Hamiltonian with z as the Independent Variable

Transcription

1 Hamiltonian with z as the Independent Variable 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ Marh 19, 2011; updated June 19, 2015) Dedue the form of the Hamiltonian when z rather than t is onsidered to be the independent variable. Illustrate this for the ase of a partile of harge q and mass m in an external eletromagneti field. 2 Solution This solution follows Appendix B of [1]. See also se. 1.6 of [2]. For simpliity we onsider only a single partile. 2.1 Use of t as the Independent Variable We reall the usual Hamiltonian desription of a partile of harge q and mass m in external eletromagneti fields E and B, whih an be dedued from salar and vetor potentials V and A in some gauge) aording to E = V 1 A, B = A, 1) H t x, y, z, p x,p y,p z ) = E meh + qv = m p 2 meh,x + p2 meh,y + p2 meh,z + qv 2) = m 2 2 +p x qa x /) 2 +p y qa y /) 2 +p z qa z /) 2 + qv, in Gaussian units, where is the speed of light in vauum, and the omponents of p = p meh + qa/ are the anonial momenta assoiated with oordinates x =x, y, z). The subsript on H t indiates that time t is the independent variable in this Hamiltonian. Hamilton s equations of motion for this ase are dx i dp i = H t p i = H t x i = 2 p meh,i = v i, 3) E meh v A q V x i x i = q = dp meh,i + q da i = dp meh,i + q A i + q v A i x, 4) using the onvetive derivative da/ = A/ +v )A forthevetorpotentialatthe position of the moving partile. Hene, dp meh,i = q V 1 A i x i + v A A ) i = q E + v ) x i x B = F Lorentz,i, 5) i 1

2 suh that the equations of motion for the mehanial momentum p meh is gauge invariant, although the Hamiltonian 2) is not. 2.2 Use of z as the Independent Variable In some appliations, suh as transport of partiles in aelerators and storage rings, it is often preferable to onsider a set of partiles at fixed values of a spatial oordinate, say z, rather than at fixed time. 1 So, we seek a Hamiltonian formalism in whih z is the independent variable, and t is the third q-oordinate, along with x and y. We must identify a anonial momentum p t that is onugate to oordinate t, and a Hamiltonian H z x, y, t, p x,p y,p t )suh that the equations of motion an be dedued from this Hamiltonian in the usual way. We antiipate that the total) energy is onugate to the time oordinate, so we tentatively identify p t? = E total = E meh + qv = H t. 6) We might then guess that, by analogy, the desired Hamiltonian H z equals the anonial momentum p z, H? z = p z = p meh,z + qa z Emeh 2 = m 2 2 p 2 2 meh,x p2 meh,y + qa z p t qv ) = 2 m 2 2 p 2 x qa ) 2 x p x qa ) 2 x + qa z. 7) The test is whether the equations of motion that follow from these identifiations are onsistent with those assoiated with H t. dx? = H z p x = p meh,x p meh,z = v x. 8) The magnitude is orret, but the sign is wrong. This suggests that there should have been a minus sign in both eqs. 6) and 7), H z = p z = p meh,z qa z = p t + qv ) = 2 m 2 2 Now, as desired, 2 p t = E total = E meh qv = H t, 9) = H z p t E 2 meh 2 p x qa x m 2 2 p 2 meh,x p 2 meh,y qa z ) 2 p x qa ) 2 x qa z. 10) = E meh 2 p meh,z = 1. 11) 1 It is often desirable that the new independent variable be the path length s along a urved, entral traetory in, say, a ring. However, only in the linear approximation an the formalism of this setion be applied to a urvilinear oordinate s. 2

3 Also, and hene, Finally, dp meh,x and hene, dp x = H z x = q V x + q = dp meh,x = q + q da x V x 1 A x + q v = dp meh,x v A i x + q A x + q v A x, 12) x A x A ) x = q E + v ) x B x = F Lorentz,x. 13) dp t de meh = H z = de meh = q v = q V + q v A q dv = de meh q V q v V, 14) V 1 ) A = q v E = F Lorentz v. 15) Thus, Hamilton s equations for H z are onsistent with the usual equations of motion dedued from H t, and it is valid to use either Hamiltonian as most onvenient. In pratie, the importane of the Hamiltonian H z is in assuring that Liouville s theorem holds for anonial oordinates x, y, t, p x,p y,p t ). When onsidering the phase spae of these oordinates, it is ommon to write p t = E meh + qv and H z = p z ), whih is not stritly orret, but auses no error unless one tries to dedue the equations of motion from this H z. 3 Liouville s Theorem Liouville s theorem [3, 4, 5] is that the phase) volume Π i dq i dp i in anonial-oordinate spae q i,p i ) is invariant under anonial transformations, if those transformations do not involve sale hanges of the oordinates. A anonial transformation operates on one set of anonial oordinates q i,p i ), for whih there exists a Hamiltonian hq i,p i ; t) and for whih the equations of motion are dq i = h dp i, p i = h, 16) q i to arrive at another set of anonial oordinates Q i,p i ) with Hamiltonian HQ i,p i ; t) for whih the equations of motion are dq i = H P i, 3 dp i = H Q i. 17)

4 Liouville s theorem is often applied to a system of N partiles, for whih anonialoordinate spae has 6N dimensions. If interations between these partiles an be ignored, we an onsider the N partiles as being within some volume in the 6-dimensional phase spae q i,p i ), i =1, 2, 3, and Liouville s theorem for the latter phase spae implies that the 6- dimensional phase volume of the set of partiles is invariant under anonial transformations of the six oordinates q i,p i ). Liouville s theorem has the orollaries that the 2-dimensions subvolumes dq i dp i and the 4-dimensional subvolumes dq i dp i dq dp have the invariants under sale-preserving anonial transformations, dq i dp i, and dq i dp i + dq dp dq k dp k, 18) i for indies i, and k all different. Evolution in time, q i t 0 ),p i t 0 )) q i t),p i t)), is an example of a anonial transformation, and Liouville s theorem is often stated in the more restrited sense that phase volume is invariant under this subset of anonial transformations. An eletromagneti gauge transformation, A A+ f, V V f/, where f is any differentiable salar funtion, is also a anonial transformation. Hene, phase volume, along with Hamilton s equations of motion, are invariant under gauge transformations although the Hamiltonian itself is not). 2 The transformation x, y, z, p x,p y,p z ) x, y, t, p x,p y,p t ) onsidered in se. 2 is also a anonial transformation in a broader sense of this term. 3 This transformation hanges the 2-dimensional phase volume dp z to J dp t = z p z z p t p z p t 0 dp t = 0 1 dp t = dp t 19) whih onfirms that Liouville s theorem holds for this anonial transformation. 4 Swann s Theorem In one of the first appliations of Liouville s theorem to a beam of partiles, Swann [8] showed that the phase volume in oordinates x, y, z, p x,p y,p z ), where the anonial momenta are those for a partile in an eletromagneti field, p = p meh + qa/, is the same as that for oordinates x, y, z, p meh,x,p meh,y,p meh,z ). The proof is straightforward, in that the determinant of the Jaobian matrix of the nonanonial) transformation, 2 In pratie, one onsiders a system in a partiular gauge. Partiularly onvenient for Hamiltonian dynamis is the so-alled Hamiltonian gauge introdued by Gibbs in 1896 [6]; see, for example, se. 8 of [7]) in whih the salar potential V is everywhere zero. For osillatory eletromagneti fields with time dependene e iωt and wave number k = ω/, the Hamiltonian-gauge vetor potential is A = ie/k; for stati eletri fields A = t t 0 )E; and for stati magneti fields the vetor potential is the same as that in the Coulomb gauge and also in the Lorenz gauge). 3 Canonial transformations that do not hange the independent variable are sometimes alled restrited anonial transformations. 4

5 x, y, z, p meh,x,p meh,y,p meh,z ) x, y, z, p x,p y,p z ), is unity, J = q A x q A x q A x x y z q A y q A y q A y x y z q A z q A z q A z x y z =1. 20) This argument learly holds if only one or two of the anonial momenta are replaed by mehanial momenta. Likewise, the argument holds for any 2-dimensional or 4-dimensional subvolume in phase spae. Furthermore, when using z as the independent variable, with t as a oordinate with anonial momentum p t = E meh qv, Swann s argument holds when p t is replaed by E meh or E meh ). Appendix: Extended Phase Spae A partile with definite mass has three degrees of freedom, so it is natural to onsider its phase spae as having six dimensions. Yet, in the relativisti view of four-dimensional spaetime, one is led to onsider the eight-dimensional extended phase spae x, p x,y,p y,z,p z,t,p t ) where p t = E, as apparently first done by Sundman in 1912 [9]. Textbook disussions are given in se of [10] and se. 5.5 of [11]. One use of extended phase spae is in deduing Hamiltonians for systems with time-dependent fores, as disussed in [12]. Referenes [1] E.D. Courant and H.S. Snyder, Theory of the Alternating-Gradient Synhrotron, Ann. Phys. NY) 3, ), [2] A.J. Dragt, Lie Methods for Nonlinear Dynamis with Appliations to Aelerator Physis Feb. 27, 2011), [3] J. Liouville, Note sur la Théorie de la Variation des onstantes arbitraires, J.Math. Pures Appl. 3, ), [4] D.D. Nolte, The Tangled Tale of Phase Spae, Phys.Today63, 4, 32 Apr. 2010), [5] L.D. Landau and E.M. Lifshitz, Mehanis, 3rd ed. Pergamon, 1976). 5

6 [6] J.W. Gibbs, Veloity of Propagation of Eletrostati Fores, Nature 53, ), [7] J.D. Jakson, From Lorenz to Coulomb and other expliit gauge transformations, Am. J. Phys. 70, ), [8] W.F.G. Swann, Appliation of Liouville s Theorem to Eletron Orbits in the Earth s Magneti Field, Phys. Rev. 44, ), [9] K.F. Sundman, Mémoire sur le Problème des Trois Corps, Ata Math. 36, ), [10] C. Lanzos, The Variational Priniples of Mehanis, 4th ed. Dover, 1986), [11] G.J. Sussman and J. Wisdom, Struture and Interpretation of Classial Mehanis, 2nd ed. MIT Press, 2014), [12] J. Strukmeier, Hamiltonian dynamis on the sympleti extended phase spae for autonomous and non-autonomous systems, J. Phys. Math. A 38, ), 6