Introduction to Classical Mechanics · Problem 3.67

Oscillations — Morin

Physics — Mechanics · polar-coordinates-derivation

Problem

Derivation of

F_r

and

F_\theta

* In Cartesian coordinates, a general vector takes the form,

\mathbf{r} = x\hat{x} + y\hat{y} = r\cos\theta\,\hat{x} + r\sin\theta\,\hat{y}. \tag{3.56}

Derive Eq. (3.51) by taking two derivatives of this expression for

\mathbf{r}

, and then using Eq. (3.46) to show that the result can be written in the form of Eq. (3.50). Note that unlike

\hat{r}

and

\hat{\theta}

, the vectors

\hat{x}

and

\hat{y}

do not change with time.