Classical Dynamics of Particles and Systems · Problem 1-32

Newtonian Mechanics—Single Particle — Marion & Thornton

Physics — Mechanics · vector-calculus-integration

Problem

Show that

\int(2a\mathbf{r} \cdot \dot{\mathbf{r}} + 2b\dot{\mathbf{r}} \cdot \ddot{\mathbf{r}})\,dt = ar^2 + b\dot{r}^2 + \text{const.}

where

\mathbf{r}

is the vector from the origin to the point

(x_1, x_2, x_3)

. The quantities

r

and

\dot{r}

are the magnitudes of the vectors

\mathbf{r}

and

\dot{\mathbf{r}}

, respectively, and

a

and

b

are constants.