This lecture note will make your life ten-fold easier in the scope of the problems it addresses. Consider it a talisman. I discovered this a couple of weeks ago when I was solving these problems for my own conceptual understanding. So I waited till I can completely enunciate the big picture. When I confirmed that its valid for all the following problems I made this note and sharing with you.
According to classical mechanics — from 400 year old knowledge of physics, space and time have been considered to be separate entities but not fundamental. Space and time were not considered fundamental quantities as these parameters did not have the mandate to change physical laws of nature. They were merely the tools of the mind or the more aptly the conscious mind. They help us visualize but not dictate what ought to happen.
All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.
First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.
fixed: which changes only if estimated wrong.
That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics.
We need to understand first that Force can be categorized into two types. One is called tangential or collinear force. This component of the force is always along the direction of motion and changes speed of an object. It can change direction once the velocity of the object has become zero. Its NOT centripetal force. It can never make an object go in a plane or 3 D trajectory, as the motion is limited to only one dimension. The object can only go back and forth.
Now look at the other component. Its called a radial force. Its always perpendicular to the direction of motion. This force is called centripetal force, always. Note that its different from what we call central forces.
In consequence, both tangential and radial forces can be central.
This fact is inherently, nothing but the concept of Quantum Entanglement. Unlike classical particles which are completely independent of each other the quantum sized objects are not. For physical intuition they are at a very small and subtle level intertwined with each other as if threads are kind of tied onto each other. We can-not toss an electron because the whole concept of one electron completely independent of all other makes sense only when all other electrons are not interacting because they are arbitrarily far away.