Simplifying Lagrangian's Equations (Classial Dynamics)

[Lavace]

Homework Statement

[PLAIN]http://img96.imageshack.us/img96/9288/lagrangiansimplifcation.jpg
"[URL

If you require more info in the derivation, it's on page 1:
http://www.ph.qmul.ac.uk/~phy304/Homework/HW3sol.pdf

Homework Equations

L = T - V

The Attempt at a Solution

I understand exactly the process of obtaining the Lagrangian... But I do not understand his simplifying process at all.

Could somebody please help/forward me into the right direction?

Thanks in advance!

 

[cristo]

If you think in terms of the action, you can always add a constant such that, when varied, the action will give the same equations of motion. This translates into meaning that you can always add to the Lagrangian (a) a constant, (b) a function that depends on time but not on the coordinates in the Lagrangian or (c) a total time derivative of a function.

The first two in the list fall into category (b). If we change the Lagrangian so that

\tilde{L}=L+F(t),

then the action is

\tilde{S}=\int_{t_1}^{t_2}{\tilde{L}}dt=\int_{t_1}^{t_2}{L}dt+\int_{t_1}^{t_2}F(t)dt=S+\int_{t_1}^{t_2}F(t)dt=S+{\rm const.}

In the third point on that list, a term is rewritten in terms of a total time derivative plus some other term; the total time derivative then being discarded. To see why, let's look at the change of the action due to the Lagrangian changing to

\tilde{L}=L+\frac{d}{dt}G(\varphi,t)

which gives

\tilde{S}=\int_{t_1}^{t_2}Ldt+\int_{t_1}^{t_2}\frac{d}{dt}G(\varphi,t)dt<br /> =S+G(\varphi(t_1),t_1)-G(\varphi(t_2),t_2)=S+{\rm const.}