# Least Action Principle

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)

$$

### Subscribe to The WOrLd

Get the latest posts delivered right to your inbox