A ball of mass moves in a vertical plane under uniform gravity

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 ball of mass

m = 0.50\,\mathrm{kg}

moves in a vertical plane under uniform gravity, with

x

horizontal,

y

upward and

L = \tfrac12 m(\dot x^2 + \dot y^2) - mgy, \qquad g = 9.8\,\mathrm{m\,s^{-2}},

air resistance ignored. It leaves the origin at

t_1 = 0

, so

x(0) = y(0) = 0

are fixed. The final time

t_2 = 2.0\,\mathrm{s}

is fixed. ((a)) Write the boundary term of

\delta S

t_2

for the two coordinates, and say what stationarity requires there when one coordinate is fixed at

t_2

and the other is free. ((b)) Suppose

y(t_2) = 0

is fixed (the ball is back at launch height) and

x(t_2)

is left free. Find the launch velocity components

\dot x(0)

\dot y(0)

and the value of

x(t_2)

. ((c)) Suppose instead

x(t_2) = 4.0\,\mathrm{m}

is fixed and

y(t_2)

is left free. Find

\dot x(0)

\dot y(0)

and

y(t_2)