# The Principle of Least Action

### From bio-physics-wiki

The most general form of the laws of motion result from the **Principle of Least Action** also called **Hamiltons Principle**. According to this principle each system is characterized by the so called Lagrange function $L$.

\begin{align} L(q, \dot{q} ,t) \end{align}

The action is defined by the integral of $L(q, \dot{q} ,t)$ from time $t=t_1$ to $t=t_2$. Hamiltons Principle states that the equations of motion are associated with a trajectory that minimizes the action along that curve.

\begin{align} \text{min } S = \int_{t_1}^{t_2} L(q, \dot{q}) dt \end{align}

We want to find the equations of motion, starting from $L$, thus our goal is to find the trajectory in phase space which minimizes the action. This is a standard problem of Variational Calculus. If we perturb the trajectory $q(t)$ by $\delta q$ and see how this variation changes the action.

\begin{align} \int_{t_1}^{t_2} L(q+\delta q, \dot{q}+\delta \dot{q}) dt-\int_{t_1}^{t_2} L(q, \dot{q}) dt \end{align}

The trajectory of minimal $S$ satisfies that the variation of the action $\delta S$ is stationary.

\begin{align} \delta S= \delta \int_{t_1}^{t_2} L(q, \dot{q}) dt=0 \end{align}

From Variational Calculus the solution called the **weak form of the Euler-Lagrange Equation** is

\begin{align} \delta S= \int_{t_1}^{t_2} \frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} dt=0 \end{align}

likewise we get for the **strong form of the Euler-Lagrange Equation**
\begin{align}
\frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}=0
\end{align}

For more than one generalized coordinate $q_i$ we derive by the same procedure the same equation for each dimension and get the **(Euler-)Lagrange Equations**

\begin{align} \frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}}=0 \end{align}

a system of differential equations from which we can derive the equations of motion.

Further Reading:

- L. D. Landau & E. M. Lifschitz - Lehrbuch der Theoretischen Physik I: Mechanik
- Herbert Goldstein - Classical Mechanics

Video Lecture:

- Stanford University, Leonard Susskind - Classical Mechanics | Lecture 3