The Action Principle

Consider a finitary, conservative dynamical system, i.e. one with a finite number of degrees of freedom, and in the absence of any dissipative effects. A fundamental question in classical dynamics is: given such a system in some initial configuration, how does it evolve in time?

The answer to this question is supplied by Newton’s laws. In particular, Newton’s second law of motion gives us a second-order differential equation that, given the configuration (i.e. a collection of generalised positions and velocities) at some time $ t _ {i} $, will uniquely determine the configuration of the system at any future time $ t _ {f} $. This is a local or differential formulation of classical dynamics: given the configuration of the system at an instant $ t $, the differential equation tells us how to construct the configuration of the system at an instant $ t + \delta t $, and so on. The path our system will trace through configuration space is composed of infinitesimal time steps, and the differential equation tells us how each step must be taken. An equivalent global or integral characterisation of this trajectory is provided by the action principle, which we now discuss.

We will momentarily restrict our attention to a system with a single degree of freedom. Say we are interested in the motion of this system from a position $ q _ {i} $ at time $ t _ {i} $ to a position $ q _ {f} $ at time $ t _ {f} $. Newton’s second law of motion, when integrated, specifies a trajectory $ q _ {c}(t) $ taken by the system. The principle of least action answers the following question: of all the possible trajectories $ q(t) $ the system could take, what makes $ q _ {c}(t) $ special?

This answer is constructed as follows. To every possible path $ q(t) $ connecting the points $ \left( q _ {i}, t _ {i} \right) $ and $ \left( q _ {f}, t _ {f}\right) $ in configuration space, we associate an action $ S[q(t)] $ defined as \begin{equation} \label{eq:action-lagrangian-defn} S[q(t)] = \int _ {t _ {i}} ^{t _ {f}} \mathrm{d} t\, L(q,\dot{q} ) \ . \end{equation} The function $ L(q,\dot{q} ) $ is called the Lagrangian. We’ll have more to say about it shortly. Writing the action (a global quantity associated to every trajectory connecting initial and final configurations) as the integral of the Lagrangian (a local quantity associated to each point in configuration space) is key to relating the local/differential and global/integral formulations of classical dynamics.

The action functional associates a number $ S[q(t)] $ to every possible trajectory $ q(t) $ in configuration space such that $ q(t _ {i}) = q _ {i} $ and $ q(t _ {f}) = q _ {f} $. The principle of least action states that the trajectory $ q _ {c}(t) $ extremises the action: \begin{equation} \delta S[q _ {c}(t)] = 0 \ . \end{equation} Writing the action $ S[q(t)] $ in terms of a Lagrangian $ L(q, \dot{q} ) $ allows us to translate the principle of least action — global principle — into a differential equation, i.e. a local formulation of the same.

Let us recall how this is done. For a general trajectory $ q(t) $, the variation of the action induced by the change $ q(t) + \delta q(t) $ is \begin{equation} \delta S = \int _ {t _ {i}} ^{t _ {f}} \mathrm{d} t\, \left( \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q} } \delta \dot{q} \right) \ . \end{equation} Integrating the second term by parts yields

\begin{equation} \delta S = \left. \frac{\partial L}{\partial \dot{q} } \delta q \right\vert_ {t _ {i}} ^{t _ {f}} + \int _ {t _ {i}} ^{t _ {f}} \mathrm{d} t\, \left( \frac{\partial L}{\partial q} - \frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial L}{\partial \dot{q} } \right) \delta q \ . \end{equation}

The boundary conditions we will impose are $ \delta q(t _ {i}) = 0 = \delta q(t _ {f}) $, which is the statement that the positions at times $ t _ {i} $ and $ t _ {f} $ are fixed to be $ q _ {i} $ and $ q _ {f} $ respectively. We find then that the first term drops out and that $ \delta S = 0 $ for arbitrary $ \delta q $ if and only if \begin{equation} \frac{\partial L}{\partial q} - \frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial L}{\partial \dot{q} } = 0 \ . \end{equation} This is the Euler-Lagrange equation of motion.

The form of the Lagrangian is decided by the requirement that when the two can be compared, the Euler-Lagrange equations of motion must reproduce Newton’s second law of motion. For example, the Lagrangian for a particle of mass $ m $ moving in one dimension subject to a potential $ V(q) $ is described by the Lagrangian \begin{equation} L = \frac{1}{2} m \dot{q} ^{2} - V(q) \ , \end{equation} and the corresponding Euler-Lagrange equation of motion is \begin{equation} m \ddot{q} + \frac{\partial V}{\partial q} = 0 \ . \end{equation} Indeed, on defining the canonical momentum conjugate to $ q $ as \begin{equation} \label{eq:canonical-momentum-particle} p = \frac{\partial L}{\partial \dot{q} } \ , \end{equation} we can rewrite this equation of motion as \begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} p = - \frac{\partial V}{\partial q} \ , \end{equation} which is Newton’s second law of motion: the rate of change of momentum of a body is determined by the force (here, the negative gradient of a potential) acting on it.

While the above discussion was for a single degree of freedom, it is easily generalised to any finite number of degrees of freedom in the following manner: let the index $ a $ run over the $ N $ degrees of freedom of a finitary system. If the generalised coordinates are independent, so too are their variations, and the Euler-Lagrange equations of motion are \begin{equation} \label{eq:el-equations} \frac{\partial L}{\partial q _ {a}} - \frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial L}{\partial \dot{q} _ {a}} = 0 \quad \mathrm{for} \quad a \in \left\lbrace 1, \cdots , N \right\rbrace\ . \end{equation}