Rotating reference frame

https://en.wikipedia.org/wiki/Rotating_reference_frame

旋转参考系是一种非惯性系，相对惯性系转动。本文仅讨论绕一根固定轴旋转的坐标系，更一般的旋转，请查看Euler angle。

所有的非惯性系都有虚拟力，旋转系中有三种Fictitious forces：离心力，科氏力，欧拉力。

旋转参考系相对静止参考系(惯性系)

旋转系 ( x ′ , y ′ , z ) (x',y',z) (x′,y′,z)，惯性系 ( x , y , z ) (x,y,z) (x,y,z)，旋转系绕z轴匀速转动 θ t = Ω \theta_t = \Omega θt=Ω， θ = Ω t + θ 0 \theta = \Omega t + \theta_0 θ=Ωt+θ0
Note： θ t = d θ d t \theta_t = \frac{d \theta}{dt} θt=dtdθ
假设起始时两个坐标系重合，即 θ 0 = 0 \theta_0=0 θ0=0

将坐标从旋转系变换回惯性系：
[ x y ] = [ c o s θ − s i n θ s i n θ c o s θ ] [ x ′ y ′ ] \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} cos\theta & - sin\theta \\ sin\theta & cos\theta \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix} [xy]=[cosθsinθ−sinθcosθ][x′y′]
X = T X ′ X = T X' X=TX′，注意这记号不是求导
T T T是一个正交矩阵， T ∗ X = X ′ T^* X = X' T∗X=X′，*表示转置
因此两个参考系的中的坐标可通过变换矩阵互相转化。

旋转系中的基矢量 i ^ , j ^ , k ^ \hat{i}, \hat{j}, \hat{k} i^,j^,k^
i ^ = ( c o s θ , s i n θ ) \hat{i} = (cos\theta, sin\theta) i^=(cosθ,sinθ)
j ^ = ( − s i n θ , c o s θ ) \hat{j} = (-sin\theta, cos\theta) j^=(−sinθ,cosθ)
i ^ t = Ω j ^ \hat{i}_t = \Omega \hat{j} i^t=Ωj^
j ^ t = − Ω i ^ \hat{j}_t = -\Omega \hat{i} j^t=−Ωi^
角速度矢量 Ω ⃗ = ( 0 , 0 , Ω ) \vec{\Omega} = (0,0,\Omega) Ω =(0,0,Ω)，则基矢量的求导可总结为：
u ^ t = Ω ⃗ × u ^ \hat{u}_t = \vec{\Omega}\times \hat{u} u^t=Ω ×u^

f ⃗ \vec{f} f 为一个矢量函数， f ⃗ = f 1 ( t ) i ^ + f 2 ( t ) j ^ + f 3 ( t ) k ^ \vec{f}=f_1(t) \hat{i} + f_2(t) \hat{j} + f_3(t) \hat{k} f =f1(t)i^+f2(t)j^+f3(t)k^
f ⃗ t = ( f ⃗ t ) r + Ω ⃗ × f ⃗ = ( ( d d t ) r + Ω ⃗ × ) f ⃗ \vec{f}_t = (\vec{f}_t)_r + \vec{\Omega} \times \vec{f} = ((\frac{d}{dt})_r + \vec{\Omega} \times) \vec{f} f t=(f t)r+Ω ×f =((dtd)r+Ω ×)f

v ⃗ = r ⃗ t = ( r ⃗ t ) r + Ω ⃗ × r ⃗ \vec{v} = \vec{r}_t = (\vec{r}_t)_r + \vec{\Omega} \times \vec{r} v =r t=(r t)r+Ω ×r
a ⃗ = v ⃗ t = ( ( d d t ) r + Ω ⃗ × ) [ ( r ⃗ t ) r + Ω ⃗ × r ⃗ ] \vec{a} = \vec{v}_t = ((\frac{d}{dt})_r + \vec{\Omega} \times)[(\vec{r}_t)_r + \vec{\Omega} \times \vec{r}] a =v t=((dtd)r+Ω ×)[(r t)r+Ω ×r ]
a ⃗ = a ⃗ r + 2 Ω ⃗ × ( r ⃗ t ) r + ( d Ω ⃗ d t ) r × r ⃗ + Ω ⃗ × ( Ω ⃗ × r ⃗ ) \vec{a} = \vec{a}_r + 2\vec{\Omega} \times (\vec{r}_t)_r + (\frac{d\vec{\Omega}}{dt})_r \times \vec{r} + \vec{\Omega}\times (\vec{\Omega} \times \vec{r}) a =a r+2Ω ×(r t)r+(dtdΩ )r×r +Ω ×(Ω ×r )
Note: 在本例中 Ω ⃗ \vec{\Omega} Ω 在惯性系与旋转系中是一样的，在其他例子中呢？

apparent weight，apparent acceleration，mass，weight

http://labman.phys.utk.edu/phys221core/modules/m3/apparent%20weight.html
https://en.wikipedia.org/wiki/Mass_versus_weight
mass：the amount of “matter” in an object
weight：the force exerted on an object by gravity
apparent weight：the force exerted on an object by gravity and inertial force
apparent acceleration

这也说明惯性力在非惯性系中可以被测出来？，电梯的加速度可通过体重计测出来。