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Pi-Bond
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Homework Statement
Show that the area enclosed by a closed curve {x(t); y (t)} is given by
[itex]A=\frac{1}{2} \int_{t_1}^{t_2} (x\dot{y}-y\dot{x})dt[/itex]
Show that the expression for the shortest curve which encloses a given area, A, may be found by minimising the expression
[itex]s=\int_{t_1}^{t_2} \sqrt{\dot{x}^2 + \dot{y}^2 }dt +\lambda(\frac{1}{2} \int_{t_1}^{t_2} (x\dot{y}-y\dot{x})dt - A)[/itex]
Hence show that [itex]x \propto \ddot{y}[/itex] and [itex]y \propto \ddot{x}[/itex].
Homework Equations
Euler-Lagrange equations
The Attempt at a Solution
I don't know how to get the equation for the area. A hint says to consider a triangle formed by (0, 0), (x, y ) and (x + dx, y + dy ) - but I'm not sure how to use it.
But assuming the expression is correct, the expression to be minimised follows simply from applying the Lagrange multiplier technique to the length mimimisation problem. The Euler-Lagrange equation applied to s for x gives
(See posts below)
A similar equation comes for y, with x and y's interchanged in the above one. For λ,
[itex]x\dot{y}=y\dot{x}[/itex]
I'm unsure on how to use these three to show the proportionality relations.
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