in English and Chinese

Keying  Guan

Department of Mathematics, School of Science, Beijing Jiaotong University

Email: keying.guan@gmail.com

Abstract: This paper will further accurately describe the equivalence relationship between force and power, point out that Einstein gave the basic principle for calculating radiation pressure based on the relationship between force and power, and point out that due to the influence of the Maxwell-Bartoli radiation pressure formula, Arthur S. Eddington underestimated the important role of radiation pressure in the interior of stars.

Contents: I. Regarding whether force and power are vectors or scalars 

              II. Electromagnetic radiation intensity and light pressure, Einstein's important contribution to the                                                    theory of light pressure

                III. Why can't the word power be found in Landau and Lifshitz's great tome of physics?

                IV. Some inner understandings of human beings on "Power" and "Force" 

                V.  Summary

In the previous blog posts

1) 没有不做功的力--马德堡半球实验、举重运动消耗的卡路里及辐射压公式 2022-07-01, or https://blog.sciencenet.cn/blog-553379-1295413.html

2) Force is equivalent to Power 2022-07-02

3) Supplementary Explanation on "Force-Power Equivalence“ 2022-07-04

the author reveals the "equivalent relationship between force and power" that is not recognized by the academic circles, although it is contained in analytical mechanics. Due to the deep-seated problems involved in physics and mechanics, it is necessary to repeatedly think and demonstrate. The author finds that there is still room for improvement and further clarification in the published arguments, and also finds that there are interesting phenomena in the relevant descriptions of the academic community. These further show that the equivalence relation does exist and must be described with further precision.

I.  Regarding whether force and power are vectors or scalars

First of all, it should be pointed out that in the derivation of the mathematical expression (10) of the equivalence relation in the first blog post, the author ignored the explanation of the vector and scalar operation processing, which led to a wrong understanding of the formula (10)

that is, the force 1 N  (Newton) on the left side of the equal sign is conventionally regarded as a vector, and the power on the right side is regarded as a scalar. In response to this problem, the author points out that equation (10) must be understood as: if the power on the right-hand side of equation (10) is a scalar, the force 1 N (Newton) on the left-hand side of the equation should be regarded as the strength (scalar) of the force, If the force on the left-hand side of (10) is a vector, then the power 1 W  (watt) on the right-hand side of (10) needs to be treated as a vector that aligns with the direction of the force.

This understanding is in line with the actual situation:

1. Some forces and powers must be described using scalars.

For example, the light emitted by an ideal point light source (such as a very small light bulb, a star seen from a distance) can be measured in watts to measure the total intensity (total power) of its overall instantaneous luminescence, and it can also be measured in Newtons the total light pressure (force). This metered total luminous power and total light pressure have no direction problem and are both scalars.

2) Some forces and powers are more accurate and convenient to use vector descriptions.

For example, if the above-mentioned point light source shoots somewhere outside, a beam flow per unit area that is locally perpendicular to the light rays must have a power (area) density and will generate a local pressure on the area. At this time, the power density and pressure of the beam flow are obviously directional and should be regarded as vectors. If the unit area is a little local part of a sphere surface centered on the light source, then the second type of surface integral can simply be used to prove that the sum of the power densities passing through the sphere surface is equal to the total power of the light source, and it can also be proved that the sum of the pressures passing through the sphere surfaces equal to the total light pressure  of the light source.  When the sphere becomes a smooth closed surface around the light source, the surface integration of the second type above can still be performed, and it can be proved that as long as there are no other light sources or other obstacles in the closed surface, the integral values, namely the total power and total light pressure, remain unchanged. If only a scalar is used to describe the power or light pressure of the local beam, it will bring a lot of troubles and even lead to errors.

3) In many literatures, when referring to force and power in general, the description of directional force and power does not care whether a scalar or a vector is used.

For example, Wikipedie explains radiation pressure (https://en.wikipedia.org/wiki/Radiation_pressure) as

The energy flux (irradiance) of a plane wave is calculated using the Poynting vector , whose magnitude we denote by S. S divided by the speed of light is the density of the linear momentum per unit area (pressure) of the electromagnetic field. So, dimensionally, the Poynting vector is S=power/area=rate of doing work/area=ΔF/ΔtΔx/area, which is the speed of light, c=Δx/Δt, times pressure, ΔF/area. That pressure is experienced as radiation pressure on the surface:

where the intensity S (scalar) of the Poyintng vector S is explicitly used to describe the power (area) density of the radiation, while the symbol <S> is used when it is used to represent the pressure, and it is not stated whether this symbol and the corresponding pressure Pincident are a vector or a scalar. So, if P_incident  is considered as a vector, then the radiated power density <S> must be considered as a vector.

For another example, when discussing solar light pressure, Maxwell stated in his important work^[1], <A Treatise on Electricity and Magnetism (1St ed.), Vol. 2>, the beginning of §793 as follows:

Thus, if in strong sunlight the energy of the light which falls on one square foot is 83.4 foot pounds per second, the mean energy in one cubic foot of sunlight is about 0.0000000882 of a foot pound, and the mean pressure on a square foot is 0.0000000882 of a pound weight.

where obviously the radiant power density of sunlight (at the Earth's surface) is expressed using the imperial system, i.e.

The corresponding part in the German translation^[2] is

Nimmt man an, dass kraftiges Sunnenlight an einem Quadratmeter in der Secunde 124.1 Kilogrammeter Energie entwickelt, so warden in einem Kubikmeter Sonnenstrahlen 124/30000000 oder 0.00000041 Kilogram Drmeter Energie entachreuck sein, eine zur Fortplanzungstrichtung der Sonnenstrahlen senkrechte Flache pro Quadratmeter erleidet, 0.00000041 Kilogrammeter betragen.

which also is expressed in the more familiar unit system

The above citation also shows that Maxwell understood the sunlight energy received per cubic meter per second as the light pressure received, which is 0.00000042 kilograms per square meter, seriously underestimating the intensity of the solar light pressure.

At the same time, Bartoli independently used his own unique idea in his work^[3] on the same problem to give the power density and pressure of electromagnetic radiation.

Inheriting the ideas of Maxwell and Bartoli, Lebedev directly gave the Maxwell-Bartoli radiation pressure formula in the paper^[4] introducing his light pressure experiment

                   (1)

where, according to the custom at the time, V  represents the speed of light.

From Maxwell to Lebedev's expressions, it can be seen that the quantities on both sides of each equation  can be either viewed as both scalars, or viewed as both vectors.

II. Electromagnetic radiation intensity and light pressure, Einstein's important contribution to the theory of light pressure

As can be seen from the introduction in the previous section, from Maxwell to Lebedev, the light pressure is calculated by dividing the incident power density of the beam by the speed of light. The rationale for this may be the common power expression

       （2）

where P (Power) is power. These scholars apparently replaced the velocity v in equation (2) with the speed of light V  to get (1). 

The author already pointed out in the first blog post at the beginning of the introduction that if the speed of light V  is used instead of v, it means that the object affected by the beam is the light itself, which is meaningless. Other problems with equation (2) are also pointed out in the third blog post.

Interestingly, the authors found that the term power was hardly seen in the important physics papers in Einstein's corpus, the leading physicist of the 20th century. Nevertheless, his relationship between power and corresponding force is clear. In particular, in his 1905 publication of one of his major works on special relativity, "Zur Elektrodynamik bewegter Körper"^[5] (<ON THE ELECTRODYNAMICS OF MOVING BODIES

>), when he demonstrated "the transformation of light energy and the radiation pressure acting on a perfectly reflecting mirror Theory", it is stated on page 915 as follows (in German)

In <ON THE ELECTRODYNAMICS OF MOVING BODIES> , the corresponding English content (in page 18-19) is

The energy (measured in the stationary system) which is incident upon unit area of the mirror in unit time is evidently A²(c cos φ−v)/8π. The energy leaving the unit of surface of the mirror in the unit of time is A'''²(−c cos φ''' + v)/8π. The difference of these two expressions is, by the principle of energy, the work done by the pressure of light in the unit of time. If we set down this work as equal to the product Pv, where P is the pressure of light, we obtain

In agreement with experiment and with other theories, we obtain to a first approximation

In this statement, Einstein referred to "the energy which is incident upon unit area of the mirror in unit time the energy per unit area of the mirror per unit time", "the energy leaving the unit of surface of the mirror in the unit of time", and "the work done by the pressure of light in the unit of time", are obviously the power density.

Einstein put forward the basic principle for the calculation of light pressure in this discussion, that is, "The difference of these two expressions is, by the principle of energy, the work done by the pressure of light in the unit of time. ”

The author points out that in the literature he has seen, unlike previous scholars, Einstein was the first physicist in the world to propose the basic principles of light pressure calculation. It is not difficult to see that the properties of light pressure derived under this principle must be consistent with the results of the Crooks radiometer experiment. Furthermore, the pressure formula derived from this principle should not underestimate light pressure like the Maxwell-Bartoli formula.

Regrettably, the author finds the relevant mathematical derivation unreasonable, because according to this statement, it is impossible to mathematically derive the resulting pressure formula

   （3）

or

  （4）

Readers are advised to check this derivation with their own hands-on calculations.

The author believes that the key to the unreasonable mathematical derivation lies in the mathematical expression: "A²(c cos φ - v)/8π " and "A'''²(-c cos φ''' + v)/8π ".

The author doubts whether these two mathematical expressions are appropriate, whether there is a mathematical or physical basis, whether A²(c cos φ - v)² is miswritten as A²(c cos φ - v), A'''²( -c cos φ''' + v)² is incorrectly written as A'''²(-c cos φ''' + v), because these mistakes will not only lead to the problem that the energy value may be negative, but also make it impossible to derive the factor (cos φ - v/c)² in (3) or the factor cos²φ in equation (4).

Another important reason why the two pressure formulas cannot be deduced in this paper is the power density expression Pv which is commonly used in textbooks or literature. This is because there are many special factors in this expression, and it is not an accurate, objective expression of the force-power relationship (see the third blog post listed above).

If the expressions A²(c cos φ - v)/8π and A'''²(-c cos φ''' + v) are respectively replaced by the corresponding power density expressions that are correctly expressed, and the expressions Pv is replaced by the objective equivalence relationship between force and power, then according to the basic principle of light pressure calculation proposed by Einstein, he will definitely derive the pressure formula obtained by the author in the first blog post above

    (5)

III. Why can't the word power be found in Landau and Lifshitz's great tome of physics?

Recently, the author noticed with surprise that it is difficult to find a description of the definition and use of power in the mechanics section of existing college physics textbooks. In particular, in the ten-volume masterpiece <Theoretical Physics Course> by the well-known physicist of the former Soviet Union, Lev Davidovich Landau and his student Yevgeny Livschitz, the author could hardly find the term "Power" in mechanics, especially the first volume <Mechanics>^[6] .

In reading this work, the authors are deeply aware that it is not the negligence of these great authors, but that they have noticed that with the development of modern physics, the forms and changes of forces and energies (including forces) are becoming more complex, there are also many related phenomena that humans do not know or understand. For example, when it comes to friction, they consider this problem is beyond pure mechanics (see Section 25 in [6]). The authors speculate that this is one of the reasons why the word "force" is rarely used in this magnum opus, even though it covers a wide range of modern physics.

Because this great tome includes very modern advances in physics, including many areas of Landau's major research results, such as in condensed matter physics ^[7], and the extremely modern volume 10 <Physical Kinetics> ^[8], although the author does not know much about most of these fields, but is convinced that "power" is a very important physical concept, and that when the theory and application of the equivalence relationship between force and power are further studied, this masterpiece has provided lots of research material.

IV.  Some inner understandings of human beings on "Power" and "Force" 

On the other hand it is not difficult to see that power is indeed covered in detail as an important physical concept in many modern literature and important textbooks. Typical are:

Wikipedia explains the related entry:

https://en.wikipedia.org/wiki/Power_(physics)

and

https://en.wikipedia.org/wiki/Force ;

<Fundamentals of Physics>^[9]  by Robert Resnick, David Halliday and Jearl Walker;

<Feynman Lectures on Physics>^[10] by Feynman R, Leighton R, and Sands M. 

It is interesting and thought-provoking to describe these concepts directly in the language of the heart, if the historical and modern knowledge is synthesized, aside from the definitions expressed through mathematical formulas.

For example，Chinese Wikipedia (https://zh.wikipedia.org/wiki/Power) explains Power as

Power (English: power) is defined as the rate at which energy is converted or used, expressed as the amount of energy per unit time, that is, the rate of work. The SI unit of power is the watt (W), named after the eighteenth-century steam engine designer James Watt. The amount of electric energy converted into heat and light energy by a light bulb in a unit time can be expressed in power. The higher the wattage, the higher the capacity (or electricity) per unit time.

The textual explanation of power in <Fundamentals of Physics> (Chapter 7, p. 166) is

The time rate at which work is done by a force is said to be the power due to the force.

The definition of power on page 13-2 of <Feynman Lectures on Physics> is

F· v is called power: the force acting on an object times the velocIty of the object (vector "dot" product) is the power being delivered to the object by that force. We thus have a marvelous theorem: the rate of change of kinetic energy of an object IS equal to the power expended by the forces acting on it.

It can be clearly seen from all the above textual explanations that in people's minds, power is regarded as a direct manifestation of the action of force, and force is inseparable from the power it consumes or outputs.

After consulting the explanation of Power in Chinese Wikipedia above, as well as https://en.wikipedia.org/wiki/James_Watt, https://en.wikipedia.org/wiki/Horsepower, the author concludes that the physics term "Power" comes from The Scottish engineer, inventor of the improved steam engine, James Watt, and the term was later widely used in mechanical theory to describe aspects such as the output capacity of engines and electricity.

Based on the above information, since the term "Power", a physical concept, first appeared in English, how do people understand this word in the English-speaking world?

According to the authoritative Oxford English Dictionary, the word power is often understood as "strength, power, energy" (see p. 562 of the 7th edition of the dictionary^[11]), that is, power is often understood as force. According to the online Google Translate https://translate.google.com/, when "power" is translated into Chinese, the result is often "force". When translated into Russian, it is "сила", which means "force". Likewise, when Google Translate translates the word into other languages, the results mostly contain the meaning of force.

In short, power and force are inseparable.

On the other hand, in the above literature, how to express the understanding of "force" only in words?

The English Wikipedia (https://en.wikipedia.org/wiki/Force) describes "Force" as follows:

In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newton (N). Force is represented by the symbol F (formerly P).

Interestingly, P  is the first letter of the word "Power". Is this pure coincidence?

<Fundamentals of Physics> textual interpretation of "Force" (Chapter 5, p. 94) is

What Is Physics? We have seen that part of physics is a study of motion, including accelerations, which are changes in velocities. Physics is also a study of what can cause an object to accelerate.That cause is a force, which is, loosely speaking, a push or pull on the object. The force is said to act on the object to change its velocity. 

Based on the explanations of "force" in the above-mentioned literature, the author has to regret to say that none of these modern explanations mention the relationship between force and power. Analyzing the reasons, these explanations are all influenced by Newton's second law, and the explanations are all overly modern. The author argues that these modern explanations forget about earlier understandings of statics, excluding statics from mechanics, just as Landau and Lifshitz dealt with friction.

There is no doubt that statics is the most classical and important part of mechanics.

In the first blog post mentioned above, the author has demonstrated that the so-called "static force" also does work, and it is not purely "static". 

The actual experience of human beings is: whether it is a human being or an inanimate matter, it must have different performances when it is subjected to static force and when it is not subjected to static force, and the longer it is subjected to static force, the more obvious these performances are. This is clear evidence that static force is doing work, but these evidences are often shown in the motion of the microscopic material world and are often ignored by people!

The author is convinced that when this neglect is faced, "static force" will no longer be regarded as an ancient and discardable concept of mechanics, but on the contrary, it should be an important object of modern physics and mechanics research. Therefore, the author emphasizes again that force is equivalent to power.

nslation results

V.   Summary

Einstein discovered the equivalence relationship between mass and energy with different dimensions, so isn’t there a similar equivalence relationship between force and power with different dimensions?

Without force there is no power, without power there is no force, force and power describe the same physical object!

If the equivalence relationship between force and power is faced, it is very meaningful for the theoretical and applied research of physics and mechanics. The author has pointed out that this relationship shows that the "static force" is also doing work, which solves the problem of calculating the energy consumption of a person with a load, and also gives a more reasonable radiation pressure formula based on this relationship.

According to this relationship, the output power of the engine can be directly regarded as the output force. In engineering matters, most engines transmit their power or force through the rotors in them and the wheels of the conveyor belt. In the past, when people calculated the "force" transmitted through the conveyor belt, it was necessary to take into account a series of special factors such as the radius of the conveyor wheel, the rotation speed (depending on the resistance encountered), the torque, etc., which undoubtedly led to the misunderstanding that two engines with the same output power could output different "forces".  In fact, the equivalence relationship between force and power can be used to simply, objectively and clearly explain how many watts the output power of the engine is, and how much Newton force the output force only corresponds to, which is the characteristic of the engine itself. 

The equivalence relationship between force and power will also be of great significance to the study of the evolution of the universe and the evolution of celestial bodies (including planets, galaxies, black holes, etc.). As a counter-example, the famous physicist, Arthur Eddington, in his famous book <The internal constitution of the stars>^[12], like Maxwell, greatly underestimated the radiation pressure (see Chapter 2 of this work, 27-28 page, §21), which undoubtedly also underestimates the role of radiation pressure in the equilibrium inside.

 "Energy" and "power" are obviously not the same physical object, and the "equivalence relationship between mass and energy" and "equivalence relationship between force and power" are obviously not the same physical object. These two types of equivalence relations are of great significance in physics.

Reference:

[1] J. C. Maxwell, A Treatise on Electricity and Magnetism (1St ed.), Vol. 2, Oxford, 1873. 

[2] J. C.  Maxwell, Lehrbuch der Elektricitiit und des Magnetismus. Deutsch von B. Weinstein, Berlin 1883. 

[3] A. Bartoli, Ezner's Bep. d. Phys. 21. p. 198. 1884, übersetzt aus Nuovo Cimento 16. p. 195. 1883. 

[4]  P. Lebedew: Untersuchen uber die druckkrafte des Lichtes, Annalen der Physik 46, 432 (1901). 

[5]  Einstein, Albert (1905d) [Manuscript received: 30 June 1905]. Written at Berne, Switzerland. "Zur Elektrodynamik bewegter Körper". Annalen der Physik (Submitted manuscript) (in German). Hoboken, New Jersey (published 10 March 2006). 322 (10): 891–921. 

[6] Landau, Lev D.; Lifshitz, Evgeny M. (1976). Mechanics. Vol. 1 (3rd ed.). Butterworth-Heinemann. ISBN 978-0-7506-2896-9.

[7] Lifshitz, Evgeny M.; Pitaevskii, Lev P. (1980). Statistical Physics, Part 2: Theory of the Condensed State. Vol. 9 (1st ed.). Butterworth-Heinemann. ISBN 978-0-7506-2636-1.

[8] Lifshitz, Evgeny M.; Pitaevskii, Lev P. (1981). Physical Kinetics. Vol. 10 (1st ed.). Pergamon Press. ISBN 978-0-7506-2635-4.

[9]  Robert Resnick, David Halliday, Jearl Walker, Fundamentals of Physics, Wiley & Sons, Incorporated, John, (2000), ISBN: 0471332356, ISBN13: 9780471332350 

[10] Feynman R, Leighton R, and Sands M. The Feynman Lectures on Physics. Three volumes 1964, 1966. Library of Congress Catalog Card No. 63-20717, ISBN 0-201-02115-3 (1970 paperback three-volume set)
[11] <Paperback Oxford English Dictionary>, seventh edition, edited by Aauice Watte, Oxfoord University Press, 2022, ISBN: 978-0-19-964094-2.

[12] Arthur S. Eddington, The internal constitution of the stars,with a new forwaord by S. Chandrasekhar, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney,1988, ISBN 0 521 33708 9

再论力-功率等价关系

管克英

北京交通大学理学院，数学系

Email：keying.guan@gmail.com

摘要：本文将进一步精确地描述力与功率的等价关系，指出爱因斯坦根据力与功率的关系对辐射压计算给出了基本原则，也指出由于受 Maxwell-Bartoli 辐射压公式的影响，爱丁顿低估了辐射压在恒星内部的重要作用。

目录：一   力与功率是矢量还是标量

           二   电磁辐射强度与光压，爱因斯坦对光压理论的重要贡献

           三   为何在朗道与栗弗席兹的物理巨著中找不到功率一词？

           四   人类对功率与力的一些内在理解

           五   总结

作者在前几篇博文

1. 没有不做功的力--马德堡半球实验、举重运动消耗的卡路里及辐射压公式 2022-07-01

2. Force is equivalent to Power 2022-07-02

3. Supplementary Explanation on "Force-Power Equivalence“ 2022-07-04

揭示了虽然在分析力学中蕴含，但没有被学界认识到的“力与功率的等价关系”。由于涉及到物理学与力学的深层次问题，还必须反复思考与论证。作者发现自己在已发布的论证中仍存在待改进与进一步说明的地方，而且还发现学界也在相关的描述上存在有趣的现象。这些进一步地表明该等价关系确实存在，而且必须被进一步精确地描述。

一      力与功率是矢量还是标量

首先需指出的是，在第一篇博文中给出了对等价关系的数学表达式 （10）

在推导中，作者忽略了对矢量与标量运算处理的解释，由此导致按通常理解 (10) 式的等号左侧的力被看作矢量，右侧的功率被看作标量的形式不匹配问题。针对此问题，作者在本文指出必须将 (10) 式理解为：如果（10）式右端的功率为标量，该式左端的力 1 N  (牛顿) 应看作是该力的强度 (标量)，如果（10）式左端的力是矢量，则需要将 (10) 式右端的功率 1 W (瓦特) 当作与力的方向一致的矢量。

这种理解是符合实际情况的：

1）有些力与功率必须使用标量描述。

例如一个理想点光源（如很小的电灯泡，远距离看到的恒星）发出的光，可以使用瓦特数计量其整体瞬时发光的总强度（总功率），也可用牛顿数计量其瞬时对外部的总光压（力）。这种计量的发光总功率与总光压不存在方向问题，都是标量。

2）有些力与功率使用矢量描述更准确、更方便。

例如，上述点光源射向外部某处一个局部垂直于光射线的单位面积光束流必定存在功率（面）密度，并会对该面积产生局部的压强。此时光束流的功率密度与压强显然是有方向的，都应该当作矢量。如果该单位面积是以光源为中心的球面的一小部分，则可简单利用第二类曲面积分证明通过该球面的功率密度总合等于光源的总功率，也可证明通过该球面的压力总和等于光源的总光压。当将球面改成将光源包在内部的光滑封闭曲面时，仍可进行上述的第二类曲面积分，而且可以证明，只要封闭曲面内没有其他光源或其他障碍物，该积分值，即总功率和总光压，保持不变。如果仅用标量描述局部射束的功率或光压的话，则会带来许多麻烦，甚至导致错误。

3）在许多文献中，在泛指力和功率时，对有方向的力与功率的描述并不在意使用标量还是矢量。

例如 Wikipedie 对辐射压 (https://en.wikipedia.org/wiki/Radiation_pressure) 的解释是 

The energy flux (irradiance) of a plane wave is calculated using the Poynting vector , whose magnitude we denote by S. S divided by the speed of light is the density of the linear momentum per unit area (pressure) of the electromagnetic field. So, dimensionally, the Poynting vector is S=power/area=rate of doing work/area=ΔF/ΔtΔx/area, which is the speed of light, c=Δx/Δt, times pressure, ΔF/area. That pressure is experienced as radiation pressure on the surface:

其中明确地用 Poyintng 矢量 S 的强度 S (标量) 描述辐射的功率（面）密度, 而当用其表示压强时则使用符号 <S> , 没有说明此符号及对应的压强 P_incident 是矢量还是标量。因此，如果将 P_incident 看作矢量，那么辐射的功率密度 <S> 则必须视作矢量。 

又如，讨论到太阳光压时，麦克斯韦在其重要著作^[1]，<A Treatise on Electricity and Magnetism (1St ed.), Vol. 2> ，§793 的开始有如下叙述：

Thus, if in strong sunlight the energy of the light which falls on one square foot is 83.4 foot pounds per second, the mean energy in one cubic foot of sunlight is about 0.0000000882 of a foot pound, and the mean pressure on a square foot is 0.0000000882 of a pound weight.

其中显然将太阳光 (在地球表面) 的辐射功率密度使用英制表示，即

在德文译本^[2]的对应部分为

Nimmt man an, dass kraftiges Sunnenlight an einem Quadratmeter in der Secunde 124.1 Kilogrammeter Energie entwickelt, so warden in einem Kubikmeter Sonnenstrahlen 124/30000000 oder 0.00000041 Kilogram Drmeter Energie entachreuck sein, eine zur Fortplanzungstrichtung der Sonnenstrahlen senkrechte Flache pro Quadratmeter erleidet, 0.00000041 Kilogrammeter betragen.

其中的功率是通过更熟悉的单位制表示

以上引文也说明麦克斯韦将每秒每立方米接收的太阳光能理解为所受到的光压，每平方米 0.00000042 千克重, 严重地低估了太阳光压的强度。

同期， 巴托利在讨论同样问题的著作^[3]中独立地用自己独特的想法，给出电磁辐射的功率密度与压强。

继承了麦克斯韦与巴托利的思想，列别捷夫在介绍他的光压实验的论文^[4]中直接给出了 Maxwell-Bartoli 辐射压公式

     （1）

其中，按当时的习惯，V 表示光速。

从麦克斯韦到列别捷夫的表达式可以看出，每个等式两边的量既可同时看作标量，也可同时看作矢量。

二    电磁辐射强度与光压，爱因斯坦对光压理论的重要贡献

由前一节的介绍可以看出，从麦克斯韦到列别捷夫，光压是用光束的入射功率密度除以光速来计算的。其中的理论依据可能是常见的功率表达式

       （2）

其中 P (Power) 表示功率。这些学者显然将公式 (2) 中的速度 v 用光速 V 代替得到了 (1)。

作者已在开始介绍的第一篇博文中指出如果用光速 V 代替 v, 即意味着被光束作用的物体是光自身，这毫无意义。还在第三篇博文指出 (2) 式存在的其它问题。

有趣的是，作者发现, 20世纪的顶级物理学家, 爱因斯坦的文集中的重要物理学论文中几乎看不到功率一词。尽管如此，他对功率与对应的力关系则描述得很清楚。 特别，在其1905年发表狭义相对论的主要著作之一, "Zur Elektrodynamik bewegter Körper"^[5] (论动体的电动力学)，当论证“光线能量的变换与作用在完全反射镜上的辐射压力理论”时，在915页有如下表述（德文）：

其中文译文为（摘自范岱年 赵中立 许良英编译的 <爱因斯坦文集> 第二卷^[6] 108页）

在这段表述中，爱因斯坦所提到的“每单位时间内射到反射镜上单位面积的能量” ，“单位时间内离开反射镜的单位面积的能量”， 以及“单位时间内光压所做的功”显然就是功率密度。

爱因斯坦在这段论述对光压计算提出了基本原则，即 “由能量原理，这两式（每单位时间内射到反射镜面上单位面积的能量，和单位时间内离开反射镜的单位面积的能量）的差就是单位时间内光压所做的功”。

作者指出，在所看到的文献中，不同于之前的学者，爱因斯坦是世界上第一个提出光压计算基本原则的物理学家。不难看出，这个原则下推导出的光压性质一定与克鲁克斯辐射计实验的结果相符。除此，按照该原则导出的压强公式不会像 Maxwell-Bartoli 公式一样，右方出现光速 V 这个大分母使得光压被严重的低估。

遗憾的是，作者发现相关的数学推导存在不合理之处，因为根据该表述，不可能通过数学推导导出所得到的压强公式

   （3）

或

  （4）

建议读者亲自动手计算检验一下这个推导。

作者认为，数学推导不合理的关键在于其中的数学表达式： “A²(V cos φ - v)/8π "，和 “A‘’‘²(-V cos φ’‘’ + v)/8π ”。

作者怀疑这两个数学表达式是否恰当，是否存在数学或物理学的依据，怀疑其中是否将 A²(V cos φ - v)² 误写成 A²(V cos φ - v)，是否将 A'''²(-V cos φ’‘’ + v)² 误写成 A'''²(-V cos φ’‘’ + v)，因为这些误写既造成能量值可能取负值的问题，也导致不可能导出的（3）式中的因子 （cos φ - e/V)² 或  (4) 式中的因子 cos²φ。

该文推导不出那两个压强公式的另一个重要原因还在于所使用的，教科书或文献常用的功率密度表达式 P●v 。

这是因为该表达式存在许多特殊因素的影响，并不是力与功率关系的准确、客观表达 （考前面列出的第三篇博文）。

如果将表达式 A²(V cos φ - v)/8π 和A'''²(-V cos φ’‘’ + v) 分别换成正确表达的相应功率密度表达式，并将表达式 P●v 换成客观的力与功率的等价关系，那么按照爱因斯坦提出的光压计算的基本原则，他一定会导出作者在上述第一篇博文得到的压强公式

    (5)

三   为何在朗道与栗弗席兹的物理巨著中找不到功率一词？

最近，作者惊奇地注意到，现有的大学物理学教材中的力学部份很难找到对功率的定义与使用的描述。特别，在前苏联知名物理学家，列夫·达维多维奇·朗道和其学生叶夫根尼·栗弗席兹的十卷巨著 <理论物理学教程> 中，作者竟然几乎找不到力学的“功率”（Power）一词，尤其第一卷 <Mechanics>^[7] （<力学>）更是如此。

在细读该著作时，作者深深地意识到这不是该巨著作者们的疏忽，而是他们注意到随着现代物理学的发展，力与能量（包括功率）的形式越来越复杂，至今仍有大量有关的现象人类尚不知道、不了解。例如，在涉及到摩檫力时，他们认为这个问题超越了纯力学的范围（参见 [7] 中的 §25)。作者猜测这也是该巨著虽然涉及到现代物理学的大量内容，却很少使用功率一词的原因之一。

由于该巨著包括了非常现代的物理研究进展，其中包括了朗道重大研究成果的诸多研究领域，如凝聚态物理^[7]，还包括极其现代的第十卷 <Physical Kinetics> ^[8](物理动力学)，虽然作者对其中多数领域并不甚了解，但确信 “功率” 是非常重要的物理概念，而且确信当进一步研究力与功率的等价关系的理论与应用时，该巨著已提供了大量的研究素材。

四     人类对功率与力的一些内在理解

另一方面，也容易看到在不少现代文献和重要的教科书中，的确将功率作为重要的物理学概念做了详细的介绍。典型的有：

Wikipedia 对相关的词条解释:

https://en.wikipedia.org/wiki/Power_(physics)

和

https://en.wikipedia.org/wiki/Force ,

Translation results

哈利迪与雷斯尼克合著的 <Fundamentals of Physics>^[9],

Translation results, 

理查德·费曼著的 <Feynman Lectures on Physics>^[10] 等。

如果综合了历史与现代知识，抛开通过数学公式表述的定义，直接用发自内心的语言描述这些概念则是趣味而且值得深思的。

例如，中文维基百科（ https://zh.wikipedia.org/wiki/功率）对功率（Power）的解释是

Jump to navigationJump to search

功率（英语：power）定义为能量转换或使用的速率，以单位时间的能量大小来表示，即是作功的率。功率的国际标准制单位是瓦特（W），名称是得名于十八世纪的蒸汽引擎设计者詹姆斯·瓦特。灯泡在单位时间内，电能转换为热能及光能的量就可以用功率表示，瓦特数越高表示单位时间用的能力（或电力）越高。

<Fundamentals of Physics> 对功率的文字解释（第七章，166页）为

The time rate at which work is done by a force is said to be the power due to the force.

中文是

力做功的时间速率称为力的功率。

<Feynman Lectures on Physics> 的第13-2页中对功率的定义是

F· v is called power: the force acting on an object times the velocIty of the object (vector "dot" product) is the power being delivered to the object by that force. We thus have a marvelous theorem: the rate of change of kinetic energy of an object IS equal to the power expended by the forces actmg on it.

中文是

F·v 称为功率：作用在物体上的力乘以物体的速度（矢量“点”积）就是该力传递给物体的功率。 因此，我们有一个奇妙的定理：物体的动能变化率等于作用在其上的力所消耗的功率。

从以上所有的文字解释可以清楚地看到，在人们的心目中，功率被看作是力作用的一种直接体现，力与其消耗或输出的功率是分不开的。

经查阅上面中文维基百科对Power的解释，以及 https://en.wikipedia.org/wiki/James_Watt ，https://en.wikipedia.org/wiki/Horsepower ，作者断定这个物理学术语 Power 来自苏格兰的工程师、改进的蒸汽机的发明人詹姆斯·瓦特，而且这个术语后来在机械理论中被广泛用于描述发动机和电力的输出能力等方面。

根据以上信息，既然物理概念 ”功率“ （Power）这个术语最早是通过英语出现的，那么在英语世界，人们是如何理解这个单词呢？

根据权威的《牛津英语词典》，“power” 一词常被理解为“力量、力、能量”（参见词典第七版第562页^[12]），即power常被理解为力。 根据在线谷歌翻译https://translate.google.com/，将“power”翻译成中文，结果往往是“force”。 翻译成俄语时，它是“сила”，也就是“力”的意思。同样，当谷歌翻译把这个词翻译成其他语言，结果大多含有力的意思。

总之，功率与力是分不开的。

另一方面，在上述文献中，又如何仅用文字表达对“力”的理解呢？

英文 Wikipedia (https://en.wikipedia.org/wiki/Force) 对 “力” （Force）如下描述：

In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newton (N). Force is represented by the symbol F (formerly P).

译成中文就是

在物理学中，力是一种可以改变物体运动的影响。 力可以使具有质量的物体改变其速度（例如从静止状态移动），即加速。 力也可以直观地描述为推或拉。 力既有大小又有方向，所以它是一个向量。 它以牛顿 (N) 的 SI 单位测量。 力由符号 F（以前的形式为 P）表示。

趣味的是，P 是单词 “Power” 的第一个字母。这是纯粹的巧合吗？

<Fundamentals of Physics> 对“力” (Force) 的文字解释（第五章，94页）为

What Is Physics? We have seen that part of physics is a study of motion, including accelerations, which are changes in velocities. Physics is also a study of what can cause an object to accelerate.That cause is a force, which is, loosely speaking, a push or pull on the object. The force is said to act on the object to change its velocity. 

对应的中文是

什么是物理学？ 我们已经看到，物理学的一部分是对运动的研究，包括加速度，加速度是速度的变化。 物理学也是研究什么可以导致物体加速的研究。那个原因是一种力，轻松地说，就是对物体的推或拉。 力被当作施加在物体上以改变其速度的作用。

根据以上文献对“力”的解释，作者不得不遗憾地说，这些现代的解释都没有提到力与功率的关系。分析其原因，这些解释都是受到牛顿第二定律的影响，解释得过分现代了。作者认为这些现代解释忘却了人类早期对静力的认识，就像朗道与栗弗席兹对待摩檫力一样，将静力也排除在力学之外。

毫无疑问，静力学是力学中最经典的重要组成部分。

作者在前面提到的第一篇博文已论证指出了，所谓的“静力”也做功，它并不是纯粹的“静”。

人类的实际经验是：无论是人还是无生命的物质，承受静力时与不承受静力时自身一定有不同的表现，而且承受静力的时间越长，这些表现就越明显。这就是静力在做功的清楚证据，只是这些证据往往表现在微观物质世界的运动，常被人们忽略！

作者确信，当这个疏忽被正视时，“静力”就不再被看成是个古老、可被抛弃的力学概念，而是恰恰相反，它应该是个现代物理学与力学研究的重要对象。因此，作者再次强调，力等价于功率。

五  总结

爱因斯坦发现了量纲不同的质量与能量之间的等价关系，那么量纲不同的力与功率之间不也是类似这样的等价关系吗？

没有力就没有功率，没有功率就没有力，力和功率描述的是同一个物理对象！

正视力与功率的等价关系其实对物理学与力学的理论与应用研究非常有意义。作者已指出，这种关系表明“静力”也在做功，从而解决了负重的人消耗能量的计算问题，也根据这种关系给出了更合理的辐射压公式。

根据这种关系，可以直接地将发动机的输出功率看成是输出的力。在工程问题上，绝大多数的发动机是通过其中的转子及传送带的轮子传送其功率或力的。以前，人们计算通过传送带传送的“力”时，需要考虑到传送轮的半径、转速（依赖于遇到的阻力）、扭矩等一系列特殊因素，这无疑会造成这样的误解，即两个相同功率的发动机却输出不同的“力”。其实使用力与功率的等价关系，可以简单、客观、明确地说明，发动机的输出功率是多大瓦特数，其输出的力唯一地对应是多大 牛顿力，这是发动机自身的性质。

力与功率的等价关系，对宇宙演化、天体（包括行星、星系、黑洞等）演化的研究也会非常有意义。作为反例，著名物理学家，亚瑟·爱丁顿在其名著 <The internal constitution of the stars>^[13] 中如同麦克斯韦一样将辐射压大大低估了（参看该著作第二章，27-28页，§21），这无疑也低估了辐射压在恒星内部的平衡中的作用。

“能量” 与 “功率” 显然不是相同的物理对象，“质量与能量之间的等价关系” 和 “力与功率的等价关系” 也显然不是同样的物理对象。这两类等价关系在物理学中有着极重要的意义。

参考文献：

[1] J. C. Maxwell, A Treatise on Electricity and Magnetism (1St ed.), Vol. 2, Oxford, 1873. 

[2] J. C.  Maxwell, Lehrbuch der Elektricitiit und des Magnetismus. Deutsch von B. Weinstein, Berlin 1883. 

[3] A. Bartoli, Ezner's Bep. d. Phys. 21. p. 198. 1884, übersetzt aus Nuovo Cimento 16. p. 195. 1883. 

[4]  P. Lebedew: Untersuchen uber die druckkrafte des Lichtes, Annalen der Physik 46, 432 (1901). 

[5]  Einstein, Albert (1905d) [Manuscript received: 30 June 1905]. Written at Berne, Switzerland. "Zur Elektrodynamik bewegter Körper". Annalen der Physik (Submitted manuscript) (in German). Hoboken, New Jersey (published 10 March 2006). 322 (10): 891–921. 

[6] 范岱年 赵中立 许良英编译，爱因斯坦文集，第二卷，商务印数馆出版，1977年第一版，统一书号：2017●184

[7] Landau, Lev D.; Lifshitz, Evgeny M. (1976). Mechanics. Vol. 1 (3rd ed.). Butterworth-Heinemann. ISBN 978-0-7506-2896-9.

[8] Lifshitz, Evgeny M.; Pitaevskii, Lev P. (1980). Statistical Physics, Part 2: Theory of the Condensed State. Vol. 9 (1st ed.). Butterworth-Heinemann. ISBN 978-0-7506-2636-1.

[9] Lifshitz, Evgeny M.; Pitaevskii, Lev P. (1981). Physical Kinetics. Vol. 10 (1st ed.). Pergamon Press. ISBN 978-0-7506-2635-4.

[10]  Robert Resnick, David Halliday, Jearl Walker, Fundamentals of Physics, Wiley & Sons, Incorporated, John, (2000), ISBN: 0471332356, ISBN13: 9780471332350 [11] Feynman R, Leighton R, and Sands M. The Feynman Lectures on Physics. Three volumes 1964, 1966. Library of Congress Catalog Card No. 63-20717, ISBN 0-201-02115-3 (1970 paperback three-volume set)
[12] <Paperback Oxford English Dictionary>, seventh edition, edited by Aauice Watte, Oxfoord University Press, 2022, ISBN: 978-0-19-964094-2.

[13] Arthur S. Eddington, The internal constitution of the stars,with a new forwaord by S. Chandrasekhar, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney,1988, ISBN 0 521 33708 9