Can Lagrangian Method be Applied to Solve Rocket Motion Equations?

[Abhishek11235]

Homework Statement

While solving equation of rocket motion with Newton's law in 1-d,I pondered to apply Lagrangian method on this. However, I didn't get correct result. Because I can eliminate last 2nd equation using last equation and get some other equation which is certainly not rockets motion.

Homework Equations

The Attempt at a Solution

The attempt is given in screenshot.

 

[BvU]

all variables and given/known data

What is $dm$ ? Dimension ?

 

[Abhishek11235]

What is $dm$ ? Dimension ?

A small mass ejected from rocket

 

[BvU]

So you don't describe a rocket but a vessel that throws some object off the ship in a backward direction ?