S7: Classical mechanics problem set 2

Transcription

1 J. Magoian MT 9, boowing fom J. J. Binney s 6 couse S7: Classical mechanics poblem set. Show that if the Hamiltonian is indepdent of a genealized co-odinate q, then the conjugate momentum p is a constant of motion. Such co-odinates ae called cyclic co-odinates. Give two examples of physical systems that have a cyclic co-odinate. Ans: Using ṗ i = / q i it s obvious that if H doesn t depend explicitly on, say, q, then p is conseved. Examples: p φ is conseved in axisymmetic potential V R, z); p z is conseved fo motion in a magnetic field B = B ˆk; p x, p y, p z ae conseved fo fee paticles, etc.. A dynamical system has genealized co-odinates q i and genealized momenta p i. Veify the following popeties of the Poisson backets: [q i, q j ] = [p i, p j ] =, [q i, p j ] = δ ij. -) Ans: Fom definition of PB Similaly, [q i, q j] and [p i, p j] ae obviously zeo. q i p j q i p j [q i, p j] = = δ ikδ jk = δ ij. -) q k p k p k q k k k k If p is the momentum conjugate to a position vecto and L = p, evaluate [L x, L y ], [L y, L x ] and [L x, L x ]. Ans: By antisymmety of PBs, [L x, L x] = and [L x, L y] = [L y, L x]. So we need only calculate [L x, L y] = [yp z zp y, zp z xp z]. One way to do this is to use the lineaity, antisymmety and chain ule fo PBs to educe the expession to something involving the canonical commutation elations see lectues). Anothe is to apply the definition of PB diectly: [L x, L y] = ypz zpy) zp z xp z) yp z zp y) zpz xpz) x p x p x x + ypz zpy) zp z xp z) yp z zp y) zpz xpz) y p y p y y + ypz zpy) z = p y) x) yp x = L z. p z zp z xp z) yp z zp y) zpz xpz) p z z -3) This is the ovely cautious way of witing out. A moe sensible answe would point out that L x is independent of x and p x and L y is independent y and p y, so the fist two lines above must clealy be zeo and we need only conside the thid.

2 The Lagangian of a paticle of mass m and chage e in a unifom magnetic field B and electostatic potential φ is L = mṙ + eṙ B ) eφ. -4) Deive the coesponding Hamiltonian and veify that the ate of change of mṙ equals the Loentz foce. Ans: To avoid ambiguity let us use index notation when we take gadients with espect to eithe o ṙ. The i th component of the momentum is then p i ) L ṙ = mṙ i + eb )i, i -5) fom which we have that p = mṙ + eb ). The Hamiltonian is the Legende tansfom of L w..t. ṙ: H = p ṙ L = mṙ + eb ) ṙ L = mṙ + eφ = m [ p eb )] + eφ. -6) To veify that the ate of change of mṙ is given by the Loentz foce, expand each side of Hamilton s equation ṗ = / componentwise: RHS: LHS: i p i = d dt mṙi + eb ṙ)i = m as usual) [ p eb )] [ p eb )] e φ. i i -7) We can expand the RHS futhe using the following emembe that, p ae independent and that B is independent of ): p B ) = [p B )] = [ p B)] = p B) i, i i i [ B ) B ] ) = B [ ) = B B ) ] = B i B )B i. i i i -8) Using p = mṙ + eb ) and -8) to expand ṗ = / gives ṗ = d dt mṙ + eb ṙ) = e [mṙ + ] m eb ) B e φ B ) B e 4m, -9) which simplifies to d mṙ = eṙ B e φ/ ) as expected. dt Show that the momentum component along B and the sum of the squaes of the momentum components ae all constants of motion. Find anothe constant of motion associated with time tanslation symmety. Ans: Notice ambiguity! Momentum could mean eithe p o mṙ. We take the latte. Dot -9) with B: d φ p B = B ṗ = eb = if φ constant. -) dt So component of p along B is conseved. So too is component of mṙ = p eb ). To show othe two components ae constant, assume φ = and dot Loentz foce equation with mṙ: d dt m ṙ ) = mṙ d mṙ = eṙ B) ṙ =. -) dt So mṙ) is conseved. Since B mṙ is along conseved, must have that sum-squae of othe two components of mṙ but not p) is conseved too. The constant of motion associated with time tanslation symmety is H itself.

3 3. Let p and q be canonically conjugate co-odinates and let fp, q) and gp, q) be functions on phase space. Define the Poisson backet [f, g]. Let Hp, q) be the Hamiltonian that govens the system s dynamics. Wite down the equations of motion of p and q in tems of H and the Poisson backet. Ans: Definition of PB: Hamilton s equations ae q = [q, H], ṗ = [p, H]. [f, g] = f g q p f g p q. 3-) In a galaxy the density of stas in phase space is fq, p, t), whee q and p each have thee components. When evaluated at the location qt), pt)) of any given sta, f is time-independent. Show that f consequently satisfies f + [f, H] =, 3-) t whee H is the Hamiltonian that govens the motion of evey sta. Ans: f is constant along obits so df/dt =. Equation 3-) follows on using the chain ule to wite df/dt = f/ t + q f/ q) + ṗ f/ q) and then substituting fo q, ṗ) fom Hamilton s equations. Conside motion in a cicula azo-thin galaxy in which the potential of any sta is given by the function V R), whee R is a adial co-odinate. Expess H in tems of plane pola co-odinates R, φ) and thei conjugate momenta, with the oigin coinciding with the galaxy s cente. Hence, o othewise, show that in this system f satisfies the equation whee m is the mass of the sta. f t + p R f m R + p φ f mr φ V R p φ mr 3 ) f p R =, 3-3) Ans: Standad pocedue: Wite down Lagangian L in tems of plane-pola co-odinates; this L defines momenta conjugate to R, φ) though p L/ q; take Legende tansfom of L to get H. The Lagangian L = m Ṙ + R φ ) V R), 3-4) so that the momenta p R = mṙ and pφ = mr φ. Taking the Legende tansfom, H = p R Ṙ + p φ φ mṙ + R φ ) + V R) [ ] = p R + p φ + V R), m R 3-5) whee the genealized velocities Ṙ and φ have been expessed in tems of the phase-space co-odinates R, φ, p R, p φ). To obtain 3-3) stat fom 3-), but witing out the [f, H] explicitly: = f f + [f, H] = t t + f R fom which the equied esult follows. p R f p R R + f φ f φ, 3-6) 3

4 4. Show that in spheical pola co-odinates the Hamiltonian of a paticle of mass m moving in a potential V x) is ) H = p + p θ m + p φ sin + V x). 4-) θ Show that p φ = constant when V/ φ and intepet this esult physically. Ans: Stating fom x = sin θ cos φ etc, we can show that the paticle s velocity satisfies ẋ = ṙ + sin θ φ + θ and so the Lagangian Using p i L/ q i, the momenta Taking the Legende tansfom of L, L = mẋ V = m [ ṙ + sin θ φ + θ ] V. 4-) p = mṙ, p θ = m θ, p φ = m sin θ φ. 4-3) H = p q L = mṙ + m θ + m sin θ φ L = m [ ṙ + sin θ φ + θ ] + V, θ, φ), 4-4) fom which 4-) follows on using 4-3) to expess the genealized velocities ṙ, θ, φ) in tems of phase-space co-odinates, θ, φ, p, p θ, p φ). If V does not depend on φ then ṗ φ = / φ = V/ φ = and so p φ the angula momentum about the z axis) is conseved. Given that V depends only on, show that [H, K] =, whee K p θ + p φ / sin θ. By expessing K as a function of θ and φ intepet this esult physically. Ans: One way of showing [H, K] = is by witing out the six tems in the Poisson backet explicitly. Altenatively, note that the Hamiltonian 4-) can be witten H = p /m + K/m + V and so [H, K] = m [p, K] + [ ] K m, K + [V, K]. 4-5) The fist tem vanishes because K does not depend on. Similaly, the final tem vanishes because V = V ) and K does not depend on p. Using the chain ule fo PBs, the middle tem [K/, K] = [K, K] + [/, K]K, 4-6) but clealy [K, K] = and [/, K] = since K is independent of p. Theefoe all tems in 4-5) vanish and so [H, K] =, meaning that K is a constant of motion. Witing K out in tems of genealized velocities θ, φ, K = m [ θ) + sin θ φ) ] = m v tangential, 4-7) which is the squae of the total angula momentum. It vanishes because the potential is spheically symmetic. Conside cicula motion with angula momentum h in a spheical potential V ). Evaulate p θ θ) when the obit s plane is inclined by ψ to the equatoial plane. Show that p θ = when sin θ = ± cos ψ and intepet this esult physically. Ans: The obit is inclined at an angle ψ, so p φ = h cos ψ. Using h = K = p θ + p φ/ sin θ, we have that ) p θ = h cos ψ, 4-8) sin θ which tends to zeo as sin θ ± cos ψ the paticle is at its tuning point in the R, z) plane, which is whee both p θ and θ ae zeo. 4

5 5. Oblate spheoidal co-odinates u, v, φ) ae elated to egula cylindical polas R, z, φ) by R = cosh u cos v; z = sinh u sin v. 5-) Fo a paticle of mass m show that the momenta conjugate to these co-odinates ae p u = m cosh u cos v) u, p v = m cosh u cos v) v, p φ = m cosh u cos v φ. 5-) Hence show that the Hamiltonian fo motion in a potential Φu, v) is H = p u + p v m cosh u cos v) + p φ m cosh + Φ. 5-3) u cos v Ans: Stat fom L = m[ṙ + R φ) + ż ] Φ. Have that R = cosh u cos v z = sinh u sin v Ṙ = u sinh u cos v v cosh u sin v ż = u cosh u sin v + v sinh u cos v 5-4) So Ṙ + ż = u [sinh u cos v + cosh u sin v] + v [cosh u sin v + sinh u cos v] = u [cosh u ) sin v) + cosh u sin v] + v [cosh u cos v) + sinh u cos v] = [ u + v ] cosh u cos v) 5-5) and L = m [ cosh u cos v) u + v ) + cosh u cos v φ ] Φ. 5-6) The momenta 5-) dop out using p u L/ u etc. Taking the Legende tansfom of L w..t. u, v, φ) we have that p u H = p u u + p v v + p φ φ L = m cosh u cos v) + p v m cosh u cos v) + p φ m cosh u cos v [ ) ] p m u + p v m ) cosh u cos v) + p φ + Φ, m cosh u cos v 5-7) using 5-) to obtain u = p u/m cosh u cos v) etc. Simplifying gives the equied expession. Show that [H, p φ ] = and hence that p φ is a constant of motion. Identify it physically. Ans: [H, p φ] = u p u pφ u + p u v p v pφ v + p v φ pφ φ, 5-8) all but one of the tems being zeo since w i/ w j = δ ij, whee w u, v, φ, p u, p v, p φ). Remembe that phase-space co-odinates ae independent of one anothe!) The emaining tem involves / φ which vanishes, since the potential and theefoe H does not depend on φ. The conseved momentum, p φ, is the angula momentum about the symmety axis. 5

6 6. A paticle of mass m and chage Q moves in the equatoial plane θ = π/ of a magnetic dipole. Given that the dipole has vecto potential A = µ sin θ 4π êφ, 6-) evaluate the Hamiltonian Hp, p φ,, φ) of the system. Ans: You might be tempted to use H = p QA) + Qφ, but that s fo Catesian p. One possibility is to make m a canonical map to new pola co-ods, but it s simple to go back to basics and edeive H fom L. Take LT of L, L = mẋ + Qẋ A φ) = m[ṙ + φ ] + Q φ) µ 4π. 6-) p = mṙ, p φ = m φ + Qµ 4π φ = p φ Qµ 4π H = p ṙ + p φ φ L = mṙ + φ ) = [p + p φ Qµ ) ]. m 4π ) / m. 6-3) 6-4) The paticle appoaches the dipole fom infinity at speed v and impact paamete b. Show that p φ and the paticle s speed ae constants of motion. Ans: H does not depend explicitly on φ, so p φ = const. We know that H is conseved, but fom 6-4) H = mv and so the speed v is constant. Show futhe that fo Qµ > the distance of closest appoach to the dipole is D = { b b a fo φ >, b + b + a fo φ <, 6-5) whee a µq/πmv. Ans: p φ = m φ + Qµ/4π is conseved. Let s look at p φ as and fo geneal :, ± depending on sign φ) ± mbv = m φ + Qµ 4π = ±mv + Qµ 4π ±b = ± + 4 a b ± 4 a = [ ] = b ± b a, geneal ) when ṙ = 6-6) choosing fist ± sign to ensue minimum >. 6

7 7. An axisymmetic top has Lagangian L = I φ sin θ + θ ) + I 3 φ cos θ + ψ) mga cos θ, 7-) whee θ, φ, ψ) ae the usual Eule angles. Show that the top s Hamiltonian H = p θ + p φ p ψ cos θ) I I sin + p ψ + mga cos θ. 7-) θ I 3 Using Hamilton s equations o othewise show that the top will pecess steadily at fixed inclination to the vetical povided θ satisfies = mga + p φ p ψ cos θ)p φ cos θ p ψ ) I sin ) θ Ans: The Hamiltonian is deived in the lectue notes. Fo the top to pecess steadily at fixed inclination we equie that θ = ṗ θ =. Using Hamilton s equation fo the ate of change of p θ I θ, = θ pφ pψ cos θ) pφ pψ cos θ) = p I sin ψ sin θ cos θ mga sin θ θ I sin 3 θ = pφ pψ cos θ)[pψ sin θ p φ p ψ cos θ) cos θ] mga sin θ I sin 3 θ pφ pψ cos θ)[pψ pφ cos θ] = mga sin θ. I sin 3 θ 7-4) 7

8 8. A point chage q is placed at the oigin in the magnetic field geneated by a spatially confined cuent distibution. Given that E = q 4πɛ 3 8-) and B = A with A =, show that the field s momentum P ɛ E B d 3 x = qa). 8-) Use this esult to intepet the fomula fo the canonical momentum of a chaged paticle in an electomagnetic field. [Hint: use B = A and then index notation easy) o vecto identities not so easy) to expand E B into a sum of two tems. To each tem apply the tenso fom of Gauss s theoem, which states that d 3 x i T = d S i T, no matte how many indices the tenso T caies. In one tem you can make use of A = and in the othe = 4πδ 3 ).] Ans: Easy to show that P = q 4π d 3 ) A). 8-3) Pemutation tenso Levi-Civita symbol) ɛ 3 = ɛ 3 = ɛ 3 = even pem,, 3) ɛ 3 = ɛ 3 = ɛ 3 = odd pem,, 3) all othe =. 8-4) Handy because summation convention) Useful identity contact ove middle index) a b) i = ɛ ijka jb k a) i = ɛ ijk ja k = ɛ ijk x j a k 8-5) ɛ ijkɛ klm = δ ilδ jm δ jlδ im 8-6) Then [ ) ] ) A) = ɛ ijk j ɛ klm la m ) i = δ ilδ jm δ jlδ im) j ) = j ia j j Integating by pats stictly, Gauss theoem) integate ) la m ) ja i. { }} ){ ) A= d 3 x j ia j = d S j iaj d 3 x {}}{ i ja j ) d 3 x j ja i integate =. = ) d S j j A i = 4πA i). d 3 x ) j j = 4πδ) A i 8-7) 8-8) To intepet, ecall that the canonical momentum fo a paticle in a magnetic field p = mṙ + qa... 8