Classical Mechanics (Kibble & Berkshire) · Problem 13.10

Hamilton's equations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · hamiltons-principle

Problem

A particle of mass 2 kg moves under uniform gravity along the

z

-axis, which points vertically downwards. Show that (in SI units) the action functional for the time interval

[0, 2]

S[z] = \int_0^2 \left(\dot{z}^2 + 20z\right) dt,

where

g

has been taken to be

10\, \text{m s}^{-2}

. Show directly that, of all the functions

z(t)

that satisfy the end conditions

z(0) = 0

and

z(2) = 20

, the actual motion

z = 5t^2

provides the least value of

S