Chapter 4 - UCSB HEP

WORK AND ENERGY
4.4 Integrating the Equation of Motion in Several Dimensions
Returning to the central problem of this chapter, let us try to
integrate the equation of motion of a particle acted on by a force
which depends on position.
In the case of one dimensional motion we integrated with respect
to position. To generalize this, consider what happens when the
particle moves a short distance Ar.
We assume that Ar is so small that F is effectively constant over
this displacement. If we take the scalar product of Eq. (4.6)
with Ar, we obtain
The sketch shows the trajectory and the force at some point
along the trajectory. At this point,
Fa Ar = F Aarcos 0.
Perhaps you are wondering how we know Ar, since this requires
knowing the trajectory, which is what we are trying to find. Let
us overlook this problem for a few moments and pretend we know
the trajectory,
Now consider the right hand side of Eq. (4.71, m(dv/dt) Ar.
We can transform this by noting that v and Ar are not independent;
for a sufficiently short: length of path, v is approximately constant.
Hence Ar = v At, where at is the time the particle requires to
travel Ar, and therefore
We can transform Eq. (4.7) with the vector identity1
The identity A (dA/dt) = +(d/dl) (Aa) is easily proved: