Solve Lagrange's Mechanics Homework on Projectiles in Gravity

[YauYauYau]

Homework Statement

Find the differential equations of motion of a projectile in a uniform gravitational field without
air resistance.

Homework Equations

For the motion of a projectile,
in horizontal, s_x = u_x t
in vertical, s_y = u_y t + 1/2 gt²
KE = 1/2 mv²
PE = mgh

The Attempt at a Solution

L = KE - PE
= 1/2 m ( v_x² + v_y² ) - mgy
Then find their partial differentiation in terms of x, x_dot, y and y_dot.

However, the answer claims that there is z-direction motion.
Why z direction takes into the case?
Thankyou.

 

[drvrm]

However, the answer claims that there is z-direction motion.
Why z direction takes into the case?

if there is a general motion of projectile i.e. there are velocities in x,y,z directions at t=0 one can have equations of motion.
first one must choose the direction of gravitational field and a natural choice will be z.

 

[YauYauYau]

if there is a general motion of projectile i.e. there are velocities in x,y,z directions at t=0 one can have equations of motion.
first one must choose the direction of gravitational field and a natural choice will be z.

Thanks I got the idea