restart:with(plots):with(DEtools):with(VariationalCalculus): Here we set up the equation for the planar rotating chain in steps. First the catenary in polar coordinates, where the angle is measured from the negative y axis, as in spherical coordinates. f:=(r(phi)*cos(phi)+lambda)*sqrt(r(phi)^2+diff(r(phi),phi)^2);

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These are the variables needed to do the EL substitutions so that Maple can differentiate with respect to a variable, and not a function. S1:={r(phi)=r,diff(r(phi),phi)=dr}; S2:={r=r(phi),dr=diff(r(phi),phi)};

PCQvLUkickc2IjYjSSRwaGlHRiZGJS8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRiRGKEkjZHJHRiY=

PCQvSSNkckc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJyR0YlNiNJJHBoaUdGJUYtL0YrRio=

The Euler-Lagrange equation L1:=subs(S2,diff(subs(S1,f),r))-diff(subs(S2,diff(subs(S1,f),dr)),phi);

LCwqJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRwaGlHRikiIiIsJiokLUkickdGKUYqIiIjRiwqJC1JJWRpZmZHRic2JEYvRitGMUYsI0YsRjFGLCooLCYqJkYvRixGJEYsRixJJ2xhbWJkYUdGKUYsRixGLSMhIiJGMUYvRixGLCooLCYqJkYzRixGJEYsRiwqJkYvRiwtSSRzaW5HRiZGKkYsRjxGLEYtRjtGM0YsRjwqKkY4RixGLSMhIiRGMUYzRiwsJiomRi9GLEYzRixGMSomRjNGLC1GNDYkRjNGK0YsRjFGLEY2KihGOEYsRi1GO0ZJRixGPA==

The Euler-Lagrange equation as calculated by Maple EulerLagrange(f,phi,[r(phi)]);

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

Put the EL eqn in the form of a standard differential equation DE1:=diff(r(phi),phi,phi)=simplify(solve(L1=0,diff(r(phi),phi,phi)));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JInJHNiI2I0kkcGhpR0YrRi1GLSooLC4qJi1JJGNvc0c2JEYlSShfc3lzbGliR0YrRiwiIiJGKSIiJCIiIyooRidGN0YpRjVGMUY1RjYqJkknbGFtYmRhR0YrRjVGKUY3RjUqJkYnRjdGOkY1RjcqKEYnRjVGKUY3LUkkc2luR0YzRixGNUY1KiZGJ0Y2Rj1GNUY1RjUsJiomRilGNUYxRjVGNUY6RjUhIiJGKUZC

We write the solution as a numerical procedure SDE1:=dsolve({DE1,r(0)=1,D(r)(0)=0},numeric,parameters=[lambda],output=listprocedure); SDE1(parameters=[lambda=-1.62]); SDE1(1);

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Now we can plot this as a curve in cartesian coordinates -- it looks just as we know it should. Note that we must choose lambda, the Lagrange multiplier, so that the endpoints land at 1, -1. Also, the initial conditions had to be chosen so that r = 1 and Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJyRzYiNiNJJHBoaUdGKUYrIiIhat phi = 0 because Maple won't solve this as a BVP. Might have to use a shooting method to get this done numerically. This restriction will give us trouble later. Luckily, it seems like the numerical procedure gives us sensible results for the next two variants, except for the unfortunate ICs necessary. evalf(Pi/2); plot({cos(k)*rhs(SDE1(k)[2]),sin(k)*rhs(SDE1(k)[2]),k=-Pi/2..Pi/2});

JCIrRmp6cTohIio=

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

Now add a centrifugal potential g:=(r(phi)*cos(phi)+lambda+(1/2)*r(phi)^2*omega^2)*sqrt(r(phi)^2+diff(r(phi),phi)^2);

KiYsKComLUkickc2IjYjSSRwaGlHRiciIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRidGKEYqRipJJ2xhbWJkYUdGJ0YqKiZGJSIiI0kmb21lZ2FHRidGMiNGKkYyRiosJiokRiVGMkYqKiQtSSVkaWZmR0YuNiRGJUYpRjJGKkY0

L2:=subs(S2,diff(subs(S1,g),r))-diff(subs(S2,diff(subs(S1,g),dr)),phi);

LCwqJiwmLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJHBoaUdGKiIiIiomLUkickdGKkYrRi1JJm9tZWdhR0YqIiIjRi1GLSwmKiRGL0YyRi0qJC1JJWRpZmZHRig2JEYvRixGMkYtI0YtRjJGLSooLCgqJkYvRi1GJUYtRi1JJ2xhbWJkYUdGKkYtKiZGL0YyRjFGMkY5Ri1GMyMhIiJGMkYvRi1GLSooLCgqJkY2Ri1GJUYtRi0qJkYvRi0tSSRzaW5HRidGK0YtRkAqKEYvRi1GMUYyRjZGLUYtRi1GM0Y/RjZGLUZAKipGO0YtRjMjISIkRjJGNkYtLCYqJkYvRi1GNkYtRjIqJkY2Ri0tRjc2JEY2RixGLUYyRi1GOSooRjtGLUYzRj9GTkYtRkA=

DE2:=diff(r(phi),phi,phi)=simplify(solve(L2=0,diff(r(phi),phi,phi)));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JInJHNiI2I0kkcGhpR0YrRi1GLSooLDIqKEYnIiIjRikiIiItSSRjb3NHNiRGJUkoX3N5c2xpYkdGK0YsRjIiIicqJkYnRjFJJ2xhbWJkYUdGK0YyIiIlKihGJ0YyRilGMS1JJHNpbkdGNUYsRjJGMSomRiciIiRGPEYyRjEqJkYzRjJGKUY/RjoqJkY5RjJGKUYxRjEqKEYnRjFGKUYxSSZvbWVnYUdGK0YxRjoqJkYpRjpGQ0YxRj9GMiwoKiZGKUYxRkNGMUYyKiZGKUYyRjNGMkYxRjlGMSEiIkYpRkg=

solve(DE2,lambda);

LCQqJiwyKigtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYnNiQtSSJyRzYiNiNJJHBoaUdGLkYwRjAiIiJGLCIiIy1JJGNvc0c2JEYoSShfc3lzbGliR0YuRi9GMSEiIyooRiZGMUYsIiIkSSZvbWVnYUdGLkYyISIiKihGKkYyRixGMUYzRjEiIicqKEYqRjFGLEYyLUkkc2luR0Y1Ri9GMUYyKiZGKkY5Rj9GMUYyKiZGM0YxRixGOSIiJSooRipGMkYsRjJGOkYyRkMqJkYsRkNGOkYyRjlGMSwoKiZGJkYxRixGMUY7KiRGKkYyRjIqJEYsRjJGMUY7I0Y7RjI=

SDE2:=dsolve({DE2,r(0)=1,D(r)(0)=0},numeric,parameters=[lambda, omega],output=listprocedure); SDE2(parameters=[lambda=-15, omega=3]); SDE2(1);

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

NyUvLUkkcGhpRzYiNiMvSStwYXJhbWV0ZXJzR0YmNyQvSSdsYW1iZGFHRiYhIzovSSZvbWVnYUdGJiIiJDckL0YsJEYtIiIhL0YvJEYwRjQvLS1JInJHRiY2I0YlRidGMS8tLUklZGlmZkclKnByb3RlY3RlZEc2JEY5RiVGJ0Yx

NyUvLUkkcGhpRzYiNiMiIiIkRigiIiEvLS1JInJHRiY2I0YlRickIjNhY3ZHMTtNdikqISM9Ly0tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRi1GJUYnJCEzR1FTR0wpKWU/PSEjPw==

Once again, the plot seems realistic, but this symmetric solution should be unstable. Unfortunately, it is forced by the initial conditions. plot({cos(k)*rhs(SDE2(k)[2]),sin(k)*rhs(SDE2(k)[2]),k=-Pi/2..Pi/2},numpoints=300);

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We can also specify different initial conditions and see we get realistic asymmetric solutions. SDE2a:=dsolve({DE2,r(Pi/2)=1,D(r)(Pi/2)=-0.32},numeric, parameters=[lambda, omega],output=listprocedure); SDE2a(parameters=[lambda=-22.4, omega=4.4]); SDE2a(1);

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Once again, the plot seems almost realistic, but we cannot easily adjust lambda to get the second endpoint to his the right spot. I conjecture that this is because the derivative and lambda have to be adjusted simultaneously. plot({cos(k)*rhs(SDE2a(k)[2]),sin(k)*rhs(SDE2a(k)[2]),k=-Pi/2..Pi/2},numpoints=300);

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Here is the BVP version for the same thing SDE2b:=dsolve({subs({lambda=-10.8, omega=2.5},DE2),r(Pi/2)=1,r(-Pi/2)=1},numeric,method=bvp,output=listprocedure);

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plot({cos(k)*rhs(SDE2b(k)[2]),sin(k)*rhs(SDE2b(k)[2]),k=-Pi/2..Pi/2},numpoints=300);

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Now we let the plane rise up by an angle alpha. Gamma1 is the scaling factor for the centrifugal potential at plane angle alpha, gamma2 is the scaling factor for the gravitational force felt by the chain element. 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gamma1:=sqrt(sin(phi)^2+(cos(phi)^2)*sin(alpha)^2); gamma2:=cos(alpha);

KiQsJiokLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJHBoaUdGKiIiIyIiIiomLUkkY29zR0YnRitGLS1GJjYjSSZhbHBoYUdGKkYtRi4jRi5GLQ==

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJmFscGhhR0Yn

h:=(gamma2*r(phi)*cos(phi)+lambda+(1/2)*(gamma1*r(phi))^2*omega^2)*sqrt(r(phi)^2+diff(r(phi),phi)^2);

KiYsKCooLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJmFscGhhR0YqIiIiLUkickdGKjYjSSRwaGlHRipGLS1GJkYwRi1GLUknbGFtYmRhR0YqRi0qKCwmKiQtSSRzaW5HRidGMCIiI0YtKiZGMkY5LUY4RitGOUYtRi1GLkY5SSZvbWVnYUdGKkY5I0YtRjlGLSwmKiRGLkY5Ri0qJC1JJWRpZmZHRig2JEYuRjFGOUYtRj0=

L3:=subs(S2,diff(subs(S1,h),r))-diff(subs(S2,diff(subs(S1,h),dr)),phi);

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

DE3:=diff(r(phi),phi,phi)=simplify(solve(L3=0,diff(r(phi),phi,phi)));

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

SDE3:=dsolve({DE3,r(Pi/2)=1,D(r)(Pi/2)=0.92},numeric,parameters=[lambda, omega, alpha],output=listprocedure); SDE3(parameters=[lambda=-11.9, omega=3, alpha=Pi/4]); SDE3(1);

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

NyUvLUkkcGhpRzYiNiMvSStwYXJhbWV0ZXJzR0YmNyUvSSdsYW1iZGFHRiYkISQ+IiEiIi9JJm9tZWdhR0YmIiIkL0kmYWxwaGFHRiYsJEkjUGlHJSpwcm90ZWN0ZWRHIyIiIiIiJTclRisvRjEkRjIiIiEvRjQkIjBbdVJqIilSJnkhIzovLS1JInJHRiY2I0YlRidGOy8tLUklZGlmZkdGNzYkRkVGJUYnRjs=

NyUvLUkkcGhpRzYiNiMiIiIkRigiIiEvLS1JInJHRiY2I0YlRickIjNVMHIyK1dkYGkhIz0vLS1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGLUYlRickIjMoPiEqKTNAJjMuciRGMg==

This time we were able to adjust lambda and the derivative at the endpoint simultaneously. plot({cos(k)*rhs(SDE3(k)[2]),sin(k)*rhs(SDE3(k)[2]),k=-Pi/2..Pi/2},numpoints=300);

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Here is the BVP solution SDE3a:=dsolve({subs({alpha=Pi/8, lambda=-6, omega=2.75},DE3),r(Pi/2)=1,r(-Pi/2)=1},numeric,method=bvp,output=listprocedure);

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plot({cos(k)*rhs(SDE3a(k)[2]),sin(k)*rhs(SDE3a(k)[2]),k=-Pi/2..Pi/2},numpoints=300);

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