Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends on any graphs == acknowledge study group member f collectve effort [All problems n ths set are from Goldbart] Problem ) Moton of a classcal partcle Consder a classcal partcle of unt mass movng along the x-axs. Suppose that the moton s free except that at tme τ = t wth 0 < t < T the partcle receves an mpulse of unt strength. a) Wrte Newton s equaton descrbng the moton of the partcle. Suppose that at tme τ = 0 the partcle s located at poston x and that at tme τ = T t s located at poston x 2. b) Sketch the poston velocty and acceleraton of the partcle as a functon of tme for 0 < τ < T. c) Compute the Green functon for the moton of the partcle.e. solve d 2 G(τ t)/dτ 2 = δ(τ t). d) Consder the appled force f (τ) (wth 0 τ T) to be a sequence of mpulses. Hence establsh the trajectory of the partcle n terms of an ntegral over the appled force. e) Suppose that the force takes the form f (τ) = τ 2 /2. Fnd the moton of the partcle. f) (optonal) Rather than solve for the Green functon drectly as you dd n part (c) construct the Green functon usng the egenfuncton expanson technque. Show by Fourer analyss that the two schemes for computng the Green functon gve equvalent results. Problem 4) Green functons Consder the second order nhomogeneous lnear dfferental equaton: y y = f (x).
a) Suppose that the boundary condtons for y are y(± ) = 0. By solvng the assocated homogeneous equaton construct the approprate Green functon for ths equaton. b) Solve the dfferental equaton when the source term s gven by f (x) = 2 e x. Problem 6) Green functons and ther nverses The purpose of ths queston s to show that f G s the Green functon assocated wth some operator L then L s the Green functon assocated wth the operator G. Consder the operator L ( d 2 /dθ 2 + q 2 /2π) n whch the real varable q does not vansh. The functons u(θ) on whch L acts are chosen to be perodc functons on the nterval 0 θ < 2π.e. u(2π) = u(0) and u (2π) = u (0). a) By usng the egenfuncton expanson show that the Green functon assocated wth the operator L s gven by G(θ θ ) = 2πq 2 + cos n(θ θ π ) n= q 2 + n 2. Now show that L s the Green functon for G n the followng sense. Consder the nhomogeneous ntegral equaton 2π 0 dθ G(θ θ ) y(θ ) = f (θ) for the unknown functon y(θ) n whch the source term f (θ) s presumed known. (b) Show that the soluton y(θ) s gven by. Optonal problems y(θ) = (L f )(θ) ( d 2 /dθ 2 + q 2 /2π) f (θ). Problem 2) Varaton of parameters Use the method of varaton of parameters to fnd the most general soluton to the second order lnear nhomogeneous dfferental equaton: x 2 y 4xy + 6y = x 4. Problem 3) General soluton to an nhomogeneous equaton Suppose y = x y = x 2 and y = x 3 each satsfy the second-order nhomogeneous equaton Ly = f (x). Fnd the general soluton. 2
Problem 5) More Green functons Fnd the Green functon for these operators: ) Ly (xy ) ; ) Ly (xy ) n 2 y/x; ) Ly x 2 y + xy + (k 2 x 2 )y. Problem 7) Egenfuncton expanson for the bowed strng Green functon In class we studed the Green functon for the response at the drvng frequency ω( ck) of a bowed stretched strng of length l by solvng drectly the equaton ( d 2 dx 2 + k2) G(x x ) = δ(x x ) subject to the boundary condtons G(x x ) x=0l = 0. Use the egenfuncton expanson to produce an alternatve dervaton of ths Green functon and demonstrate that the verson thus obtaned s equvalent to the verson G(x x ) = sn(kx <) sn k(x > l) k sn(kl) obtaned n class n whch x < mn(x x ) and x > max(x x ). Problem 8) Drac delta functon In ths queston we shall explore some of the propertes of the Drac delta functon (the generalsaton to the contnuum of the Kronecker delta symbol). The Drac delta functon δ(x) has as ts argument the real varable x. It has the followng rather strkng propertes:() δ(x) = 0 for x = 0; and dx δ(x) =. From these propertes you can see that δ(x) has an nfntely hgh spke at the orgn s zero elsewhere and has unt area. a) Gve a heurstc (.e. sloppy) proof that f f (x) s a suffcently smooth functon then dx δ(x) f (x) = f (0). b) Show that dx δ(x a) f (x) = f (a). Often we shall neglect to wrte the lmts on ntegrals. c) Usng ntegraton by parts show that dx δ (x) f (x) = f (0) where f (x) d f (x)/dx and δ (x) dδ(x)/dx. d) Consder the famly of gaussan functons ξ (x) { exp x2 } 2πξ 2 2ξ 2 parametrsed by the wdth ξ. Show that the defnng propertes of δ(x) are satsfed by lm ξ 0 ξ (x) and hence that ths lmt represents δ(x). Hnt: recall that dy exp( y2 /2) = 2π. 3
e) By consderng the Fourer transform and ts nverse establsh that f (y) = Thus demonstrate that f (q) = f (x) = δ(y) = dx f (x) dx f (x) e qx 2π dq f (q) eqx 2π dq 2π eqy. dq 2π eq(y x). Ths ntegral representaton of the Drac delta functon s known as the Fourer representaton of the Drac delta functon. f) By consderng the Fourer representaton show that δ(ax) = a δ(x). g) Prove ths result startng wth the gaussan representaton of part (d). We can also have hgher dmensonal delta functons by whch we mean delta functons wth vector arguments. For example f r s a three-dmensonal cartesan vector wth cartesan components x y and z then the delta functon δ (3) ( r) s defned to be the product δ (3) ( r) = δ(x) δ(y) δ(z). Often the superscrpt (3) s omtted. h) Wrte down the Fourer representaton of δ (3) ( r) n terms of the vector r and an ntegral over the vector q. ) Wrte down a representaton for δ (3) ( r) n terms of the gaussan functon. j) A real functon of a sngle varable f (x) has zeros at the set of ponts {x }. At these zeros the gradent of f s non-vanshng and takes the values { f () }. Show that δ ( f (x) ) = δ(x x ) f (). k) A real symmetrc 3 matrx A s non-sngular (.e. all ts egenvalues are non-zero real numbers). Prove usng ether the gaussan or the Fourer representaton that δ (3) (A x) = deta δ(3) ( x). 4
A thrd representaton of the delta functon s very convenent n the context of complex varables. Consder the real varables ω and. If Im denotes the magnary part use the followng strategy to show that l.) Show that l.) Show that lm lm ω 0 0 lm 0 ω 2 + 2 = 0 π Im ω = δ(ω). Im ω = ω 2 + 2. lm lm 0 ω 0 ω 2 + 2 =. l.) Usng the substtuton ω = tan θ or otherwse show that dω ω 2 + 2 = π. l.v) Put together these peces to argue that ndeed lm 0 π Im ω = δ(ω) s a representaton of the delta functon. 5