# Lagrange Multipliers for finding Geodesics on a Cylinder

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!

• It will help to translate the terms "geodesics" and "helices" into mathematical equations/inequalities that must be satisfied. – Michael Jul 9 at 19:59
• 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? – Conifold Jul 9 at 20:03
• See "Example: Problem 6.4 – Part III" of teacher.pas.rochester.edu/PHY235/LectureNotes/Chapter06/… – nmasanta Jul 10 at 5:09

When you have a functional $$$$J[\gamma]=\int_a^b F(t,\gamma(t),\dot{\gamma}(t))dt$$$$ restricted to the conditions $$\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]$$ such that satisfies the following differential equation: $$$$\begin{split} F_x-\lambda g_x=\frac{d}{dt}F_{\dot{x}}\\ F_y-\lambda g_y=\frac{d}{dt}F_{\dot{y}}\\ F_z-\lambda g_z=\frac{d}{dt}F_{\dot{z}}. \end{split}$$$$
Since the geodesics are the critical points of energy functional, we have $$F=\dot{x}^2+\dot{y}^2+\dot{z}^2$$ and $$g=x^2+y^2-1$$. It follows that $$$$\begin{split} -2\lambda x=2\dot{x}\\ -2\lambda y=2\dot{y}\\ 0=2\dot{z}. \end{split}$$$$