Recently we have added $\LaTeX$ functionality (again) to this blog, and to demonstrate (show off) the easiness comes along, I shall write a short class note on the Principle of least action - from from physical analysis and mathematical exploration point of view.


The way we think about 'action' is a bit abstract. We know from middle school physics that $\vec{F} = m\vec{a}$ and work, as a consequence of motion along direction of force, is:
$$
dW = \vec{F} \cdot d\vec{x}
$$

Action is similar to work, but the integrand of action is Lagrangian $\times$ Time. Least action simply means, mechanical systems evolves so that the action is minimized (in fact we just need it to be a statinary point). Which mathematically would look like:
$$
S = \int_{t_1}^{t_2} \mathcal{L}dt
$$
Where Lagrangian $\mathcal{L}= T-U$, kinetic energy $-$ potential energy of the system.


d'Alembert Principle

One way of getting your head around the concept of action is to make sense of d'Alembert Principle first.

We shall start doing so by defining a few terms:

  • Degree of Freedom: any particle has 3 degree of freedom in vacumn, but constrains imposed to them might reduce that number. If we have a system of N particles, we shall have 3N degree of freedom without constrains.
  • Constrains and constrains force: Think of these as an extened catagory of normal force, they are there to prevent objects to move into certain direction, and their magnetude depends on the situation given.
  • Virtual displacement: $\delta\vec{r}$ is a vector of displacement. Imagine you freeze the time when a particle is traveling along a trajectory, now you imagine the particle is $displaced$ from its original path, that imaginary displacement can be expressed using this virtual displacement vector.
    • Virtual work $\delta W = \vec{F}_\text{total} \cdot \delta\vec{r}$

Now consider some particle going under some motion, we can denote the position of the particle $\vec{r}(t)$ for any moment in time. This is the true path of the particle, now imagine instead we let the particle to travel through a near-by trajectory instead the true path, we can express the new trajectory by:
$$
\mathbf{R}(t) = \vec{r}(t) + \vec{\epsilon}(t)
$$
Where $\vec{\epsilon}(t)$ goes from point on the true path to point our near-by path at any given moment.

Since the particle is going on a specified trajectory, we say it is constrained. The easiest way to depict this is to imaging a frictionless bob sliding on a curved surface. Now we want the near-by path to be a physically possible path, which means that $\vec{\epsilon}(t)$ has to be consistent with the constrains given. In our imaginary case of bob sliding on a surface, we can easily see that $\vec{\epsilon}(t)$ has to be parallel to the surface at any given moment. (Otherwise the particle either goes into the surface or fly away from it).

Recall the definition of the action and express Lagrangian using parameters we just defined:
$$
S = \int_{t_1}^{t_2} \mathcal{L}(\mathbf{R},\dot{\mathbf{R}},t)dt
$$
We want to show this integral is stationary respect to the change of $\vec{\epsilon}(t)$ when the path $\mathbf{R}$ is the true path we started with. Thus
$$
\delta S=S-S_0
$$
From here we can see that upon substitution and evaluation:
$$
\delta \mathcal{L} = \frac{1}{2}m((\dot{\mathbf{r}}+\dot{\mathbf{\epsilon}})^2-\dot{\mathbf{r}}^2)-(U(\mathbf{r}+\mathbf{\epsilon},t)-U(\mathbf{r},t)
$$