Newtonian Mechanics: Historical View
Newton started physics, but there's a little story behind it.
Ordinary Introduction
$$\vec{F}=\vec{P}$$
The above equation defines two new things.
1. Vector: The physical quantities force and are defined as vectors, which are a collection of three numbers.
2. Differential equation: An equation in which differential operators are introduced.
To make the above possible, Newton first introduces an inertial coordinate system.
This is a textbook introduction, and we will approach it in a different way.
Of course, the final destination is the same.
How did humans invent physics?
Physics is often called ‘the law of everything.’
It refers to a simple, most fundamental law that applies to all things.
Look at the world. It's so complicated.
The strategy we will adopt:
1. Isolate the object of interest from the world.
2. Let's find out the essence of the object, and then how it interacts with the world.
3. By repeating this process, you can understand everything in the world.
(Here, the belief that the essence of an object lies within oneself is based on the Western analytical way of thinking. It is the opposite of the Eastern harmonious way of thinking.)
What do we want to see?
What characteristics apply to everything?
Let's say an unknown object appears. Are you afraid of this object?
Let's change the question: Are you afraid if it stay still or if it move?
Movement is a phenomenon that humans instinctively try to understand.
Because understanding the movements of animals, the movement of throwing stones, the movement of falling fruit, and the movement of the sun and moon is essential for survival.
This is not just a characteristic of humans; it is a characteristic that all animals have.
Because humans evolved to use tools, a deeper understanding was needed.
Be thankful to your ancient ancestors.
Mathematics
The discovery of the axioms that make up the ideal world of geometry led to the search for something similar in the real world.
But it wasn't easy.
It didn't appear until mathematics became much more advanced, effectively blending algebra and geometry.
Algebra is the study of creating equations with unknowns and solving them. It was invented to manage resources.
The invention of numbers is great, but the invention of unknowns is so great. This is because the ‘number you want to know’ is written as an unknown number and the equation is written first.
The ability to solve equations is the ability to find out what you don't know, and is equivalent to the level of human knowledge.
Geometry is the study of logic. It cannot be understood without deep thought, and it logically reveals the relationships between abstract points, lines, and planes.
But Euclidean geometry was difficult for everyone to handle.
Descartes' analytic geometry linked algebra and geometry. It turns geometry into equations. A situation with two unknowns is expressed on a coordinate plane, and the problem of finding a point is converted into a problem of solving an equation.
Just add calculus and we have the math to express the laws of motion.
Natural Philosophy
Humans has struggled for 2,000 years to find the axioms of nature.
Unlike mathematics, science is established through consensus.
Socrates, the originator of consensus, realized one thing during the debate: clear definitions of concepts are important.
His disciple Platon wandered around the world in shock after Socrates' execution.
Then, I came across leaked data from the ‘Pythagorean School’ created by Pythagoras, a disciple of Thales (forced open source).
Plato realizes the existence of Ideas (a complete world that contains the essence of the world somewhere outside this world).
He creates the Platonic Academy and begins providing education.
Mechanics was one of the essences he discovered. This was the first natural law based on observation.
But this is not a good rule. Because it's useless.
Usability is based on ‘exact value’. In other words, you have to use numbers. However, mathematics exists within ideas. The philosophy that the essence of nature is expressed through mathematics is 'Neo-Platonism'.
After a long time, Galileo succeeded in an accurate experiment and found an 'equation' that describes free fall. He also came up with the primitive concept of ‘inertia’ through the oblique plane thought experiment. But he thought that the most basic motion was circular motion. Celestial bodies move in circles in empty space. Since the Earth rotates, objects on the Earth's surface that remain still also move in a circular motion.
Descartes Again!
Descartes appears again. While thinking about the universe, he came up with a causal worldview. Everything has a cause and effect.
He also thought about exercise. In our view, movement is not a different type of movement just because it is still, flying, or stopping. Movement is a relative concept, and is just a phenomenon of changing relative positions. Even if it appears stationary to some, it appears to be moving to others.
He also thought about the causes of the movement. However, there is no specific cause. Without external interference (cause), an object will move in a straight line at a constant speed, and there is no cause for that motion. (In fact, Descartes said that the cause is God outside the world.) He completed the concept of inertia. .
Descartes also envisions the concept of momentum. It is said that when objects collide, the motion is transferred to the object that collided, and that much is lost. This is because if you look at the objects as a group, the overall momentum must be constant. It defines momentum, which represents the ‘degree of movement’ that an object inherently possesses.
Descartes made progress because he was able to think without gravity, but at the same time, he had limitations. He thought that changes in the state of motion occurred only through collisions, so he failed to account for the 'continuous' change in momentum caused by gravity.
Newton finally apear: Gravity changes momentum
Descartes' philosophy can be summarized as follows.
$$P(t')=P(t)+I$$ when $P$ is momentum and $I$ is external action, also lose of external momentum.
Descartes thought that changes in motion were discontinuous.
On the other hand, Newton saw the apple accelerating and thought that gravity was also an external force and that it 'continuously' changed the apple's momentum. In other words, the falling motion measured by Galileo was thought to be a motion in which momentum increases steadily under gravity. Recognizing the existence of gravity, which transfers momentum even without contact, it was natural that celestial bodies floating in the sky would also be subject to gravity. That's how Newtonian mechanics began.
Momentum is Vector
$$P=m v\vec{\alpha}$$
when $P$ is momentum of object, $m$ is amount of object, $v$ is speed, $\vec{\alpha}$ is moving derection.
Force is differential of Momentum
$$P(t+o)=P(t)+oA + O(o^2)$$
when $o$ is very small, $$[P(t+o)-P(t)]/o = A $$
Define $A$ as force, $F$.
Equation of motion connect state
State is defined as locatin $x$ and momentum $p$ for each time.
EOM $F(x,p,t) = dp/dt$ connect phase space $(x,p)$ with time changes.
Solving EOM means find that path connection.