ARCHIVOS TEÓRICOS DEL MUSEO MATEMÁTICO.

ÁREA- ANÁLISIS-

TEMA-CÁLCULO DE VARIACIONES.ENFOQUE GLOBAL.

[Maple Bitmap]

////// 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

[Maple Math]

Quedando para resolver:

[Maple Math]

> restart;

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

[Maple Math]

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

[Maple Math]

>

> ans5 := dsolve(ode5);

[Maple Math]
[Maple Math]

> with(DEtools): odeadvisor(ode5);

[Maple Math]

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

[Maple Math]

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

[Maple Math]

> ans6 := dsolve(ode6);

[Maple Math]
[Maple Math]

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);

[Maple Plot]

>

> 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(%);

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple OLE 2.0 Object]

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]);

[Maple Plot]

>