A particle of mass moves about a fixed centre in the central potential

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Lagrangian mechanics

Action and Symmetry, Ch Symmetry, Noether's theorem and what is conserved

Problem

A particle of mass

m

moves about a fixed centre in the central potential

V(r) = -k'/r^n

, with

k'>0

and

n

a positive real number,

n\neq1

(the case

n=1

is the inverse-square force). Following the same steps used for the inverse-square force, attempt to build a constant vector

\mathbf A = \mathbf p\times\mathbf L - mk\hat{\mathbf r}

for some constant

k

. ((a)) Using

\dot{\mathbf p} = f(r)\hat{\mathbf r}

with

f(r) = -\mathrm dV/\mathrm dr

, and

\mathrm d\hat{\mathbf r}/\mathrm dt = (\mathbf v - \hat{\mathbf r}(\hat{\mathbf r}\cdot\mathbf v))/r

, show that

\frac{\mathrm d\mathbf A}{\mathrm dt} = -m\big(f(r)r^2 + k\big)\dot{\hat{\mathbf r}},

and evaluate

f(r)r^2

for this potential as a function of

n

k'

and

r

. ((b)) Explain why no constant

k

can make

\mathbf A

constant along a generic (non-circular) orbit unless

n=1

. ((c)) For

n=3

, find the factor by which the magnitude of the leftover term

f(r)r^2

changes when

r

doubles, and state this as a percentage change.