The Principle of Least Action is the most dense form of classical mechanics. This rule is omnipresent across all spheres of physics, including classical physics, quantum mechanics and so on.
In order to understand this principle, there are a few prerequisites we will need to cover first. Over the next few articles, we will be touching upon the following:
- Stationary points and minimizing functions
- System of more than one particle
- Energy
- Calculus of Variations
- Euler-Lagrangian Equation
Today, we are going to take a look at stationary points and minimizing functions.
- Stationary Points and Minimizing Functions
To begin with, consider a function F of y (Figure 1.1),
You notice that there are points on the curve where a shift in y always translates in the upward direction and these points are known as Local Minima. Each dot is at the bottom of the little depression. The lowest possible place on the curve is global minimum.
The condition for a local minimum is that the derivative of the function with respect to the independent variable at that point is zero. This is a necessary condition, but not a sufficient condition. This condition defines stationary point.
The second test is to calculate the second derivative to examine the character of the stationary point. If the second derivative is greater than zero, then the nearby points will be above the stationary point and we would have a local minimum.
If the second derivative is less than zero, then the nearby points will be below the stationary point and we would have a local maximum (Figure 1.2).
If the second derivative is equal to zero, then the derivative changes from positive to negative at the stationary point. This point is called the point of inflection (Figure 1.3).
Stationary points in higher dimensions :
Local minima, Local maxima and stationary point can happen for function of more than one variable. Imagine a hilly terrain (Figure 1.4), the altitude A is a function that depends on the two variables, latitude and longitude, A(x,y). The top of the hills and the bottom of valleys are local maxima and local minima of A(x,y), but they are not the only places where the terrain is locally horizontal. Saddle points (The very top of hills; these are places where no matter which way you move, you soon go down and vice-versa) occur between two hills.
Let’s take a slice along the x axis through our space so that the slices pass through a local minimum of A (Figure 1.5),
It’s very clear that at the minimum the derivative of A with respect to x vanishes. We write this as:
On the other hand, had the slice been oriented along the y axis, then we would have concluded with:
If there were more directions of A, in which it could vary, then the condition for a stationary point is given by:
These equations can be summarized. The change in a function when the point x is varied is given by:
These sets of equations are equivalent to a condition that
for any small variations in x.
If suppose, we found such a point then “How do we tell whether it is maximum, minimum or saddle?”. The answer is to look at the second derivatives. However there are several second derivatives. For the case of 2 dimensions, the last two being the same.
These partial derivatives are often arranged on a spatial matrix called the Hessian Matrix.
Important quantities such as determinants and trace can be calculated as below:
Using the below definitions we can determine the local minimum, local maximum and saddle points:
- If Det H and Tr H are positive, then the point is a local minima.
- If Det H is positive and Tr H is negative, then the point is local maxima.
- If Det H is negative, then irrespective of the value of trace, the point is a saddle point.
I have illustrated the above with a simple example.
In the next article, we will discuss “System of more than one particle”. Thank you for reading.
