1.Classical Mechanics
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an elementary depiction about Newton’s mechanics: 质点力学
1.1. Basic Notions and Their Presupposition
space and time
At present, when we talk about something occur, means that it must contains the properties of time and space, or it must occur in the space-time specifically. But, we never say that a time event or a space event occurs alone. We don’t care such a event at present.
No matter what time and space is, however, we can get a sufficient description of them that correspond with our physical concept, because we can always describe phenomenon definitely, in despite of we don’t know the essential of it: (ref: Geometry)
1. an affine space.
2. a translation group act to the affine space
3. Euclid structure upon the translation vector space.
4. Galileo structure upon the affine space
Each step imply some physical presuppositions:
1. point: time point and space point.
Just a point is undistinguishable.
2. there is a distance between two different points.
How to know the distance? or how to describe the distance?
There are four forms of distance: the time distance at the same space points; the time distance at different space points; the space distance at the same time point; the space distance at different time points.
To measure these forms of distance, we have to induct some physical presuppositions:
the time distance at the same space point: translation invariant of time.
->the time distance at different space points: simultaneity.
->the space distance at the same time point: rigid body.
->the space distance at different time points: …
3. there is the property of direction in space and time.
We can ask a question: what does direction means? Just like distance, since we can’t consider a space event or a time event at present, we have to consider other occurrences that embody the direction of space or time.
Let’s study the distance and the direction from a simplest occurrence: a point x(x, t_1), and a point y(y. t_2).
x . . y
we say “at the time point t_1, we find a particle is at the space point x, at the time point t_2, we find the same particle is at the space point y, if we can identify they are the same”.
or, we can describe the same occurrence as “a particle move from x to y”. then, this is become a typical physical occurrence! Means that we can definite the direction of time and space in such a occurrence, furthermore, we can ask the first physical question: how the particle move from x to y?
Just experiences can answer it!
If the time and the space satisfy our primary rational number describe on them, then we can use interpolation to detect the process of the particle movement: Let’s repeat that occurrence, but add a screen to isolate x and y absolutely, and drilling a hole z on the screen, no matter where is z, when the occurrence is repeat successfully, we can claim that the point z is an intermediate occurrence between x and y. This is been required by our basic causality.
x . | z . y
From this experience, we get Newtonian mechanics and quantum mechanics respectively when we use bullet and photon to realize such a occurrence.
4. reference system
realization
5. velocity and acceleration
The velocity is the derivative of space over time.
The acceleration is the derivative of velocity over time.
momentum and force :: the conservation of momentum
As Newton’s second definition, momentum is used to measure the movement property of object, and as the fourth definition of Newton, the force is used to describe the cause of the movement of object.
the conservation of energy :: kinetic energy and potential energy
why Newton is right?
1.2 General Mechanics
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