BoJone has already spent a considerable amount of space in the previous “Natural Extremes” series to describe how universal “extremes” are in nature. They can guide our thinking on certain problems, leading to simple and quick solutions. Next, I want to mention an even more surprising “fact”: “Extremes” do not only provide us with creative thinking on certain mathematical or physical problems; they even construct the entirety of classical mechanics and even the whole of physics! This is no exaggeration. This is a principle in physics known as the “Principle of Least Action.” Many physicists (such as Feynman) were deeply attracted to it, even considering it the “ultimate formula of God’s creation of the world”! (Regarding the Principle of Least Action, you might want to read the series of articles “The Principle of Least Action and the Beauty of Physics” written by Fan Xiang).
In the 18th century, Euler and Lagrange opened a unique path, which was to use the calculus of variations to study classical mechanics. This breathed new life into classical mechanics and gave rise to a branch called “Theoretical Mechanics” or “Analytical Mechanics.” There are many benefits to studying mechanics using variational methods. The objects of variation are generally scalar functions. We only need to write the expressions for the kinetic and potential energy of a dynamical system to conduct a series of studies, such as listing the equations of motion of particles, judging the stability of equilibrium points, finding periodic orbits, and so on. (Since BoJone’s research into theoretical mechanics is not yet deep enough, I cannot provide too many examples, but please believe that its role far exceeds these). It saves a lot of tedious vector analysis, which was difficult to study before the invention of the variational method.
The content of theoretical mechanics is very extensive, and its foundation is the “Principle of Least Action.” For any type of mechanics, as long as a “least action” is found, the variational method can be used to construct the entire mechanical system! Therefore, in addition to studying classical mechanics, quantum mechanics, special relativity, and general relativity can all be studied using similar methods because we can write their respective actions. For example, the action of classical mechanics is very simple and can be written as: S=\int_{t_1}^{t_2} L(\vec{r},\dot{\vec{r}},t)dt
It should be noted that here \vec{r} no longer has the meaning of a vector; it is just a set of generalized coordinates (x_1, x_2, \dots, x_n). You might ask: What is L? L is called the “Lagrangian function.” Since this is just a “brief overview,” we only consider conservative systems, where the function L does not explicitly contain time t. In this case, we have: L=T-U T is the total kinetic energy of the system, and U is the total potential energy. Generally speaking, T=T(\dot{\vec{r}}) and U=U(\vec{r}), meaning T does not explicitly contain \vec{r}, and U does not explicitly contain \dot{\vec{r}}.
I have already written a lot of descriptions about theoretical mechanics, but introducing theoretical mechanics is not the main purpose of this article. For more content on variations and mechanics, you can refer to books such as “Variational Methods, Finite Element Methods and Extrapolation Methods” and “Mathematical Methods of Classical Mechanics.” This article mainly introduces some forms and special solutions of various “Euler-Lagrange equations” in the calculus of variations without proof (the proof method can be analogized to the last article of “Natural Extremes”), and briefly looks at their applications in mechanics.
I. Variational calculus of functionals containing derivatives of various orders (Higher-order)
If S=\int_{t_1}^{t_2} F(x,\dot{x},\ddot{x},\dots,x^{(n)},t)dt reaches an extremum (i.e., its variation \delta S=0), then the Euler-Lagrange equation is: \sum_{i=0}^n (-1)^n \frac{\partial^n}{\partial t^n}\left(\frac{\partial F}{\partial x^{(n)}}\right)=0 \tag{1} For n=1, we have: \frac{\partial F}{\partial x}-\frac{d}{dt}\left(\frac{\partial F}{\partial \dot{x}}\right)=0 For n=2, we have: \frac{\partial F}{\partial x}-\frac{d}{dt}\left(\frac{\partial F}{\partial \dot{x}}\right)+\frac{d^2}{dt^2}\left(\frac{\partial F}{\partial \ddot{x}}\right)=0 …
Since variational calculus containing higher-order derivatives is not very widely used in physics, we will not discuss it further here.
II. Variational calculus of functionals containing multiple variables (Multivariate)
Let \vec{r}=(x_1,x_2,\dots,x_n), and denote F(x_1,x_2,\dots,x_n,\dot{x}_1,\dot{x}_2,\dots\dot{x}_n,t) as F(\vec{r},\dot{\vec{r}},t). If S=\int_{t_1}^{t_2} F(\vec{r},\dot{\vec{r}},t) takes an extremum, then: \frac{\partial F}{\partial \vec{r}}-\frac{d}{dt}\left(\frac{\partial F}{\partial \dot{\vec{r}}}\right)=0 \tag{2}
It is necessary to give a simple explanation of the symbol \frac{\partial }{\partial \vec{r}}. Here \frac{\partial F}{\partial \vec{r}}=(\frac{\partial F}{\partial x_1},\frac{\partial F}{\partial x_2},\dots,\frac{\partial F}{\partial x_n}), representing a vector. Obviously, \frac{\partial F}{\partial \vec{r}} \cdot d\vec{r}=dF (\frac{\partial }{\partial \dot{\vec{r}}} can be analogized).
Therefore, the operation \frac{\partial }{\partial \vec{r}} can be seen as the inverse operation of the path integral, i.e., if \int f(\vec{r}) \cdot d\vec{r}=F(\vec{r}), then \frac{\partial F}{\partial \vec{r}}=f(\vec{r}), where f is a vector function.
Below are some very simple operational formulas: \frac{\partial}{\partial \vec{r}} (\vec{c} \cdot \vec{r})=\vec{c} (\vec{c} is a constant vector) \begin{aligned}\frac{\partial }{\partial \vec{r}}(\vec{r}\cdot \vec{r})=2\vec{r} \\ \frac{\partial |\vec{r}|}{\partial \vec{r}}=\frac{\partial }{\partial \vec{r}}(\sqrt{\vec{r}\cdot \vec{r}})=\left(\frac{1}{2\sqrt{\vec{r}\cdot \vec{r}}}\right)\times \frac{\partial }{\partial \vec{r}}(\vec{r}\cdot \vec{r})=\frac{\vec{r}}{|\vec{r}|}\end{aligned} Most other forms of operations can be analogized to general differentiation operations, so there is no need to explain them in detail ^_^
I specifically mentioned these three because the original functions and the results of these three operations can all be expressed in terms of \vec{r}, without the need for coordinate decomposition, which appears more regular.
Finally, we have: \frac{\partial }{\partial \vec{r}}[f(g(\vec{r}))]=\left(\frac{df}{dg}\right)\frac{\partial }{\partial \vec{r}}[g(\vec{r})]
III. Classical Mechanics
With the variational method, Newton’s classical mechanics can be rewritten as: \frac{\partial L}{\partial \vec{r}}=\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\vec{r}}}\right) \tag{3} The meaning of L has already been explained at the beginning of the article. Here \vec{r}=(x_1,x_2,\dots,x_n) is a set of generalized coordinates, meaning they do not necessarily have to be in a Cartesian coordinate system; polar coordinates, elliptical coordinates, etc., are all fine.
If you find the above form unattractive, we can rewrite it as: \begin{aligned}\vec{p}=\frac{\partial L}{\partial
\dot{\vec{r}}} \\ \dot{\vec{p}}=\frac{\partial L}{\partial
\vec{r}}\end{aligned} This is such a
symmetrical form of expression!! This shows the beauty of
physics!!
(For the proof method, please refer to tutorials such as “Theoretical
Mechanics”).
IV. Some special solutions
There is no general method to solve (2), but when certain restrictions are given to the function F, (2) can yield some first integrals.
The first case is when the function F does not explicitly contain time t. We can write: \begin{aligned} \frac{\partial F}{\partial \vec{r}}-\frac{d}{dt}\left(\frac{\partial F}{\partial \dot{\vec{r}}}\right)=0 \\ \frac{\partial F}{\partial \vec{r}}\cdot \dot{\vec{r}}-\frac{d}{dt}\left(\frac{\partial F}{\partial \dot{\vec{r}}}\right)\cdot \dot{\vec{r}}=0 \\ \left[\frac{\partial F}{\partial \vec{r}}\cdot \dot{\vec{r}}+\frac{\partial F}{\partial \dot{\vec{r}}}\cdot \ddot{\vec{r}}\right]-\left[\frac{d}{dt}\left(\frac{\partial F}{\partial \dot{\vec{r}}}\right)\cdot \dot{\vec{r}}+\frac{\partial F}{\partial \dot{\vec{r}}}\cdot \ddot{\vec{r}}\right]=0 \end{aligned}
The two terms in the brackets can be “contracted” into: \frac{dF}{dt}-\frac{d}{dt}\left(\frac{\partial F}{\partial \dot{\vec{r}}}\cdot \dot{\vec{r}}\right)=0
Integrating gives: F-\frac{\partial F}{\partial \dot{\vec{r}}}\cdot \dot{\vec{r}}=C \tag{4} where C is a constant. If we replace F with the Lagrangian L of a classical mechanical system, then this integral corresponds to the “Law of Conservation of Energy” in classical mechanics (interested readers can calculate it themselves).
The second case is when F does not explicitly contain any coordinate x_i (but allows \dot{x}_i to exist), then obviously: \frac{\partial F}{\partial x_i}=0, i.e., \frac{d}{dt}\left(\frac{\partial F}{\partial \dot{x}_i}\right)=\frac{\partial F}{\partial x_i}=0 Thus we obtain the integral: \frac{\partial F}{\partial \dot{x}_i}=C \tag{5} In this case, x_i is called a cyclic coordinate. The appearance of a cyclic coordinate corresponds to a first integral. Thus, solving the dynamics of a system becomes a problem of finding cyclic coordinates.
V. Conclusion
BoJone is limited in ability and has only briefly browsed the relevant textbooks on theoretical mechanics and the calculus of variations. Coupled with limited writing skills, I cannot fully display the beauty of mechanics to all readers. Therefore, readers are welcome to give advice, exchange ideas, and progress together to appreciate the infinite beauty of physics!! We want to tell everyone: Physics is not as boring or unfathomable as imagined, but is everywhere overflowing with a beautiful perspective!!
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