A bead slides on a frictionless wire shaped as ( z = \alpha r^2 ) (paraboloid of revolution), rotating about the vertical axis with constant angular speed ( \Omega ). Find the Lagrangian and the equation of motion for the radial coordinate ( r ).
Yet, mastering the Lagrangian method requires practice. Theory alone is insufficient. You need —step-by-step examples that reveal how to set up coordinates, write the Lagrangian, apply the Euler-Lagrange equation, and interpret the results. lagrangian mechanics problems and solutions pdf
( r ) (distance from rotation axis) Kinetic energy: ( T = \frac{1}{2} m (\dot{r}^2 + r^2\omega^2) ) – note the centrifugal term emerges naturally. Potential energy: ( U = 0 ) (horizontal plane) Lagrangian: ( L = \frac{1}{2} m (\dot{r}^2 + r^2\omega^2) ) A bead slides on a frictionless wire shaped
[ \ddot{r} - \omega^2 r = 0 \quad \Rightarrow \quad r(t) = A e^{\omega t} + B e^{-\omega t} ] Theory alone is insufficient
[ \frac{d}{dt}(m l^2 \dot{\theta}) + mgl \sin\theta = 0 \quad \Rightarrow \quad \ddot{\theta} + \frac{g}{l}\sin\theta = 0 ]