Question

Write the Lagrangian of this particle. Show that the Euler-Lagrange equa-

tions of motion for this particle is identical to what one would obtain from

Newton’s second law.

Answer 1

The lagrangian of the system

$$L=T-V=\frac{m({\dot x}^2+{\dot y}^2)}{2}-mgy$$

The Euler-Lagrange equations

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot x}}\right)-\frac{\partial L}{\partial {x}}=0,\\\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot y}}\right)-\frac{\partial L}{\partial {y}}=0,$$

give

$$m{\ddot x}+0=0,\\ m{\ddot y}+mg=0.$$

Finally

$$m{\ddot x}=0,\\ m{\ddot y}=-mg.$$