That's because the Hamiltionian function is defined as the "Legendre Transformation" of the Lagrangian function or, in practice:
$$H = \sum \frac{\partial L}{\partial \dot{q_i}} \dot{q_i} - L $$ so the integrand function in (2.78) is nothing but the Lagrangian function. As you should know and...
Hi, I'm new here (and I'm not an english native speaker, so I apologize for any mistake), but I'm pretty sure you have to be more specific about your question. If you have so many doubts maybe a textbook could be more useful than a post written here.