 #1
dRic2
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Homework Statement:

I have to minimize the integral
$$\int_{\tau_1}^{\tau_2} y \sqrt{{x'}^2+{y'}^2} d \tau$$
with the isoperimetrical condition
$$\int_{\tau_1}^{\tau_2} \sqrt{{x'}^2+{y'}^2} = c$$
where ##c## is a constant.
Relevant Equations:
 EulerLagrange equations.
Using Lagrange multiplier ##\lambda## (only one is needed) the integral to minimize becomes
$$\int_{\tau_1}^{\tau_2} (y + \lambda) \sqrt{{x'}^2+{y'}^2} d \tau = \int_{\tau_1}^{\tau_2} F(x, x', y, y', \lambda, \tau) d\tau $$
Using EL equations:
$$\frac {\partial F}{\partial x}  \frac d {d \tau} \frac {\partial F}{\partial x'} = 0  \frac d {d \tau} \left( (y+\lambda) \frac {x'}{\sqrt{{x'}^2+{y'}^2}} \right) = 0$$
$$\frac {\partial F}{\partial y}  \frac d {d \tau} \frac {\partial F}{\partial y'} = \sqrt{{x'}^2+{y'}^2}  \frac d {d \tau} \left( (y+\lambda) \frac {y'}{\sqrt{{x'}^2+{y'}^2}} \right) = 0$$
$$\frac {\partial F}{\partial \lambda}  \frac d {d \tau} \frac {\partial F}{\partial \lambda'} = \sqrt{{x'}^2+{y'}^2}  0 = 0$$
If I sum the equation for ##y## and ##\lambda## I get:
$$\frac d {d \tau} \left( (y+\lambda) \frac {y'}{\sqrt{{x'}^2+{y'}^2}} \right) = 0$$
Integrating the first equation I get:
$$(y+\lambda) \frac {x'}{\sqrt{{x'}^2+{y'}^2}} = \text{const.} \rightarrow y+\lambda = \frac {\sqrt{{x'}^2+{y'}^2}} {x'}$$
Putting together these last two equations I get:
$$\frac d {d \tau} \left( \frac {y'}{x'} \right) = \frac d {d \tau} \left( \frac {dy}{dx} \right) = 0$$
which is wrong!
I think I'm missing something fundamental because it is a pretty straightforward exercise.
$$\int_{\tau_1}^{\tau_2} (y + \lambda) \sqrt{{x'}^2+{y'}^2} d \tau = \int_{\tau_1}^{\tau_2} F(x, x', y, y', \lambda, \tau) d\tau $$
Using EL equations:
$$\frac {\partial F}{\partial x}  \frac d {d \tau} \frac {\partial F}{\partial x'} = 0  \frac d {d \tau} \left( (y+\lambda) \frac {x'}{\sqrt{{x'}^2+{y'}^2}} \right) = 0$$
$$\frac {\partial F}{\partial y}  \frac d {d \tau} \frac {\partial F}{\partial y'} = \sqrt{{x'}^2+{y'}^2}  \frac d {d \tau} \left( (y+\lambda) \frac {y'}{\sqrt{{x'}^2+{y'}^2}} \right) = 0$$
$$\frac {\partial F}{\partial \lambda}  \frac d {d \tau} \frac {\partial F}{\partial \lambda'} = \sqrt{{x'}^2+{y'}^2}  0 = 0$$
If I sum the equation for ##y## and ##\lambda## I get:
$$\frac d {d \tau} \left( (y+\lambda) \frac {y'}{\sqrt{{x'}^2+{y'}^2}} \right) = 0$$
Integrating the first equation I get:
$$(y+\lambda) \frac {x'}{\sqrt{{x'}^2+{y'}^2}} = \text{const.} \rightarrow y+\lambda = \frac {\sqrt{{x'}^2+{y'}^2}} {x'}$$
Putting together these last two equations I get:
$$\frac d {d \tau} \left( \frac {y'}{x'} \right) = \frac d {d \tau} \left( \frac {dy}{dx} \right) = 0$$
which is wrong!
I think I'm missing something fundamental because it is a pretty straightforward exercise.
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