Time-Dependent Classical Lagrangian with variation of time

Hello everyone!

I was reading the following review:

http://relativity.livingreviews.org/...ticlesu23.html

And I got stuck at the first equation; (10.1)

So how I understand this is that there are two variations,

[itex]\tilde{q}(t)=q(t)+\delta q(t) \hspace{1cm} \text{and} \hspace{1cm} \tilde{t}=t+\delta t [/itex]

Further we also have a `total variaton' for q at first order:
[itex]\tilde{q}(\tilde{t})=q(t)+\delta q(t)+\dot{q}(t)\delta t [/itex]

and its derivative,
[itex]\dot{\tilde{q}}(\tilde{t})=\dot{q}(t)+\delta\dot{q}(t)+\ddot{q}(t) \delta t [/itex]

So now how is [itex]\delta L(q,\dot{q},t)[/itex] defined?