Prove the Langrangian is not unique

[It's me]

Question:
If L is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equations show by direct substitution that

http://qlx.is.quoracdn.net/main-74d090d14ee4fea0.png

also satisfies Lagrange's equations where F is any arbitrary but differentiable function of its arguments.
Attempt at a solution:
I'm not really sure how to solve this problem by direct substitution; I found a way to do it using the action integral but not direct substitution. Any clues or help on how to approach the problem?

 

[robphy]

Use L' as the new lagrangian in Lagrange's equation.
Does L' satisfy that equation? Under what condition?

 

[It's me]

I got that L' satisfies that equation when the derivative for Lagrange's equation of F is 0.

 

[robphy]

You should probably show your work.
You can use the LaTeX button below to help you typeset your equations.

 

[It's me]

I don't know how to write LaTeX

 

[samalkhaiat]

Question:
If L is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equations show by direct substitution that

http://qlx.is.quoracdn.net/main-74d090d14ee4fea0.png

also satisfies Lagrange's equations where F is any arbitrary but differentiable function of its arguments.

Your question is wrong. The function $F$ can not depend on the velocity $\dot{q}(t)$. If it does, then $d F /d t$ will depend (at least linearly) on the acceleration $\ddot{q}(t)$. This in turns mean that the two Lagrangians are not equivalent to each other. So, try to solve the exercise for the function $F(t) = F( q(t) , t )$.

This is one part of well know theorem which states

A function $\Lambda$ of $q(t)$, $\dot{q}(t)$ and $t$ satisfies Lagrange’s equations identically (i.e., independent of the $q_{a}(t)$) if, and only if, it is the total time derivative $d F / dt$ of some function $F ( q(t) , t )$.