The Principle Of Least Action

The principle of least action

The principle of least action is a powerful idea in classical physics that offers a new perspective compared to Newton’s laws. Instead of focusing on forces acting at an instant, I look at the trajectory a system follows as one that makes a particular quantity, the action, stationary.

For a system described by generalized coordinates q_1(t), \dots, q_n(t), I define the action S as:

S = \int_{t_1}^{t_2} \mathcal{L}(q_1, \ldots, q_n, \dot{q}_1, \ldots, \dot{q}_n, t)\, \mathrm{d}t

where \mathcal{L} is the Lagrangian, usually given by the difference between kinetic and potential energy.

To see how equations of motion come from this, I consider a small variation \delta q(t) to the path, where the endpoints are fixed. The first-order variation in the action is

\delta S = \int_{t_1}^{t_2} \left( \frac{\partial \mathcal{L}}{\partial q} \delta q + \frac{\partial \mathcal{L}}{\partial \dot{q}} \delta \dot{q} \right) \mathrm{d}t

Applying integration by parts and requiring the action to be stationary for arbitrary \delta q(t), I arrive at Lagrange’s equation:

\frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \frac{\partial \mathcal{L}}{\partial q} = 0

One of the main advantages is the ease of handling constraints. Instead of eliminating specific variables, I introduce Lagrange multipliers to enforce constraints directly in the action and keep all coordinates independent during the variation.

Using the principle of least action not only unifies many aspects of classical mechanics, but also connects to other areas in physics, including quantum mechanics and field theory.

For more insights into this topic, you can find the details here.