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Thread: orthogonal trajectaries

  1. #1
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    orthogonal trajectaries

    Find the orthogonal trajectory of the system $\displaystyle r^2cos^2 2\theta=c^2 sin 2 \theta$.

    ans: $\displaystyle r^4(3-cos \theta)=k$.
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  2. #2
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    Quote Originally Posted by dimuk View Post
    Find the orthogonal trajectory of the system $\displaystyle r^2\cos^2 2\theta=c^2 \sin 2 \theta$.

    ans: $\displaystyle r^4(3-\cos \theta)=k$. I'm getting $\displaystyle \color{red}r^4(3-\cos4\theta)=k$.
    In polar coordinates, the quantity $\displaystyle \tfrac1r\tfrac{dr}{d\theta}$ represents the slope of the tangent of a curve with respect to the radial direction. (Informally, a point on the curve moves an infinitesimal distance dr in the radial direction and rdθ in the transverse direction.) So the orthogonal system should have "radial slope" $\displaystyle -r/\tfrac{dr}{d\theta}$ at that point.

    Write $\displaystyle r^2\cos^2 2\theta=c^2 \sin 2 \theta$ as $\displaystyle r^2 = c^2\sec2\theta\tan2\theta$, and differentiate with respect to θ: $\displaystyle 2rr' = 2c^2(\sec2\theta\tan^22\theta + \sec^32\theta)$, where $\displaystyle r' = \tfrac{dr}{d\theta}$. Substitute $\displaystyle c^2 = \frac{r^2\cos^22\theta}{\sin2\theta}$ and simplify a bit, to get $\displaystyle \frac{r'}r = \frac{1+\sin^22\theta}{\sin2\theta\cos2\theta}$.

    The orthogonal trajectory will therefore have $\displaystyle \frac{r'}r =-\frac{\sin2\theta\cos2\theta}{1+\sin^22\theta}$. Integrate this: $\displaystyle \int\frac{dr}r = -\int\frac{\sin2\theta\cos2\theta}{1+\sin^22\theta} d\theta$.

    To find the integral on the right-hand side, substitute $\displaystyle u = \sin2\theta$, and it becomes $\displaystyle -\int\frac{u}{2(1+u^2)}du = -\tfrac14\ln(1+u^2)$.

    That gives the orthogonal trajectory as $\displaystyle \ln r = -\tfrac14\ln(1+\sin^22\theta)\; +$ const., or $\displaystyle r^4(1+\sin^22\theta) = $ const.

    Finally, the trig identity $\displaystyle \sin^2\phi = \tfrac12(1-\cos2\phi)$ can be used to write the answer as $\displaystyle r^4(3-\cos4\theta)=$ const.
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