Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.3 SPACE CURVES

so we finally obtain

dˆb

ds

=−τˆn. (10.14)

Taking the dot product of each side withnˆ, we see that the torsion of a curve is

given by

τ=−nˆ·

dˆb

ds

.

We may also define the quantityσ=1/τ, which is called theradius of torsion.

Finally, we consider the derivativednˆ/ds.Sincenˆ=ˆb׈twe have

dnˆ

ds

=

dbˆ

ds

׈t+ bˆ×

dˆt

ds

=−τnˆ×ˆt+ bˆ×κˆn

=τˆb−κˆt. (10.15)

In summary,ˆt,nˆandbˆand their derivatives with respect tosare related to one

another by the relations (10.13), (10.14) and (10.15), theFrenet–Serret formulae,

dˆt

ds

=κˆn,

dnˆ

ds

=τˆb−κˆt,

dbˆ

ds

=−τnˆ. (10.16)

Show that the acceleration of a particle travelling along a trajectoryr(t)is given by

a(t)=

dv

dt

ˆt+v

2

ρ

nˆ,

wherevis the speed of the particle,ˆtis the unit tangent to the trajectory,nˆis its principal

normal andρis its radius of curvature.

The velocity of the particle is given by

v(t)=

dr

dt

=

dr

ds

ds

dt

=

ds

dt

ˆt,

whereds/dtis the speed of the particle, which we denote byv,andˆtis the unit vector
tangent to the trajectory. Writing the velocity asv=vˆt, and differentiating once more
with respect to timet,weobtain

a(t)=

dv

dt

=

dv

dt

ˆt+vd

ˆt

dt

;

but we note that
dˆt
dt

=

ds

dt

dˆt

ds

=vκnˆ=

v

ρ

nˆ.

Therefore, we have

a(t)=

dv

dt

ˆt+v

2

ρ

nˆ.

This shows that in addition to an accelerationdv/dtalong the tangent to the particle’s
trajectory, there is also an accelerationv^2 /ρin the direction of the principal normal. The
latter is often called thecentripetalacceleration.