Βιβλιογραφία
A particle with position $x(t)$ moving from time $t_1$ until time $t_2$ between two points $x_1=x(t_1),\, x_2=x(t_2)$ is following a trajectory that minimizes the functional
$$ I = \int_{t_1}^{t_2} L(x,\dot{x},t)\,dt $$
called the action.
Lagrangian
The function $L$ is called the Lagrangian and it is defined through the energy of the system
$$ L = T - V $$
where $T$ is the kinetic energy and $V$ is the potential energy.
Example
Assume the energy of a free particle. Its Lagrangian is
$$ L(\dot{x}) = T = \frac{1}{2} m \dot{x}^2.
$$
Its equation of motion is derived by minimizing the action