Volume 68A, number I PHYSICS LETTERS 18 September 1978
NOETHER’S THEOREM AND THE TIME-DEPENDENT HARMONIC OSCILLATOR
M. LUTZKY
10111 QuinbySt., Silver Spring,MD 20901, USA
Received 6 July 1978
Noether’s theorem is used to derive the Lewis invariant for the time-dependent harmonic oscillator. The application of
the method to non-linear dynamical systems is discussed.
It has recently been shown by Lewis [1] that a ated by
conserved quantity for the time-dependent harmonic a a
oscillator is given by G = ~(x, t) ~- + n(x, t) ~—
leave the action integral f L(x, ~, t) dt invariant. Then
I ~[(x2/p~) ÷ ~~x)2]
—
where x satisfies the harmonic oscillator equation aL aL . aL
x+w2(t)x=0, (2) +n~+(ñ—~)-~+~L=f, (5)
and p is any solution of the auxiliary equation where f= f(x, t), and
/~+w2(t)p~—l/p3 . (3) ~ n=~+~.~! [÷~~L
- at ax ‘ at ax ‘ at ax
This was accomplished by demonstrating that a pre-
viously known adiabatic invariant [2] was in fact an Furthermore, a constant of the motion for the system
exact invariant. Subsequently Leach [3] , in the course is given by
of treating the more general equation ai.
i~+g(t)i +w2(t)x =0, (4) ~ib ‘(tx —n)-~ ~L +f. — (6)
found the Lewis invariant by utilizing a time-dependent The lagrangian L = ~(~2 w2x2) leads to the equa-
—
canonical transformation which reduced the hamil- tion of motion (2); using this lagrangian in (5), and
tonian to a constant. This transformation was a mem- equating coefficients of powers of i to zero, we ob-
ber of the symplectic group, and was deduced by a tam a set of equations for ~, n, f which can easily be
somewhat involved procedure. We demonstrate here shown to imply that ~ is a function of t alone, and
that the invariant is obtainable by a straightforward satisfies
application of Noether’s theorem [4], which relates .
-
the conserved quantities of a lagrangian system to the ~+4~ww+4~ ~0. (7)
symmetry groups which leave the action invariant. We In addition, we obtain the results
also discuss the relation between solutions of the
- . - ..
equation of motion and solutions of the auxthary
equation, and briefly consider the potentialities of ~-~- -~— ~ 2 ,-, T 2 —
- . - j~X,tj
4cX +~,X+t~, ~+W —
the method for the solution of non-linear equations.
We use the following formulation of Noether’s Choosing C = 0, ‘P = 0, and substituting these values
theorem [5]:let the one-parameter Lie group gener- in (6), we find that
3
Volume 68A, number 1 PHYSICS LETTERS 18 September 1978
2 + ~] x2 — ~x1) (8) where W = 1
= 1(~2 + [~w
1x2 x112, and f~and ‘2 are constants
—
Thus, a general solution of (3) is known if two in-
is a conserved quantity for (2) if ~ satisfies (7). Note dependent solutions of (2) can be found. (Since the
that (7) has the2w2first
= C
integral wronskian W is constant for independent solutions
i~2+ 2~ 1 (9) of(2), we may set I~= 1,12 = so that (13) be-
2 in (8) and (9), and let C comes p = \/~+(i/J4/2)x~ this result was first given,
if we set ~ = p 1 = 1, we ob- without a derivation, by Pinney [61). Furthermore,
tam the Lewis invariant (~ we see from (10) that two independent solutions of
Eq. (1) may be considered to be a differential (2) are x1 = ~cos ‘P~x2 = ~ sin ‘P, where ~ is any partic-
equation for x, which is easily solved by introducing ular solution of (3). Then W = 1, and (13) becomes
a ne~, ariable for x/p; the result may be put in the
form = sin Ø+J2C0S ‘P+ ~1~2 ~
x=p[Acosø+Bsin’P] , ‘P=’P(t), (10) (14)
2 and A and B are arbitrary constants. ~= l/~.
where 0 = general
Thus, the 1/p solution of (2) can be found if one We have thus used Noether’s theorem to obtain a corn-
particular solution of (3) is known. plete solution of (3) in terms of any particular solution
We may also consider (1) as providing a conserved of (3). It is suggested that this approach may be of use
quantity for (3), whenever x satisfies the auxiliary for solving certain non-linear dynamical systems, inas-
equation (2). This viewpoint is interesting because it much as it may lead to a conserved quantity containing
furnishes an example of the use of Noether’s theorem an auxiliary function which satisfies an auxiliary dif-
to replace the problem of solving a non-linear equation ferential equation. Even if the auxiliary equation is not
by the problem of solving a linear equation. Thus, if linear, it may be simpler than the original equation; in
we had initially set ourselves the task of solving (3), any case, relations may be established between solutions
we could have used Noether’s theorem, with of the original equation and solutions of the auxiliary
L(p, ~, t) = ~2 — w2p2 — (lip2)), equation.
In conclusion, we point out that the method may
to show that be applied to eq. (4), which generalizes (2); in this way
cii~ = + C
2 -~-~-
÷(p1 — ~x)2] we may obtain the results stated by Eliezer and Gray
(11) 171 The relevant lagrangian
- 2(t)x2 }, where
for (4)dF/dt
is given
= g(t).
by
p
is a conserved x
quantity for (3), provided that L = ~ eJ~(t)~12 w —
i+w2xzc 3 . (12)
2/x References
The quantity C
2 is an arbitrary constant; by choosing
C2 = 0, we reduce the solution of(3) to the solution [1] H.R. Lewis Jr., J. Math. Phys. 9(1968)1976.
of (2), while (11) becomes the Lewis invariant (I). f2] M. Kruskal, J. Math. Phys. 3 (1962) 806.
Ifwe write (1) for two different solutions of(2), [3] P.G.L. Leach, Siam J. Appl. Math. 34(1978)496.
x1 and x2, while keeping p the same, and eliminate [4] E. Noether, Nachr. Ges. Wiss. Gottingen 253 (1918) 57.
[5] M. Lutzky, J. Phys. All (1978) 249.
~ifrom the resulting two equations, we obtain [6] E. Pinney, Proc. Amer. Math. Soc 1(1950)681.
[7] C.J. Eliezer and A. Gray, Siam J. AppI. Math. 30 (1976)
= ~ ~/~2x1 2~12~}l, (13) 463.
