ActionIntegral

In physics, the action integral $S$ is a functional that takes a trajectory $q(t)$ (or a set of trajectories $q_i(t)$ for multiple degrees of freedom) and maps it to a real number. The action integral is defined as the integral of the Lagrangian $L$ over time $t$:

$$ S[q(t)] = \int_{t_1}^{t_2} L(q(t), \dot{q}(t), t) dt $$

Here:

• $S[q(t)]$ is the action, a scalar quantity.
• $L(q(t), \dot{q}(t), t)$ is the Lagrangian of the system, a function of the generalized coordinates $q(t)$, their time derivatives $\dot{q}(t)$, and possibly time $t$ itself.
• $t_1$ and $t_2$ are the initial and final times, respectively.
• $\dot{q}(t)$ represents the time derivative of $q(t)$.

For systems with multiple degrees of freedom, the Lagrangian $L$ is a function of multiple coordinates $q_i(t)$ and their time derivatives $\dot{q}_i(t)$, and the action integral becomes:

$$ S[q_1(t), q_2(t), \ldots, q_n(t)] = \int_{t_1}^{t_2} L(q_1(t), \dot{q}_1(t), q_2(t), \dot{q}_2(t), \ldots, q_n(t), \dot{q}_n(t), t) dt $$

The principle of least action (or Hamilton's principle) states that the actual trajectory $q(t)$ taken by the system between times $t_1$ and $t_2$ is the one that makes the action integral $S$ stationary (usually a minimum). Mathematically, this is expressed by the Euler-Lagrange equations:

$$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0 $$

for each generalized coordinate $q_i$.