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Given a right circular cylinder: \begin{align} g(x,y,z) = x^2+y^2-1 = 0 \end{align} Use Lagrange multipliers to show that the geodesics on the cylinder are helices.

The Euler-Lagrange Equation is easy to set up, but how would one go about setting up the Lagrange Multiplier for this?

Any tips would be appreciated!

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  • $\begingroup$ It will help to translate the terms "geodesics" and "helices" into mathematical equations/inequalities that must be satisfied. $\endgroup$ – Michael Jul 9 at 19:59
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    $\begingroup$ It is not very clear what the context of this problem is. How are the geodesics defined? Is it just for surfaces in $\mathbb{R}^3$? Do you have the metric already written in coordinates on the cylinder? $\endgroup$ – Conifold Jul 9 at 20:03
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    $\begingroup$ See "Example: Problem 6.4 – Part III" of teacher.pas.rochester.edu/PHY235/LectureNotes/Chapter06/… $\endgroup$ – nmasanta Jul 10 at 5:09
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When you have a functional \begin{equation} J[\gamma]=\int_a^b F(t,\gamma(t),\dot{\gamma}(t))dt \end{equation} restricted to the conditions $\gamma(a)=p$, $\gamma(b)=q$, and $g(\gamma(t))=0$ for all $t\in [a,b]$, if $g_x,g_y,g_z$ do not vanish simultaneously in $S:g(x,y,z)=0$ then there exists $\lambda$ a function in $[a,b]$ such that satisfies the following differential equation: \begin{equation} \begin{split} F_x-\lambda g_x=\frac{d}{dt}F_{\dot{x}}\\ F_y-\lambda g_y=\frac{d}{dt}F_{\dot{y}}\\ F_z-\lambda g_z=\frac{d}{dt}F_{\dot{z}}. \end{split} \end{equation}

Since the geodesics are the critical points of energy functional, we have $F=\dot{x}^2+\dot{y}^2+\dot{z}^2$ and $g=x^2+y^2-1$. It follows that \begin{equation} \begin{split} -2\lambda x=2\dot{x}\\ -2\lambda y=2\dot{y}\\ 0=2\dot{z}. \end{split} \end{equation}

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