# Can we justify the Principle of Least Action through considerations of symmetry?

The Principle of Least Action is quite important in Classical Mechanics and Classical Field Theory as it asserts which is the path of time evolution the system follows. In Classical Mechanics, for simplicity, it can be states as follows:

Let a classical system be given whose configuration manifold is $Q$ and whose lagrangian is $L : TQ\times \mathbb{R}\to \mathbb{R}$. The time evolution of the system between instants $t_1$ and $t_2$ is the path $\gamma : [t_1,t_2]\to Q$ which makes the action functional

$$S[\gamma]=\int_{t_1}^{t_2}L(\gamma(t),\gamma'(t),t)dt$$

stationary to first order.

This principle is very important as it directly affords the equations of motion and furthermore, provide as one immediate corolary the important result of Noether's Theorem, which is heavily connected to the study of symmetries.

Now, it is usually hard to justify the principle, other than saying: "we use it because it works".

On the other hand, in Classical Mechanics, it is possible to "arrive at the principle" by taking another route, starting from D'Alembert's principle.

My question here is: can we starting from considerations of transformations and symmetries justify the Principle of Least Action? Can we justify and arrive at this principle just by dealing with transformations and symmetries?

• Hot off the press. You may be interested in "The Lazy Universe: An Introduction to the Principle of Least Action" by Jennifer Coopersmith. It tackles your question and many others. It's a beast. – Sedumjoy Nov 16 '17 at 23:28