Suppose I have a configuration space $ M $ and a lagrangian $ L: TM \rightarrow \mathbb{R} $. The action functional is defined as $$ \mathcal{S}[q(t)] = \int_0^T L(\dot q(t) ) \ dt , $$ The principle of least action states that the trajectory starting from $ q_i $ at time $ t_i $ and arriving at $ q_f $ at time $ t_f $ is the path minimizing the action subject to the boundary conditions $ q(t_i) = q_i $ and $ q(t_f) = q_f $.
My question is: when do solutions to the principle of least action exist? What conditions on $ M $ and $ L$ ensure that solutions exist for all boundary conditions $ (q_i, t_i) , (q_f, t_f) $?