Mathematical Methods for Physicists: A concise introduction - Site Map

MOTION IN A PLANE
Figure 1.11.
Parametric representation of a curve.
is a vector in the direction of r, and its limit (if it exists) dr=du is a vector in the
direction of the tangent to the curve at …x 1 ; x 2 ; x 3 †.Ifu is the arc length s measured
from some ®xed point on the curve C, then dr=ds ˆ ^T is a unit tangent vector to
the curve C. The rate at which ^T changes with respect to s is a measure of the
curvature of C and is given by d ^T/ds. The direction of d ^T/ds at any given point on
C is normal to the curve at that point: ^T ^T ˆ 1, d… ^T ^T†=ds ˆ 0, from this we
get ^T d ^T=ds ˆ 0, so they are normal to each other. If ^N is a unit vector in this
normal direction (called the principal normal to the curve), then d ^T=ds ˆ ^N,
and is called the curvature of C at the speci®ed point. The quantity ˆ 1= is
called the radius of curvature. In physics, we often study the motion of particles
along curves, so the above results may be of value.
In mechanics, the parameter u is time t, then dr=dt ˆ v is the velocity
of the particle which is tangent to the curve at the speci®c point. Now we
can write
v ˆ dr
dt ˆ dr ds
ds dt ˆ v ^T
where v is the magnitude of v, called the speed. Similarly, a ˆ dv=dt is the acceleration
of the particle.
Motion in a plane
Consider a particle P moving in a plane along a curve C (Fig. 1.12). Now r ˆ r^e r ,
where ^e r is a unit vector in the direction of r. Hence
v ˆ dr
dt ˆ dr
dt ^e r ‡ r d^e r
dt :
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