Action-angle perturbation expansion, lecture example, 5 December 2005 restart; The perturbed Hamiltonian H:=p^2/2+q^2/2+epsilon/3*q^3;

LCgqJiMiIiIiIiNGJSlJInBHNiJGJkYlRiUqJkYkRiUpSSJxR0YpRiZGJUYlKigjRiUiIiRGJUkoZXBzaWxvbkdGKUYlKUYsRi9GJUYl

The simple-harmonic oscillator canonical transformation to unperturbed action angle variables q:=sqrt(J/Pi)*cos(2*Pi*w);

KiYpKiZJIkpHNiIiIiJJI1BpRyUqcHJvdGVjdGVkRyEiIiNGJyIiI0YnLUkkY29zRzYkRilJKF9zeXNsaWJHRiY2IywkKihGLEYnRihGJ0kid0dGJkYnRidGJw==

p:=-sqrt(J/Pi)*sin(2*Pi*w);

LCQqJikqJkkiSkc2IiIiIkkjUGlHJSpwcm90ZWN0ZWRHISIiI0YoIiIjRigtSSRzaW5HNiRGKkkoX3N5c2xpYkdGJzYjLCQqKEYtRihGKUYoSSJ3R0YnRihGKEYoRis=

H;

LCgqKiMiIiIiIiNGJUkiSkc2IkYlSSNQaUclKnByb3RlY3RlZEchIiIpLUkkc2luRzYkRipJKF9zeXNsaWJHRig2IywkKihGJkYlRilGJUkid0dGKEYlRiVGJkYlRiUqKkYkRiVGJ0YlRilGKyktSSRjb3NHRi9GMUYmRiVGJSoqI0YlIiIkRiVJKGVwc2lsb25HRihGJSkqJkYnRiVGKUYrI0Y7RiZGJSlGN0Y7RiVGJQ==

The transformed Hamiltonian H:=J/2/Pi+epsilon/3*q^3;

LCYqKCMiIiIiIiNGJUkiSkc2IkYlSSNQaUclKnByb3RlY3RlZEchIiJGJSoqI0YlIiIkRiVJKGVwc2lsb25HRihGJSkqJkYnRiVGKUYrI0YuRiZGJSktSSRjb3NHNiRGKkkoX3N5c2xpYkdGKDYjLCQqKEYmRiVGKUYlSSJ3R0YoRiVGJUYuRiVGJQ==

Zeroth-order action-angle solutions (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 ) assuming that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEicUYnRi9GMi1JI21vR0YkNjBRMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZULyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkobWZlbmNlZEdGJDYjLUYjNiMtSSNtbkdGJDYkUSIwRidGPi1GOzYwUSI9RidGPkZARkNGRUZHRklGS0ZNRk8vRlNRL3RoaWNrbWF0aHNwYWNlRicvRlZGZG9GV0ZaLUZdbzYkRllGPg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEicEYnRi9GMi1JI21vR0YkNjBRMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZULyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkobWZlbmNlZEdGJDYjLUYjNiMtSSNtbkdGJDYkUSIwRidGPi1GOzYwUSI9RidGPkZARkNGRUZHRklGS0ZNRk8vRlNRL3RoaWNrbWF0aHNwYWNlRicvRlZGZG9GV0ZaRlxv . J0:=Pi;w0:=t/2/Pi;

SSNQaUclKnByb3RlY3RlZEc=

LCQqKCMiIiIiIiNGJUkidEc2IkYlSSNQaUclKnByb3RlY3RlZEchIiJGJQ==

First-order Hamilton equations, using LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIkpGJ0YvRjItSSNtbkdGJDYkUSIwRicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIndGJ0YvRjItSSNtbkdGJDYkUSIwRicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== in the small terms on the right-hand sides eqw1:=diff(w1(t),t)=subs(w=w0,J=J0,diff(H,J));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSN3MUc2IjYjSSJ0R0YpRissJiomIyIiIiIiI0YvSSNQaUdGJSEiIkYvKipGLkYvSShlcHNpbG9uR0YpRi8pLUkkY29zRzYkRiVJKF9zeXNsaWJHRilGKiIiJEYvRjFGMkYv

s:=dsolve({eqw1,w1(0)=0},w1(t));

Ly1JI3cxRzYiNiNJInRHRiUsJCooIyIiIiIjQ0YrLCgqJkkoZXBzaWxvbkdGJUYrLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJComIiIkRitGJ0YrRitGK0YrKigiIipGK0YvRistRjFGJkYrRisqJiIjN0YrRidGK0YrRitJI1BpR0YzISIiRis=

w1:=rhs(s);

LCQqKCMiIiIiI0NGJSwoKiZJKGVwc2lsb25HNiJGJS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKjYjLCQqJiIiJEYlSSJ0R0YqRiVGJUYlRiUqKCIiKkYlRilGJS1GLDYjRjRGJUYlKiYiIzdGJUY0RiVGJUYlSSNQaUdGLiEiIkYl

w1:=expand(w1);

LCgqLCMiIiIiIidGJUkoZXBzaWxvbkc2IkYlKS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjSSJ0R0YoIiIjRiVJI1BpR0YtISIiLUkkc2luR0YsRi9GJUYlKiojRiUiIiRGJUYnRiVGMkYzRjRGJUYlKigjRiVGMUYlRjBGJUYyRjNGJQ==

eqJ1:=diff(J1(t),t)=-subs(J=J0,w=w0,diff(H,w));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSNKMUc2IjYjSSJ0R0YpRissJCosIiIjIiIiSShlcHNpbG9uR0YpRi8pLUkkY29zRzYkRiVJKF9zeXNsaWJHRilGKkYuRi8tSSRzaW5HRjRGKkYvSSNQaUdGJUYvRi8=

s:=dsolve({eqJ1,J1(0)=J0},J1(t));

Ly1JI0oxRzYiNiNJInRHRiUsKCoqIyIiIyIiJCIiIkkoZXBzaWxvbkdGJUYtKS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYmRixGLUkjUGlHRjNGLSEiIkY1Ri0qKEYqRi1GLkYtRjVGLUYt

J1:=rhs(s);

LCgqKiMiIiMiIiQiIiJJKGVwc2lsb25HNiJGJyktSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2I0kidEdGKUYmRidJI1BpR0YuRichIiJGMkYnKihGJEYnRihGJ0YyRidGJw==

Check that 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 behaves itself under this perturbation expansion plot(subs(epsilon=.2,sqrt(J1/Pi)*cos(2*Pi*w1)),t=0..10); It does -- it still has a turning point on the right at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JEZURjk= and on the left, where the cubic term in the potential makes the potential weaker, it goes further out.

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

Now do the next order in the perturbation expansion by using LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIkpGJ0YvRjItSSNtbkdGJDYkUSIxRicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIndGJ0YvRjItSSNtbkdGJDYkUSIxRicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== on the right-hand sides of Hamilton's equations. eqw2:=diff(w2(t),t)=subs(w=w1,J=J1,diff(H,J));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSN3Mkc2IjYjSSJ0R0YpRissJiomIyIiIiIiI0YvSSNQaUdGJSEiIkYvKixGLkYvSShlcHNpbG9uR0YpRi8pKiYsKCoqI0YwIiIkRi9GNEYvKS1JJGNvc0c2JEYlSShfc3lzbGliR0YpRipGOkYvRjFGL0YyRjFGLyooRjlGL0Y0Ri9GMUYvRi9GL0YxRjJGLkYvKS1GPTYjLCQqKEYwRi9GMUYvLCgqLCNGLyIiJ0YvRjRGLylGPEYwRi9GMUYyLUkkc2luR0Y+RipGL0YvKiojRi9GOkYvRjRGL0YxRjJGS0YvRi8qKEYuRi9GK0YvRjFGMkYvRi9GL0Y6Ri9GMUYyRi8=

Whoa, this is horrible! But recall that at this level in the perturbation procedure we are only accurate to order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21uR0YkNiRRIjJGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yr . So we may expand the right-hand side through this order and throw away higher order terms eqw2:=lhs(eqw2)=convert(taylor(rhs(eqw2),epsilon=0,3),polynom);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSN3Mkc2IjYjSSJ0R0YpRissKComIyIiIiIiI0YvSSNQaUdGJSEiIkYvKipGLkYvSShlcHNpbG9uR0YpRi8pLUkkY29zRzYkRiVJKF9zeXNsaWJHRilGKiIiJEYvRjFGMkYvKipGLkYvLCYqLCIiJ0YvKUY2RjBGLy1JJHNpbkdGOEYqRi9GMUYvLCYqKiNGL0Y+Ri9GP0YvRjFGMkZARi9GLyooI0YvRjpGL0YxRjJGQEYvRi9GL0YyKiYsJiomRkZGL0Y1Ri9GMkZGRi9GL0Y1Ri9GL0YvRjFGMilGNEYwRi9GLw==

We could keep going, but what we really want is the perturbed frequency. And since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIndGJ0YvRjItSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== is the angle variable, its time derivative is the frequency (in cycles per second). If we time average LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkmbWZyYWNHRiQ2KC1GIzYlRistSSVtc3ViR0YkNiUtRiw2JVEjZHdGJ0YvRjItSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GLDYlUSNkdEYnRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Yr, then, we should have the average frequency. There is a potential problem here, however: over what period should we average? To lowest order the period is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRJSZwaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRidGLw== , but we are about to find out that this is incorrect, that there is a shift in the period. But this shift will turn out to be of order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21uR0YkNiRRIjJGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yr , but the terms in 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 which we will be averaging are all of order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21uR0YkNiRRIjJGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yr . Hence, the effects of using the wrong period will be of order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21uR0YkNiRRIjNGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yr and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21uR0YkNiRRIjRGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yr , so it's OK to just average over the lowest order period, 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This average frequency is nuavg:=int(rhs(eqw2),t=0..2*Pi)/2/Pi;

LCQqKCMiIiIiIiNGJSwmKiYjIiImIiM3RiUpSShlcHNpbG9uRzYiRiZGJSEiIkYlRiVGJUkjUGlHJSpwcm90ZWN0ZWRHRi9GJQ==

nuavg:=expand(%);

LCYqKCMiIiYiI0MiIiJJI1BpRyUqcHJvdGVjdGVkRyEiIilJKGVwc2lsb25HNiIiIiNGJ0YqKiYjRidGLkYnRihGKkYn

from which we may find the average angular frequency omega:=expand(nuavg*2*Pi);

LCYqJiMiIiYiIzciIiIpSShlcHNpbG9uRzYiIiIjRichIiJGJ0Yn

Now let's check our result and see if this gives the correct perturbed period by numerically solving the problem in the original variables. restart;H:=p^2/2+q^2/2+epsilon/3*q^3;

LCgqJiMiIiIiIiNGJSlJInBHNiJGJkYlRiUqJkYkRiUpSSJxR0YpRiZGJUYlKigjRiUiIiRGJUkoZXBzaWxvbkdGKUYlKUYsRi9GJUYl

Hamilton's equations eqq:=diff(q(t),t)=subs(q=q(t),p=p(t),diff(H,p));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJxRzYiNiNJInRHRilGKy1JInBHRilGKg==

eqp:=diff(p(t),t)=-subs(q=q(t),p=p(t),diff(H,q));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJwRzYiNiNJInRHRilGKywmLUkicUdGKUYqISIiKiZJKGVwc2lsb25HRikiIiIpRi0iIiNGMkYv

Set LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and solve them numerically epsilon:=.1;

JCIiIiEiIg==

s:=dsolve({eqq,eqp,q(0)=1,p(0)=0},{q(t),p(t)},type=numeric);

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

Check that the procedure gives numbers back s(1);

NyUvSSJ0RzYiJCIiIiIiIS8tSSJwR0YlNiNGJCQhM1VCYiU0QmwjeiopISM9Ly1JInFHRiVGLCQiMz02czFKUGY+XUYv

with(plots): Plot LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEicEYnRi9GMi1JI21vR0YkNjBRMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZULyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkobWZlbmNlZEdGJDYjLUYjNiMtRiw2JVEidEYnRi9GMkYr to see if it comes back to zero after one period odeplot(s,[t,p(t)],t=0..2*Pi/(1-5/12*epsilon^2)); It looks pretty good.

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

Look at the numbers to see how precise it is s(2*Pi/(1-5/12*epsilon^2));.

NyUvSSJ0RzYiJCIvTSNHeXUlNGohIzgvLUkicEdGJTYjRiQkIjM3eDIvMTslb0wjISM/Ly1JInFHRiVGLCQiMmsncEYkMyQpKioqKiohIzw=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== comes quite close to zero, but misses by a little. Let's experiment numerically to see by what factor we must multiply the period to have 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 so that we can see the order of our error. s(2*Pi/(1-5/12*epsilon^2)*1.000337);

NyUvSSJ0RzYiJCIvS2N2NWc2aiEjOC8tSSJwR0YlNiNGJCQhM3NeQic9MCp6IjMjISNCLy1JInFHRiVGLCQiM0BhI3kyeisrKyIhIzw=

So the relative error in the period is about 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 , which is of order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRLSZ2YXJlcHNpbG9uO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21uR0YkNiRRIjRGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yr , consistent with our answer being accurate through second order.