For the various ways of deriving the kinematic equations of simple harmonic vibration, there are high school textbooks based on scientific inquiry, there are also Feynman's way of reasoning from dynamic ordinary differential equations combined with intuition, and Mr. Zhao Kaihua's mathematical calculus based on the law of conservation of mechanical energy.
However, the beauty of the simple harmonic vibration is not only reflected in the multiple derivations of the formula, it also has some strange relationships, which we will talk about today.
We have already seen that the relationship used to describe the change in oscillator displacement over time can be written in the form of the following sine (or cosine) function, i.e
Not only that, but we will also see another statement in the book that the kinematic equations of simple harmonic vibration can also be expressed in the following complex exponential form, that is
Seeing the above two statements, I don't know if everyone feels strange, I think at least two aspects are quite strange:
The simple harmonic vibration of the spring oscillator is a one-dimensional problem, and the trigonometric function is born from trigonometry, which is a geometric problem with a two-dimensional "gene" in the bones. Is there any geometric connection between them?
The displacement of the spring oscillator away from the equilibrium position should be a real number, how can there be an imaginary number in the expression?
In fact, this matter has to start with Euler's formula and the complex plane.
Beautiful Euler formula
Let's start with Euler's formula. Euler's formula is generally expressed as:
It is known as one of the most beautiful formulas in the history of mathematics. Why? Because when we make the above equation, we will get a magical equation, that is, .
Where is its beauty? Because it relates the two most important irrational numbers ( , ) , the most important imaginary numbers ( ), the most important two elements ( 0 , 1 ) , the most important operator ( ) , and the most important relational character ( ) in an incomparably reduced way. It's truly remarkable, and we can't help but wonder if this is a precious page in God's creation book.
After admiring Euler's formula, we have to see what kind of relationship it has with simple harmonic vibration.
Complex numbers versus points on the complex plane
Now let's focus on the part to the right of the equal sign of Euler's formula. As you can see, to the right of Euler's formula is the sum of two terms, one is a real number and the other is an imaginary number. Mathematically, we call numbers of this form ( where all are real numbers ) complex numbers. Where the real part called the complex number, the imaginary part called the complex number, is called the imaginary unit.
We all know that any real number can be represented by a corresponding point on the axis of real numbers, so can a complex number not be? Plurals are also possible. However, complex numbers are not the same as real numbers. Real numbers only need points on one dimension, and to determine a complex number, it is necessary to determine both the real part and the imaginary part (that is, the sum), so the complex number must be represented by two number axes to form a plane, one of which is used to determine the real part, and the other number axis is used to determine the imaginary part. We call such a special plane a complex plane.
In this way, any complex number can find a point-to-one correspondence on the complex plane, as shown in the following figure.
The figure above uses two coordinates to represent complex numbers: one is in the form of cartesian coordinates such as points; the other is in the form of points, which are polar coordinates, where the modulo called complex numbers is called the amplitude angle of complex numbers.
Two coordinates describe the same thing, so the two coordinates have to be related, right. It is easy to see that the correspondence between the two coordinates can be reflected by the following two equations, i.e
Let this point go in circles
The dead spot is not fun, it is interesting to make it move. Let's say we make this point spin in a circle at a uniform speed. Like this below.
Since the horizontal and vertical coordinates of the points are marked, we can intuitively see that the points change periodically during the process of uniform rotation. They take positive values, negative values, larger, and smaller, and the size of the two is reversed, a bit of a sine function and a cosine function. Isn't simple harmonic vibration expressed in sine or cosine functions? Good fellow, is it vaguely felt that the uniform circumferential motion seems to have a little ambiguous taste with the sine and cosine function.
In order to characterize the uniform circular motion of this point in more detail, we use polar coordinates to express Cartesian coordinates. Suppose that at the point of time at any position in the circumference of the radius, the spoke angle is , so that the angular velocity of the uniform rotation of the point is . Then after the time has passed, the angle value of the point turn is. In this case, the point should be at the position of the Cartesian coordinate after any time has passed
The connection between circular motion and simple harmonic vibration
Haha, isn't the above equation an expression of simple harmonic vibration? In order to visualize the changes in the horizontal and vertical axes of uniform circular motion, I made the following picture and GIF. The picture shows the components of the position coordinates of the particles at any time on the respective horizontal and vertical axes, and it can be seen that these components are changing in the manner of the sine and cosine function over time.
Even so, we can make a GIF that relates the uniform circular motion to the simple harmonic vibration. It looks like this:
As can be seen from the movement:
When the point is moved at a uniform speed around the circumference of the radius, the horizontal and vertical axes change with time, which is consistent with the simple harmonic vibration;
There is a real intuitive sense in the vibration function, that is, the initial spoke angle corresponding to the uniform circumferential motion at the zero moment, which we call the initial phase, that is, the initial position of the particle. It is the angle of rotation on the basis of the initial spoke angle corresponding to the uniform circumferential movement;
The so-called amplitude is actually the trajectory radius corresponding to the uniform circular motion.
In the past, when learning simple harmonic vibrations, because there is no such GIF to assist in understanding, we are always confused about what phase and initial phase mean, and why they should take this name. Now, with intuitive cognition, the understanding of concepts can become deeper.
Complex exponential representation of simple harmonic vibration
Let's talk about another function of Euler's formula, that is, it can be used as a plural to represent simple harmonic vibrations. Let's replace it with Euler's formula to get the following form:
What does this equation mean? It shows that, as a complex number of points on a complex plane, its real or imaginary parts can be used to represent a simple harmonic vibration. Why? Because the calculation encountered when using the exponent to discuss some complex problems will be more convenient than using trigonometric functions, such as damped vibration, forced vibration and other issues.
Final words
From the small talk above, we can see that Euler's formula is like an adhesive, connecting simple harmonic vibrations with circular motion, allowing us to establish connections between different types of motion. Moreover, Euler's formula also provides a chic idea for the study of vibration problems, which can be analyzed with trigonometric functions in some problems, and perhaps more convenient with complex indices in others.
The method of combining numbers and shapes has always been a magic weapon to understand the laws and concepts of physics, and sometimes the concept that is difficult to express in a sentence only needs to be given a scene and an animation, and the meaning of the concept can be easily obtained, giving people a sense of pleasure that silence is better than sound at this time.
The reproduced content represents the views of the author only
Does not represent the position of the Institute of Physics, Chinese Academy of Sciences
Source: Bear said slowly
EDIT: just_iu