The principle of least action is not merely a mathematical tool or a historical insight—it represents one of the most elegant ideas in physics. Rather than describing nature’s behaviour through brute force calculations, it tells us something profound: nature follows the most efficient path, optimising a quantity known as the action.

Whether you’re new to theoretical physics or brushing up on classical mechanics, this post guides you through a modern and accessible derivation of this powerful principle, concluding with the celebrated Euler-Lagrange equations.

1. What is the Action?

Suppose we have a system described by a coordinate $q(t)$ that evolves over time. The dynamics are encoded in a function called the Lagrangian, written as: \(\mathcal{L}(q, \dot{q}, t)\) which typically represents the difference between kinetic and potential energy.

The action functional $\mathcal{A}[q]$ is the time integral of this Lagrangian: \(\mathcal{A}[q] = \int_{t_1}^{t_2} \mathcal{L}(q(t), \dot{q}(t), t)\, \mathrm{d}t\)

Hamilton’s principle states that the path a system actually takes makes this action stationary (i.e. a minimum, maximum, or saddle point).

We assume:

• The functions $q(t)$ and $\dot{q}(t)$ are sufficiently smooth (continuously differentiable).
• The endpoints $q(t_1)$ and $q(t_2)$ are fixed for the purpose of variation. This leads to the condition: \(\delta q(t_1) = \delta q(t_2) = 0.\)

2. Wait – Why do we Fix the Endpoints?

This is a common source of confusion. Setting $\delta q(t_1) = \delta q(t_2) = 0$ is not a physical statement about knowing both endpoints in a real-world problem. Rather, it is part of the mathematical structure of the variational derivation. Let us clarify this in the context of commonly encountered problems in science and engineering:

• Initial Value Problems (IVP): The initial position and velocity are specified, e.g., $q(t = 0)$, $\dot{q}(t = 0)$.
• Boundary Value Problems (BVP): The value of $q$ is prescribed at two spatial boundaries, e.g., $q(x = x_1, t)$, $q(x = x_2, t)$, at all times.
• Boundary-Initial Value Problems (BIVP): Both initial conditions $q(x, t = 0)$, $\dot{q}(x, t = 0)$ and boundary conditions $q(x_1, t)$, $q(x_2, t)$ are given.

Regardless of the type of problem, variational theory always considers two arbitrary time instants, $t_1$ and $t_2$, and assumes the system lies on the physical (equilibrium) path at those times. The goal is to explore what happens in between.

In IVPs or BIVPs, as a special case where $t_1 = 0$, we actually know the state of the system from the initial conditions. Meanwhile, boundary conditions impose further constraints: since $q(x = {x_1, x_2}, t)$ is known for all $t$, the variation $\delta q(t)$ must vanish at those boundaries.

In this way, the variational formulation and the principle of least action naturally generalise to and remain consistent with the kinds of problems typically encountered in science and engineering.

3. Variation of the Action and Euler-Lagrange Equation

Imagine nudging the path $q(t)$ slightly: \(q(t) \rightarrow q(t) + \epsilon \eta(t)\) Here, $\epsilon$ is a small parameter and $\eta(t)$ is any smooth function that vanishes at the endpoints: $\eta(t_1) = \eta(t_2) = 0$. This keeps the start and end points fixed while testing neighbouring paths.

Then the varied action becomes: \(\mathcal{A}[q + \epsilon \eta] = \int_{t_1}^{t_2} \mathcal{L}(q + \epsilon \eta, \dot{q} + \epsilon \dot{\eta}, t)\, \mathrm{d}t\)

Expanding to first order in $ \epsilon $: \(\delta \mathcal{A} = \left. \frac{\mathrm{d}}{\mathrm{d}\epsilon} \mathcal{A}[q + \epsilon \eta] \right|_{\epsilon=0} = \int_{t_1}^{t_2} \left( \frac{\partial \mathcal{L}}{\partial q} \eta + \frac{\partial \mathcal{L}}{\partial \dot{q}} \dot{\eta} \right) \mathrm{d}t\)

Integrate the second term by parts: \(\int_{t_1}^{t_2} \frac{\partial \mathcal{L}}{\partial \dot{q}} \dot{\eta}\, \mathrm{d}t = \left[ \frac{\partial \mathcal{L}}{\partial \dot{q}} \eta \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) \eta(t) \mathrm{d}t\)

Due to the boundary condition $\eta(t_1) = \eta(t_2) = 0$, the boundary term vanishes, yielding: \(\delta \mathcal{A} = \int_{t_1}^{t_2} \left[ \frac{\partial \mathcal{L}}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) \right] \eta(t)\, \mathrm{d}t\)

Since this must hold for all admissible $\eta(t)$, the integrand must vanish identically: \(\boxed{\frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \frac{\partial \mathcal{L}}{\partial q} = 0}\)

This is the Euler-Lagrange Equation, an elegant expression which encodes the full dynamics of the system. It is derived not by pushing forward in time, but by requiring that the entire trajectory be optimal.

4. Why This Matters

The principle of least action unifies the laws of physics under a single, beautifully general framework. It works not just in mechanics, but in optics, electromagnetism, relativity, and even quantum field theory.

It doesn’t just explain how things move. It tells us why they move that way.

And that, in its own quiet way, is extraordinary.