Dynamics[0] | The Treasure Chest That Holds the Crown of Physics

In 1644, in order to decide who are going to lead the formulation of the new calendar, officials of the Imperial Observatory of the Qing government and missionaries from the West made a bet to see who could more accurately predict the upcoming solar eclipse. Of course, the answer had to be given in advance and verified when the solar eclipse occurred.

This does not mean that the missionaries and officials of the Imperial Observatory at that time were proficient in the falsifiability of scientific philosophy, but it is enough to illustrate the credibility of time and the screening power of Mr. Nan Guo who immediately said "Ah, yes, yes, I thought so" when the conclusion of science was given——

Standard form of dynamic equations

The ultimate goal of dynamics is to give a function of the physical quantity of interest changing with time $z=z(t,\dots)$, where time is at least one of the independent variables of this function.

To deal with the above time-varying functions, it is necessary to write the quantitative relationship that the above functions should satisfy based on experimental phenomena. Continuous functions generally contain derivatives with respect to time, and discrete series generally contain recursive relationships (such as Markov chains). The equation written is the dynamic equation.

The standard form of the dynamic equation is a combination of several differential equations, all of which are first-order derivatives with respect to time, and all appear only on the same side of the equal sign:

$$\begin{cases} \dot z_1=\mathrm dz_1/\mathrm dt = f_1(z_1,z_2,\dots z_n) \\ \dot z_2=\mathrm dz_2/\mathrm dt = f_2(z_1,z_2,\dots z_n) \\ \dots \\ \dot z_n=\mathrm dz_n/\mathrm dt = f_n(z_1,z_2,\dots z_n) \\ \end{cases}$$

If all the f on the right side of the equal sign are linear combinations of physical quantities, such a dynamical system is called a linear system, and the entire set of equations can be written as matrix multiplication $\dot{\vec z} = \frac{\mathrm d}{\mathrm dt} \vec z = \mathbb J\ \vec z$; otherwise it is called a nonlinear system.

The minimum number of standard form equations required to describe a dynamical system is called the degrees of freedom of the system, and a system with n degrees of freedom is called an n-dimensional dynamical system. A one-dimensional dynamical system is a single scalar equation $\dot z = \mathrm dz/\mathrm dt = f(z)$

When writing the above sentence, it feels a bit wrong, because the degrees of freedom of a general physical system is the number of variables minus the number of equations. Later, I thought of an explanation: $\dot z_n$ are also variables (generalized momentum), so n equations have 2n variables, and the degree of freedom is 2n - n = n

Another point to note is that the space discussed here is phase space (phase space), that is, the mathematical space spanned by the generalized coordinates of the dynamic system, and no longer the space where real matter exists as ordinary people say in daily life.

This difference is the reason why most physics popular science works fail to convey correct information to the public. Among them, there are also deliberate misleaders. To deal with such people, when you hear them talking about "space" and "dimension", just keep your hands firmly over your wallet...

What to do if the form is not standard

Of course, not all dynamic systems can be written in such a standard form——

If the dynamic function f itself contains time as a variable, then add a dimension $\dot t=\mathrm dt/\mathrm dt=1$ to the above equation

If there is a high-order derivative of a quantity with respect to time in the dynamic equation, n-1 intermediate variables can be introduced to turn an n-order differential equation into n first-order differential equations——

For example, Newton's second law, F = ma, acceleration is the second-order derivative of position with respect to time, so in order to fit into our discourse system, it is necessary to additionally define the first-order derivative of position with respect to time as an intermediate variable, that is, velocity v.

$$\begin{cases} \dot v=\mathrm d v/\mathrm d t = F/m \\ \dot x = \mathrm d x/\mathrm d t = v \end{cases}$$

If the above simplifications don't work, the higher-order derivatives with respect to time are still entangled with f. The function that z satisfies is called an implicit function. It may not look good to mathematicians, but for physicists, there is no need to bother with it since we are going to do numerical simulations anyway...

Why does mechanics stop at acceleration?

Because Newton's second law is a second-order differential equation, and we generally only discuss conservative force fields.

These two together lead to the fact that its standard form contains only two first-order time differential equations, and each real space dimension only needs two generalized coordinates (position $x$, velocity $v$), two generalized velocities (velocity $\dot x$, acceleration $\dot v$), and higher-order derivatives do not appear.

On the other hand, if the force field is not conservative, and the potential energy field itself contains higher-order derivatives of position with respect to time, then this dynamic system does require higher-order derivatives than acceleration. Such a system does not satisfy the law of energy conservation (there is external energy input or external work), but it does not mean that it cannot exist.

Previously, I saw a Zhihu article forwarded by the Institute of Physics of the Chinese Academy of Sciences on its WeChat account, "Newton's Second Law: There's Really No More to Say". The original article spent a lot of time in Hamiltonian mechanics and came up with a proof by contradiction that "energy has no lower bound".

This kind of theoretical game to prove Newton's Second Law gives people a feeling of being obsessed with theory...

Newton's Second Law is an axiom of classical mechanics, and axioms are self-evident (the original target "But when we ask our mechanics teacher questions, the answer we get is often 'This is a law summarized by experiments'" is the most correct explanation~)

As the starting point for proving other theorems, it is impossible to prove it within the logical system, and the proof must have a circular reasoning fallacy. (It seems that many people still think that logical closure is awesome, right~)

I'm sorry that I can't tell at which step the original article completed this smuggling. (The initial suspicion is that the direct use of the Euler-Lagrange equations for high-order systems is an analogy fallacy.)

This Zhihu article is the last of a trilogy, but the first two articles that the Institute of Physics did not forward are more educationally meaningful...

"The Four 力学s"

"Theoretical Mechanics", "Quantum Mechanics", "Electrodynamics", and "Thermodynamics and Statistical Physics" are four courses. Because the two Chinese characters "力 (force)" and "学 (study)" are in the course titles, they are collectively called "The Four 力学s" in China.

Previously, Dr. Cao Zexian wrote an article "The erroneous statement of "Four Major Mechanics" can be stopped" to try to get to the bottom of it.

These four courses are certainly not "the study of force", but these four courses are not without commonality. They are all dynamic systems discussed in this article.

• Theoretical mechanics studies the dynamics of position and velocity (momentum is used to connect with quantum mechanics and relativity more smoothly),
• Quantum mechanics studies the dynamics of probability amplitudes of position and velocity,
• Electrodynamics studies the dynamics of electric and magnetic fields,
• Thermodynamics and statistical physics should study the dynamics of thermal motion, but due to limited statistical tools, information theory is rarely taught in physics majors, and what can be written into textbooks is basically about the stationary points of dynamic systems, which does not seem to be very "dynamic".

As for saying that it can be stopped, I think it is just the opposite. The more such incomplete statements are, the more you can say after correction -

After all, who says that completely correct things are correct, so what point can I make by saying it?

It is the statements that are not completely correct and cannot be avoided that shows one's power of speech. Why are they called the four major mechanics? Because in Chinese, their roots are the same.

Of course, you should know that the "力学" here is dynamics, not study of forces, so it is right to persist, and it is also right to correct. Correction is for better persistence~

What's next

Some jargons in dynamics