# Deriving Lagrange's Equations

1. Sep 9, 2010

### Zorba

I am having a problem with deriving Lagrange's Equations.

It's the derivation starting from Hamilton's Principle, the part where you consider the deviation from the path:

Suppose $$q=q(t)$$ is the function for which the action is a minimum, and now consider a deviation from $$q$$ where we replace it with $$q(t)+\delta q(t)$$ and then consider $$\delta S$$:

[itex]\delta S = \int^{t_2}_{t_1} L(q+\delta q,\ \dot{q}+\delta \dot{q},\ t)\ dt-\int^{t_2}_{t_1}L(q,\ \dot{q},\ t)\ dt[/tex]

And then the argument is made that $$\delta S = 0$$ as a necessary condition for $$S$$ to have an extremum.

So I don't entirely follow why this last part is. Hand & Finch write as the reason for this:

So now it seems sort of apparent to me mathematically why we must have this condition, but intuitively I'm still lost. I can see obviously in either case that if we have a maximum why it can't have $$\delta S>0$$ and likewise if it's a minimum it can't be that $$\delta S<0$$, but the other cases don't seem to follow immediately for me. Can anyone perhaps give an intuitive reason why this is? I'm sure I'm overlooking something simple.