Classical Mechanics (Kibble & Berkshire) · Problem 13.6

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

Physics — Mechanics · euler-lagrange-equation

Problem

Cost functional A manufacturer wishes to minimise the cost functional

C[x] = \int_0^4 \left((3 + \dot{x})\dot{x} + 2x\right) dt

subject to the conditions

x(0) = 0

and

x(4) = X

, where

X

is volume of goods to be produced. Find the extremal of

C

that satisfies the given conditions and prove that this function provides the global minimum of

C

. Why is this solution not applicable when

X < 8