Physics+ Introduction: UNIZOR.COM - Physics+

Notes to a video lecture on UNIZOR.COM

Classic Physics+ Introduction

This course contains material not usually addressed in high school course of Physics. However, it's still a part of Classic Physics, and it's essential to understand the concepts presented here, as they play a very important role in contemporary Physics.

In the previous course Physics 4 Teens, part Waves, chapter Phenomena of Light, lecture Angle Refraction we have mentioned the intuitively understandable and natural Fermat's Principle of the Least Time.
Based on this principled we derived the optimal trajectory and angle of refraction of the ray of light going from one medium to another with a different refraction index.

Briefly speaking, if a ray of light moves along certain trajectory from point A to point B, the time it spends during this movement should be less than if it moved along any other trajectory.
If both points A and B are in empty space, the trajectory will be a straight line.
If, however, there are different media between them and, consequently the light propagates there with different speed, the Principle of Least Time can help to determine the angle of refraction on each change of medium along a trajectory to minimize the time to travel.

What's important to pay attention to in this phenomena is that there are many trajectories to reach point B from point A, but the ray of light chooses the one that minimizes certain numerical characteristic that depends on an entire trajectory - time of travel.

There is nothing wrong with application of Newton's Laws to find the trajectory of movement, but in many practical cases the complexity of such an approach is very high, so it would be quite a challenging endeavor.

Generalizing from the above example, the points A and B might not be real points in our three-dimensional Euclidean space, but some numerical characteristics of a state of a physical system under our observation. It can be a combination of spherical coordinates and velocities, for example, or positions relative to the center of our galaxy and impulses etc.
In any case, it's intuitively easy to accept that the change of a system from one state to another should be going along such a trajectory that minimizes or maximizing some numerical characteristic of an entire trajectory.

We will introduce a quantity that depends on an entire trajectory of movement of a physical system from one state to another. Stationary value (minimum, maximum, saddle point) of this quantity characterizes the trajectory of movement, which is similar to the Fermat's principle that a trajectory of the ray of light is the one that minimizes the time light travels from one point to another.

At the end of 18th century Italian-French mathematician Joseph-Louis Lagrange has developed exactly this theory, suggested a function that depends on system's characteristics and shown that finding the stationary point of this function leads to system of differential equations identical to Newtonian Laws.
This function was called action.
What was quite important, this approach to finding the trajectory of complex systems significantly simplified the calculations comparing to directly applying Newton's Laws.

Yet another approach, based on Lagrange work, was suggested by Irish mathematician and astronomer William Hamilton in 1833. His approach allowed to build a system that successfully bridged Classic Physics with Quantum one.

Details of both Lagrangian and Hamiltonian approach to formulate Classic Mechanics are the subject of this course.