A particle of mass moves in a plane under an attractive central force with potential ,

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

Action and Symmetry, Ch The action, the first variation and the boundary term

Problem

A particle of mass

m = 1.0\,\mathrm{kg}

moves in a plane under an attractive central force with potential

V(r) = -k/r

k = 2.0\,\mathrm{N\,m^2} > 0

, so that

L = \tfrac12 m(\dot r^2 + r^2\dot\phi^2) + \frac{k}{r}.

Both endpoints of

r

and

\phi

are held fixed in the variation. ((a)) Derive the two Euler-Lagrange equations for

r

and

\phi

. ((b)) At an instant when

r = 0.50\,\mathrm{m}

\dot\phi = 2.0\,\mathrm{rad\,s^{-1}}

and

\dot r = 0

, evaluate

\ddot r

. ((c)) State the quantity conserved as a consequence of the

\phi

equation, and give its value at this instant.