User:Egm6321.f11.team2.kim/PR1

First total time derivative

$$\displaystyle S=Y^1(t)$$ $$ \displaystyle \frac {d}{dt}f(Y^1(t),t)=\frac {\partial f(S,t)}{\partial S}\frac {\partial S}{\partial t}+\frac {\partial f(S,t)}{\partial t}

= \frac {\partial f(Y^1(t),t)}{\partial S} \frac {\partial}{\partial t}(Y^1(t))+\frac {\partial f(S,t)}{\partial t} = \frac {\partial f(Y^1(t),t)}{\partial S} \dot Y^1 +\frac {\partial f(Y^1(t),t)}{\partial t} $$ Second total time derivative $$ \displaystyle \frac {d}{dt}(\frac {\partial f(Y^1(t),t)}{\partial S} \dot Y^1 +\frac {\partial f(Y^1(t),t)}{\partial t})= \frac {d}{dt}(\frac {\partial f(S,t)}{\partial S}\frac {\partial S}{\partial t}+\frac {\partial f(S,t)}{\partial t}) $$ $$ =\displaystyle \frac {\partial^2 f(S,t)}{\partial S \partial S}\frac {\partial S}{\partial t} \frac {\partial S}{\partial t} + \frac {\partial^2 f(S,t)}{\partial S \partial t} \frac {\partial S}{\partial t} + \frac {\partial f(S,t)}{\partial S} \frac {\partial^2 S}{\partial t^2} + \frac {\partial^2 f(S,t)}{\partial S}{\partial t} \frac{\partial S}{\partial t} + \frac {\partial^2 f(S,t)}{\partial t^2}

$$ $$ =\displaystyle \frac {\partial^2 f(S,t)}{\partial S \partial S}\dot Y^1\dot Y^1 + \frac {\partial^2 f(S,t)}{\partial S \partial t} \dot Y^1 + \frac {\partial f(S,t)}{\partial S} \ddot Y^1 + \frac {\partial^2 f(S,t)}{\partial S \partial t} \dot Y^1 + \frac {\partial^2 f(S,t)}{\partial t^2} $$ $$ =\displaystyle \frac {\partial^2 f(Y^1(t),t)}{\partial S \partial S}(\dot Y^1)^2 + \frac {\partial^2 f(Y^1(t),t)}{\partial S \partial t} \dot Y^1 + \frac {\partial f(Y^1(t),t)}{\partial S} \ddot Y^1 + \frac {\partial^2 f(Y^1(t),t)}{\partial S \partial t} \dot Y^1 + \frac {\partial^2 f(Y^1(t),t)}{\partial t^2} $$ $$\displaystyle \frac{d^2f}{dt^2}=f_{,S}(Y^1,t)\ddot Y^1 + f_{,SS}(Y^1,t)(\dot Y^1)^2 + 2f_{,St}(Y^1,t)\dot Y^1 + f_{,tt}(Y^1,t) $$

$$ \frac{d^2\mathbf{A}}{dt^2} = \frac{d}{dt}(\frac{d'\mathbf{A}}{dt} +\mathbf{\Omega} \times \mathbf{A}) $$
 * $$ = \frac{d}{dt}(\frac{d'\mathbf{A}}{dt}) + \mathbf{\Omega} \times \frac{d\mathbf{A}}{dt} + \frac{d\mathbf{\Omega}}{dt} \times \mathbf{A} $$
 * $$ = \frac{d'^2\mathbf{A}}{dt^2} + \mathbf{\Omega} \times \frac{d'\mathbf{A}}{dt} + \mathbf{\Omega} \times (\frac{d'\mathbf{A}}{dt} +\mathbf{\Omega} \times \mathbf{A}) + \frac{d\mathbf{\Omega}}{dt} \times \mathbf{A} $$
 * $$ = \frac{d'^2\mathbf{A}}{dt^2} + 2\mathbf{\Omega} \times \frac{d'\mathbf{A}}{dt} + \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{A}) + \frac{d\mathbf{\Omega}}{dt} \times \mathbf{A} $$

$$\frac{d\mathbf{A}}{dt} =  \frac{d\mathbf{A'}}{dt} + V $$
 * $$=\mathbf{v'}+ V $$
 * $$=\mathbf{\dot{A_x'}} \hat \mathbf{x'} +\mathbf{\dot{A_y'}} \hat \mathbf{y'} +\mathbf{\dot{A_z'}} \hat \mathbf{z'}+ +\mathbf{A_x'}\frac{d\mathbf{\hat x'}}{dt} + \mathbf{A_y'}\frac{d\mathbf{\hat y'}}{dt}+\mathbf{A_z'}\frac{d\mathbf{\hat z'}}{dt} $$

이를 바탕으로 회전좌표계에서 x', y', z' 의 단위벡터의 회전을 적용하여 표현하면 다음과 같다.

$$\frac{d\mathbf{A}}{dt} =  \frac{d'\mathbf{A}}{dt} +\mathbf{A_x'}(\mathbf{\Omega} \times \mathbf{\hat x'}) + \mathbf{A_y'}(\mathbf{\Omega} \times \mathbf{\hat y'}) + \mathbf{A_z'}(\mathbf{\Omega} \times \mathbf{\hat z'})$$
 * $$ = \frac{d'\mathbf{A}}{dt} +\mathbf{\Omega} \times (\mathbf{A_x'}\mathbf{\hat x'} + \mathbf{A_y'}\mathbf{\hat y'} + \mathbf{A_z'}\mathbf{\hat z'})$$
 * $$ = \frac{d'\mathbf{A}}{dt} +\mathbf{\Omega} \times \mathbf{A} $$