Problem 2: (a) Consider a right circular cylinder of radius R centered on the z axis. Findthe relation betweenpand z that describes the geodesics (stationary paths) on the surfaceof this cylinder. (b) Specifically, for an initial point at (p, z) = (0,0) and an endpoint atarbitrary (pf, zf), write down an equation for the stationary paths between these two points.[There are many such paths. Why?]

In the study of geodesics on the surface of a right circular cylinder with radius and centered on the z-axis, we aim to derive the relation between the azimuthal angle and the vertical position , describing the stationary paths or geodesics.The resulting equation for geodesics between an initial point at and an arbitrary endpoint at takes the form , illustrating helical paths along the cylinder's surface.The periodic nature of contributes to the multitude of such paths, allowing for various helices with different winding numbers around the cylinder.The geodesics on the surface of a right circular cylinder can be described by the equation , where and are constants, and is the azimuthal angle measured around the z-axis.To derive this, consider the metric for cylindrical coordinates . For geodesics, the Euler-Lagrange equation for the Lagrangian gives us the geodesic equations.For the cylindrical coordinates, the Lagrangian becomes . Applying the Euler-Lagrange equations, we get:1. For : gives .2. For : gives , implying , a constant of motion.3. For : gives .Solving these equations, we find that , where and is another constant. This represents a helical path along the surface of the cylinder, with being the azimuthal angle and the vertical position.For the geodesics on the surface of a right circular cylinder, the equation for stationary paths between an initial point and an endpoint at arbitrary is given by , where is a constant of motion.This equation describes a helical path around the cylinder. The azimuthal angle increases linearly with height , resulting in a helix.There are many such paths because the azimuthal angle has periodicity. Specifically, is periodic with due to the cylindrical symmetry.As a result, for any given , there are infinitely many paths corresponding to different values of in the equation .Each represents a different turn around the cylinder.So, the stationary paths between the specified initial and endpoint are helices with varying values, representing different winding numbers around the cylinder.In conclusion, the derived equation elegantly characterizes the stationary paths or geodesics on the surface of a right circular cylinder.The helical nature of these paths, influenced by the azimuthal angle and the constant of motion , provides a concise representation of the intricate trajectories between an initial point and an arbitrary endpoint on the cylinder.The periodicity of further underscores the diversity of such paths, allowing for a myriad of helices with different winding numbers.