ARCHIVOS TEÓRICOS DEL MUSEO MATEMÁTICO.

ÁREA- ANÁLISIS-

TEMA-CÁLCULO DE VARIACIONES.ENFOQUE GLOBAL.

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Problemas variacionales ///////////

Autor- Leonard Echagüe-Mate UBA Museum- lechague@dm.uba.ar

Cicloide como minimal de tiempo de caída.(Brachistócrona).

Planteo por cálculo de variaciones:

Por Euler-Lagrange simplificada

f-diff(f, [Maple Math] )* [Maple Math] =a

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Quedando para resolver:

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> restart;

> z:=diff(y(x),x);

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> ode5 := a=y(x)+y(x)*z^2;

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>

> ans5 := dsolve(ode5);

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> with(DEtools): odeadvisor(ode5);

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> z:=diff(y(x),x);

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> ode6 := 10=y(x)+y(x)*z^2;

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> ans6 := dsolve(ode6);

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Copiando, pegando y adaptando constantes....

> plot({sqrt(-y^2+10*y)-5*arcsin(-1+1/5*y),-10/2*Pi-sqrt(-y^2+10*y)+5*arcsin(-1+1/5*y)},y=0..10);

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>

> y1:=0;sqrt(-y1^2+10*y1)-5*arcsin(-1+1/5*y1);evalf(%);y1:=10;sqrt(-y1^2+10*y1)-5*arcsin(-1+1/5*y1);evalf(%);

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Verificación gráfica.....

> with(plots):

> p1:=plot({sqrt(-y^2+10*y)-5*arcsin(-1+1/5*y),-10/2*Pi-sqrt(-y^2+10*y)+5*arcsin(-1+1/5*y)},y=0..10):

> p2:=plot([-5*cos(t)+5,5*t-5*sin(t)-15/2*Pi,t=0..2*Pi],color=blue,style=point):

> display([p1,p2]);

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>