The Euler-Lagrange equation and the principle of least action

In classical, non-relativistic particle mechanics the action of a system is given by
$$ S[\vec{q}(t)] = \int_{t_1}^{t_2} L(\vec{q}, \dot{\vec{q}}, t) ~ dt $$
which is a functional $S: F^n \rightarrow \mathbb{R}$, i.e. it maps the $n$ functions $\vec{q}(t) = (q_1(t),\dots,q_n(t)) \in F^n$ of some function space $F$ onto the real numbers. Continue reading “The Euler-Lagrange equation and the principle of least action”