每日彙整: 2018-03-21

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰慣性系【一】

300px-Standard_conf

伽利略變換

\begin{bmatrix} x^{\prime} \\ t^{\prime} \end{bmatrix} = \begin{pmatrix} 1 & -v \\0 & 1 \end{pmatrix} \begin{bmatrix} x \\ t \end{bmatrix}

220px-Light_cone

時空圖

Galilean_transform_of_world_line

Lorentz_transforms_2.svg

勞侖茲變換

\begin{bmatrix} x^{\prime} \\ t^{\prime} \end{bmatrix} = \frac{1}{\sqrt{1 - {(\frac{v}{c})}^2}} \begin{pmatrix} 1 & -v \\ -\frac{v}{c^2} & 1 \end{pmatrix} \begin{bmatrix} x \\ t \end{bmatrix}

220px-Hyperbo

300px-Minkowski_lightcone_lorentztransform_inertial.svg

300px-Minkowski_lightcone_lorentztransform.svg

Lorentz_transform_of_world_line

『運動是第一義』它意指什麼的呢？如果考察人們對『時間』的『認識』，總離不開對物體『運動』的『觀察』。之前在《時間是什麼？？》一文裡，我們談到了『古典物理』是以『牛頓第一運動定律』所指稱的『慣性座標系觀察者』之『時空觀』為『基礎』的。『牛頓』假設『存在』一個對所有的『慣性座標系』中『觀察者』都『相同』並且『恆定恆速』的『時間之流』，自此『時間』就成為『第一義』的了。也就是說如果『□觀察者』說『兩事件』『同時發生』，『○觀察者』也講那『兩事件』『同時發生』。因而『第一運動定律』──  假使沒有外力作用，靜者恆靜，動者作等速直線運動，在『第二運動定律』的強大光芒『覆照』下，反倒顯得晦暗不明的了，宛如是個『力等於零』的『特例』一般。於是『速度』 v 的『定義』 v = \frac{\Delta x}{\Delta t} 與『相對速度』是 v 的『兩』個『慣性座標系』彷彿是『同義語』。殊不知這個『相對速度』是『兩』個『觀察者』之『互見』，而且『運動方向』相反，並不能『自見』的啊！要是說果真能夠『自見』又豈會自己『無法度量』的呢？於是乎有『無窮多』個『慣性觀察者』各以『無限種』之『相對速度』『運動』，然而他們所『觀察到』的『自然律』都是一樣的，這就是『慣性』的『本義』。其實『觀察者』之『概念』有一點像『抽象擬人化』的說法，比方說，一個『對我而言』運動中的『粒子』，在『粒子』自己的『慣性座標系』裡，『自然律』一樣的『適用』。如此『對我而言』可用『我的時空』將那個『粒子』標示在『我的時空圖』 (x_{\Box}, t_{\Box}) 上，一個與『粒子偕行』相對『靜止』的『觀察者』，就把『我的運動』畫在『他的時空圖』 (x_{\bigcirc}, t_{\bigcirc}) 上的了。這個『互為動靜』的『論述』就是『相對運動』的『實質』，並不存在『絕對運動』的啊。所以『我說』『那個粒子』在 t_{p^{-}} 『時刻』『接近』 x_{p^{-}} 『位置』，當 t_p 『時』『到達』 x_p 『處』，於 t_{p^{+}} 『之後』『離開』 x_{p^{+}} 『之地』，『我』將此『等速運動』歸之於『粒子』的『運動慣性』；那個與『粒子偕行』相對『靜止』的『觀察者』亦將此『等速運動』歸之於『我』的『運動慣性』，這就是『運動』之『慣性』的『第一義』。所謂『飛鳥之景未嘗動也，鏃矢之疾而有不行不止之時』是不了解『慣性之意』『跳躍』於『互為動靜』之間，事實上對『任一方』而言，那個『相對運動』都是『存在的』，根本不會有『瞬時速度』存不存在的問題，所以才名之為『慣性定律』︰ v = \frac{- \delta x}{- \delta t} = \frac{ \delta x}{ \delta t} = \frac{+ \delta x}{+ \delta t}，或者比喻的說︰在牛頓力學裡，沒有任何東西能夠阻擋『恆定恆速』之『時間之流』的啊！！

當『愛因斯坦』假設了『光速』對所有的『慣性觀察者』都是『一樣的』之後，引申出了『同時性的破壞』、『運動的鐘會變慢』、『運動的尺會縮短』…等等的『大哉論』，人們開始恍然大悟所謂的『相對』、所見的『運動』…之種種必須以『量測方法』為依據，面對『大自然』的『事實』並沒有『純粹思辯』所得之理『一定對』之『位置』的吧！

─── 《【SONIC Π】電聲學之電路學《四》之《 !!!! 》下》

 

面對『基元概念』，說文解字往往不足矣！

咀嚼思辨可為其法乎？

Inertial frame of reference

An inertial frame of reference, in classical physics, is a frame of reference in which bodies, whose net force acting upon them is zero, are not accelerated; that is they are at rest or they move at a constant velocity in a straight line.[1] In analytical terms, it is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner.[2] Conceptually, in classical physics and special relativity, the physics of a system in an inertial frame have no causes external to the system.[3] An inertial frame of reference may also be called an inertial reference frame, inertial frame, Galilean reference frame, or inertial space.[citation needed]

All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration. Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, in any region small enough for the curvature of spacetime and tidal forces[4] to be negligible, one can find a set of inertial frames that approximately describe that region.[5][6]

In a non-inertial reference frame in classical physics and special relativity, the physics of a system vary depending on the acceleration of that frame with respect to an inertial frame, and the usual physical forces must be supplemented by fictitious forces.[7][8] In contrast, systems in non-inertial frames in general relativity don’t have external causes, because of the principle of geodesic motion.[9] In classical physics, for example, a ball dropped towards the ground does not go exactly straight down because the Earth is rotating, which means the frame of reference of an observer on Earth is not inertial. The physics must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force.

……

Background

A brief comparison of inertial frames in special relativity and in Newtonian mechanics, and the role of absolute space is next.

A set of frames where the laws of physics are simple

According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform translation: [18]

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K’ moving in uniform translation relatively to K.

— Albert Einstein: The foundation of the general theory of relativity, Section A, §1

This simplicity manifests in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames have external causes.[3] The principle of simplicity can be used within Newtonian physics as well as in special relativity; see Nagel[19] and also Blagojević.[20]

The laws of Newtonian mechanics do not always hold in their simplest form…If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a ‘force’ pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton’s laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity:
The laws of mechanics have the same form in all inertial frames.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 4

In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment (otherwise the differences would set up an absolute standard reference frame).[21][22]According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup.[23] In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the Galilean group of symmetries.

Absolute space

Main article: Absolute space and time

Newton posited an absolute space considered well approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then one in uniform translation relative to absolute space. However, some scientists (called “relativists” by Mach[24]), even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced.

Indeed, the expression inertial frame of reference (German: Inertialsystem) was coined by Ludwig Lange in 1885, to replace Newton’s definitions of “absolute space and time” by a more operational definition.[25][26] As translated by Iro, Lange proposed the following definition:[27]

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

A discussion of Lange’s proposal can be found in Mach.[24]

The inadequacy of the notion of “absolute space” in Newtonian mechanics is spelled out by Blagojević:[28]

  • The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out.
  • Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.
  • Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.
— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5

The utility of operational definitions was carried much further in the special theory of relativity.[29] Some historical background including Lange’s definition is provided by DiSalle, who says in summary:[30]

The original question, “relative to what frame of reference do the laws of motion hold?” is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.

— Robert DiSalle Space and Time: Inertial Frames

 

或能借尚不熟悉之程式庫反覆嘗試表達體會耶！？