The Action Principles in Mechanics

Abstract

We begin this chapter with the definition of the action functional as time integral over the Lagrangian $L(q_{i}(t),\dot{q}_{i}(t);t)$ of a dynamical system:

$$\displaystyle{ S\left \{[q_{i}(t)];t_{1},t_{2}\right \} =\int _{ t_{1}}^{t_{2} }dt\,L(q_{i}(t),\dot{q}_{i}(t);t)\;. }$$

(2.1)

Here, q _i , i = 1, 2, …, N, are points in N-dimensional configuration space. Thus q _i (t) describes the motion of the system, and $\dot{q}_{i}(t) = dq_{i}/dt$ determines its velocity along the path in configuration space. The endpoints of the trajectory are given by q _i (t ₁) = q _i1, and q _i (t ₂) = q _i2.