also of the form (12). Let us denote by 1 “integrated terms,” such as the first term on the right hand side of (ll), which vanish at t = f ~0. Since J?, / Q2’“’ / dt = 0( Ed) me have from (11)
mQ(P) cosVT dr
st
sin
d r st m sin cm
= x!z
The first term on the right is an I, and the second is easily seen to be of the form (12)) but with all m 2 P + 1. By successive stages (giving increasing m) it is clear that we shall arrive at a result of the form log $$
00
= c
I + c [- QCrn)dt + O(;). m t
(13)
This is closely parallel to the situation at the top of p. 242 of ref. 1, and we find by a parallel application of Liouville’s theorem that in (13) we have JY, QCn’ dt = 0 for each term of cm . Since the I vanish at t = - 00 we accordingly have log
~J(-~1 J(a)
= O(E%),
and so J( 00) -
J( - 00) = 0(2&e”),
as in Section V. I take this opportunity of correcting some mistakes in I. p. 234, line 8, for a, read 8, ; LEMMA 1, line 3 should read 1A j = . . . 5 BH 1~j (; line 4, for j CGdt read 1 ~j 1dt. p. 235, Eq. (3), for E read +; Eq. (4), for & read --h ; Eq. (5), for Q = read
18 -
LITTLEWOOD q1 = ; LEMMA
p. read p. p.
2, the proof should read: t xy dt = -x2 s to 2
[11t to
= O(1).
238, line 11, for ,$I read -& ; line 13, for q1 read -ql ; line 14, for d/dt * * * = -d/dt ..* = ; line 19, for @ read CL. 240, Section 9, line 5, for z = read XI = . 242, line 7 up, for Y’ read VI ; line 8 up, for iken read k,e2”.
RECEIVED:
March
6, 1964 REFERENCE
I. J. E. LITTLEWOOD,
Lorentz’s
pendulum
problem.
Ann.
Phys.
(N.Y.)
21, 233-242
(1963).