A stone of mass is thrown straight up from ground level at

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

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

Problem

A stone of mass

m = 0.20\,\mathrm{kg}

is thrown straight up from ground level at

t_1 = 0

, so

x(0) = 0

is fixed, with

x

measured upward and

L = \tfrac12 m\dot x^2 - mgx, \qquad g = 9.8\,\mathrm{m\,s^{-2}},

air resistance ignored. Its height at

t_2 = 1.5\,\mathrm s

is left free (unconstrained) in the variation. ((a)) Derive the natural boundary condition that stationarity of

S

imposes at

t_2

when

x(t_2)

is free. ((b)) Use this condition together with the equation of motion to find the launch speed

\dot x(0)

and the height

x(1.5\,\mathrm s)

. ((c)) Suppose instead one used

L' = L + \dfrac{\mathrm d}{\mathrm dt}(\alpha x t)

with

\alpha = 3.0\,\mathrm N

a constant. Show that the equation of motion is unchanged, and find the momentum

p'

that enters the natural boundary condition for

L'

t_2

, in terms of

\alpha

t_2

and

\dot x(t_2)

. State whether

p'(t_2) = 0

automatically once

p(t_2) = 0

for the original

L