Question

1. Derive and solve the equation of motion of a particle, in a uniform gravitational field, projected with initial horizontal velocity v 0 at a height h.
2. Write the Lagrangian of this particle. Show that the Euler-Lagrange equations of motion for this particle is identical to what one would obtain from Newton’s second law.

Answer 1

The Newton's second law says

$$m{\bf a}={\bf F}$$

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

Equations of motion:

$${\ddot x}=0,\quad {\dot x}(0)=v_0,\quad x(0)=0,\\{\ddot y}=-g,\quad {\dot y}(0)=0,\quad y(0)=h.$$

Solutions:

$$x(t)=v_0t,\\ y(t)=h-\frac{gt^2}{2}$$

2. 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

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