A particle of mass moves in a plane under the potential with

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 = 0.50\,\mathrm{kg}

moves in a plane under the potential

V(r) = \tfrac12 kr^2

with

k = 8.0\,\mathrm{N\,m^{-1}}

, so that

L = \tfrac12 m(\dot r^2 + r^2\dot\phi^2) - \tfrac12 kr^2.

At some instant

r = 0.20\,\mathrm m

\dot r = 0

and

\dot\phi = 5.0\,\mathrm{rad\,s^{-1}}

. ((a)) Explain why

L

has no explicit dependence on the independent variable

t

, and use this to write down the conserved quantity

h = \dot r\,\dfrac{\partial L}{\partial\dot r} + \dot\phi\,\dfrac{\partial L}{\partial\dot\phi} - L

, simplifying it in terms of the kinetic energy

T

and potential energy

V

. ((b)) Evaluate

h

at the given instant, in joules.