F＝ｍ（ｄ＾2ｒ／ｄｔ＾2）
（Fｘ Fy）＝ｍ（ｘ２回微分 ｙ２回微分）
（Fr FΘ）＝（cosΘ sinΘ）（Fx Fy)
-sinΘ cosΘ
Fr=m(x２回微分cosΘ＋ｙ２回微分sinΘ)
=m［ｒ２回微分-ｒ(Θ１回微分\()^{2}\)］
FΘ=m(-x２回微分sinΘ＋ｙ２回微分)
=m(rΘ２回微分＋２ｒ１回微分Θ１回微分)
=m×\(\frac{1}{r}\)×\(\frac{d}{d}\)t(\(r^{2}\)Θ１回微分)
問い
Fｒ=m[r２回微分-r(Θ１回微分\()^{2}\)］、
FΘ=m×\(\frac{1}{r}\)×\(\frac{d}{d}\)t(\(r^{2}\)Θ１回微分）を証明せよ
まったくわかりません。ホント困ってます。よろしくおねがいします。
Θはシータです。２～４行目は行列と考えてください。
x=r*cosθ
d\(\frac{x}{d}\)t=d\(\frac{r}{d}\)t*cosθ-r*sinθ*dθ/dt
\(d^{2}\)\(\frac{x}{d}\)\(t^{2}\)=\(d^{2}\)\(\frac{r}{d}\)\(t^{2}\)*cosθ-d\(\frac{r}{d}\)t*sinθ*dθ/dt-d\(\frac{r}{d}\)t*sinθ*dθ/dt-r*cosθ*(dθ/dt\()^{2}\)
-r*sinθ*\(d^{2}\)θ/d\(t^{2}\)
=cosθ[\(d^{2}\)\(\frac{r}{d}\)\(t^{2}\)-r*(dθ/dt\()^{2}\)]-sinθ[2*d\(\frac{r}{d}\)t*dθ/dt+r*\(d^{2}\)θ/d\(t^{2}\)]
y=r*sinθ
d\(\frac{y}{d}\)t=d\(\frac{r}{d}\)t*sinθ+r*cosθ*dθ/dt
\(d^{2}\)\(\frac{y}{d}\)\(t^{2}\)=\(d^{2}\)\(\frac{r}{d}\)\(t^{2}\)*sinθd\(\frac{r}{d}\)t*cosθ*dθ/dt+d\(\frac{r}{d}\)t*cosθ*dθ/dt-r*sinθ*(dθ/dt\()^{2}\)
+r*cosθ*\(d^{2}\)θ/d\(t^{2}\)
=sinθ[\(d^{2}\)\(\frac{r}{d}\)\(t^{2}\)-r*(dθ/dt\()^{2}\)]+cosθ*[2*d\(\frac{r}{d}\)t*dθ/dt+r*\(d^{2}\)θ/d\(t^{2}\)]
だから
Fr=[\(d^{2}\)\(\frac{r}{d}\)\(t^{2}\)-r*(dθ/dt\()^{2}\)]
Fθ=[2*d\(\frac{r}{d}\)t*dθ/dt+r*\(d^{2}\)θ/d\(t^{2}\)]