III Space and Time in Classical Mechanics

 The purpose of mechanics is to describe how bodies change their position in space with “time.” I should load my conscience with grave sins against the sacred spirit of lucidity¹ were I to formulate² the aims of mechanics in this way, without serious reflection and detailed³ explanations. Let us proceed to disclose these sins.

 

It is not clear what is to be understood here by “position” and “space.” I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend⁴ in a straight line. A pedestrian who observes the misdeed from the footpath⁵ notices that the stone falls to earth in a parabolic curve. I now ask: Do the “positions” traversed by the stone lie “in reality” on a straight line or on a parabola? Moreover, what is meant here by motion “in space”? From the considerations of the previous section the answer is self-evident. In the first place we entirely⁶ shun⁷ the vague word “space,” of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by “motion relative to a practically rigid⁸ body of reference.” The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section. If instead of “body of reference” we insert “system of co-ordinates,” which is a useful idea for mathematical description, we are in a position to say: The stone traverses a straight line relative to a system of co-ordinates rigidly⁹ attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory¹⁰ (lit. “path-curve”1), but only a trajectory relative to a particular body of reference.

 

In order to have a complete description of the motion, we must specify¹¹ how the body alters its position with time; i.e. for every point on the trajectory it must be stated at what time the body is situated¹² there. These data must be supplemented by such a definition of time that, in virtue¹³ of this definition, these time-values can be regarded essentially¹⁴ as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction; the man at the railway-carriage window is holding one of them, and the man on the footpath the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity¹⁵ of propagation of light. With this and with a second difficulty prevailing¹⁶ here we shall have to deal in detail later.

 

1 That is, a curve along which the body moves.

1 lucidity   

n.明朗，清晰，透明

参考例句：

• His writings were marked by an extraordinary lucidity and elegance of style.他的作品简洁明晰，文风典雅。
• The pain had lessened in the night, but so had his lucidity.夜里他的痛苦是减轻了，但人也不那么清醒了。

2 formulate   

v.用公式表示；规划；设计；系统地阐述

参考例句：

• He took care to formulate his reply very clearly.他字斟句酌，清楚地做了回答。
• I was impressed by the way he could formulate his ideas.他陈述观点的方式让我印象深刻。

3 detailed   

adj.详细的，详尽的，极注意细节的，完全的

参考例句：

• He had made a detailed study of the terrain.他对地形作了缜密的研究。
• A detailed list of our publications is available on request.我们的出版物有一份详细的目录备索。

4 descend   

vt./vi.传下来，下来，下降

参考例句：

• I hope the grace of God would descend on me.我期望上帝的恩惠。
• We're not going to descend to such methods.我们不会沦落到使用这种手段。

5 footpath   

n.小路，人行道

参考例句：

• Owners who allow their dogs to foul the footpath will be fined.主人若放任狗弄脏人行道将受处罚。
• They rambled on the footpath in the woods.他俩漫步在林间蹊径上。

6 entirely   

ad.全部地，完整地；完全地，彻底地

参考例句：

• The fire was entirely caused by their neglect of duty. 那场火灾完全是由于他们失职而引起的。
• His life was entirely given up to the educational work. 他的一生统统献给了教育工作。

7 shun   

vt.避开，回避，避免

参考例句：

• Materialists face truth,whereas idealists shun it.唯物主义者面向真理，唯心主义者则逃避真理。
• This extremist organization has shunned conventional politics.这个极端主义组织有意避开了传统政治。

8 rigid   

adj.严格的，死板的；刚硬的，僵硬的

参考例句：

• She became as rigid as adamant.她变得如顽石般的固执。
• The examination was so rigid that nearly all aspirants were ruled out.考试很严，几乎所有的考生都被淘汰了。

9 rigidly   

adv.刻板地，僵化地

参考例句：

• Life today is rigidly compartmentalized into work and leisure. 当今的生活被严格划分为工作和休闲两部分。
• The curriculum is rigidly prescribed from an early age. 自儿童时起即已开始有严格的课程设置。

10 trajectory   

n.弹道，轨道

参考例句：

• It is not difficult to sketch the subsequent trajectory.很容易描绘出它们最终的轨迹。
• The path followed by a projectile is called its trajectory.抛物体所循的路径称为它的轨道。

11 specify   

vt.指定，详细说明

参考例句：

• We should specify a time and a place for the meeting.我们应指定会议的时间和地点。
• Please specify what you will do.请你详述一下你将做什么。

12 situated   

adj.坐落在...的，处于某种境地的

参考例句：

• The village is situated at the margin of a forest.村子位于森林的边缘。
• She is awkwardly situated.她的处境困难。

13 virtue   

n.德行，美德；贞操；优点；功效，效力

参考例句：

• He was considered to be a paragon of virtue.他被认为是品德尽善尽美的典范。
• You need to decorate your mind with virtue.你应该用德行美化心灵。

14 essentially   

adv.本质上，实质上，基本上

参考例句：

• Really great men are essentially modest.真正的伟人大都很谦虚。
• She is an essentially selfish person.她本质上是个自私自利的人。

15 velocity   

n.速度，速率

参考例句：

• Einstein's theory links energy with mass and velocity of light.爱因斯坦的理论把能量同质量和光速联系起来。
• The velocity of light is about 300000 kilometres per second.光速约为每秒300000公里。

16 prevailing   

adj.盛行的；占优势的；主要的

参考例句：

• She wears a fashionable hair style prevailing in the city.她的发型是这个城市流行的款式。
• This reflects attitudes and values prevailing in society.这反映了社会上盛行的态度和价值观。