Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.6 EXERCISES 21.18 A sphere of radiusaand thermal conductivityk 1 is surrounded by an infinite medium of conductivityk 2 in wh ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS 21.22 Point chargesqand−qa/b(witha<b) are placed, respectively, at a pointP,a ...

21.7 Hints and answers inVand takes the specified formφ=fonS, the boundary ofV.The Green’s function,G(r,r′), to be used satisfie ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS 21.25 The terms inG(r,r 0 ) that are additional to the fundamental solution are ...

22 Calculus of variations In chapters 2 and 5 we discussed how to find stationary values of functions of a single variablef(x), ...

CALCULUS OF VARIATIONS y a b x Figure 22.1 Possible paths for the integral (22.1). The solid line is the curve along which the i ...

22.2 Special cases to these variations, we require dI dα ∣ ∣ ∣ ∣ α=0 = 0 for allη(x). (22.3) Substituting (22.2) into (22.1) and ...

CALCULUS OF VARIATIONS A (a, y(a)) dx dy ds B (b, y(b)) Figure 22.2 An arbitrary path between two fixed points.
Show that the s ...

22.2 SPECIAL CASES dx y ds dy x Figure 22.3 A convex closed curve that is symmetrical about thex-axis. 22.2.2Fdoes not containxe ...

CALCULUS OF VARIATIONS we can use (22.8) to obtain a first integral of the EL equation fory,namely y(1−y′^2 )^1 /^2 +yy′^2 (1−y′ ...

22.3 SOME EXTENSIONS (a) (b) (c) b −b z ρ a Figure 22.4 Possible soap films between two parallel circular rings. surface area be ...

CALCULUS OF VARIATIONS 22.3.1 Several dependent variables Here we haveF=F(y 1 ,y′ 1 ,y 2 ,y 2 ′,...,yn,yn′,x)whereeachyi=yi(x). ...

22.3 SOME EXTENSIONS ∆x ∆y y(x) h(x, y)=0 y(x)+η(x) b Figure 22.5 Variation of the end-pointbalong the curveh(x, y)=0. that we r ...

CALCULUS OF VARIATIONS A y x=x 0 B x Figure 22.6 A frictionless wire along which a small bead slides. We seek the shape of the w ...

22.4 CONSTRAINED VARIATION wherekis a constant. Lettinga=k^2 and solving fory′we find y′= dy dx = √ a−y y , which on substitutin ...

CALCULUS OF VARIATIONS −a y O a x Figure 22.7 A uniform rope with fixed end-points suspended under gravity. which, together with ...

22.5 Physical variational principles wherekis a constant; this reduces to y′^2 = ( ρgy+λ k ) 2 − 1. Making the substitutionρgy+λ ...

CALCULUS OF VARIATIONS θ 1 θ 2 n 1 n 2 A B x y Figure 22.8 Path of a light ray at the plane interface between media with refract ...

22.5 PHYSICAL VARIATIONAL PRINCIPLES y O dx l x Figure 22.9 Transverse displacement on a taut string that is fixed at two points ...

CALCULUS OF VARIATIONS Using (22.13) and the fact thatydoes not appear explicitly, we obtain ∂ ∂t ( ρ ∂y ∂t ) − ∂ ∂x ( τ ∂y ∂x ) ...