The Principle of Least Action

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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

Stanford Video Lecture of Leonhard Susskind on the Principle of Least Action