Basic Hamiltonian mechanics - CERN

1 Potential V, independent of velocity, the Lagrangian takes the specific form: OCR OutputIn the simplest, non-relativistic case where the forces can be derived from a scalar2 . OUTLINE OF LAGRANGIAN AND Hamiltonian FORMALISMis then a function of 2k dynamical (qkvqk>t)necessarily, the time variable. The "Lagrangian" or Lagrange function L of the form:"velocity" co-ordinates qk = dqk / dt and the independent variable t, which is often, but notfreedom is characterised by a set of generalised "position" co ordinates qk, generalisedIn the Lagrangian formulation the dynamical behaviour of a system with lc degrees ofmotion of single particles, taking no account of the forces due to space covered by Jackson [2] in Chapter 12.

2 Furthermore we shall be dealing with theconditions, namely the motion of charged particles in electromagnetic fields, a domain which isaccelerator physics and the Hamiltonian formulation it is sufficient to consider a restricted set ofdiscussion of these topics is given by Goldstein [1] in Chapter 1; for the application todepending on the nature of the problem and the form of the dynamical constraints. A detailedThe Lagrange equations of motion can be presented in a number of different versions,wherever the need is specially presentations of the subjects can be found in the Bibliography and are cited in the textcoverage here of Lagrangian and Hamiltonian dynamics can only be rather limited.

3 MoreThe range of topics is so large that even in the restricted field of particle accelerators ourbecome an important part of the framework on which quantum mechanics has been remains not only a broad and fundamental part of classical physics as a whole but hasGeneralised classical mechanics has developed considerably since the time of Lagrangespecial of a four-dimensional space, which was in a sense a precursor of the structure oftradition, that mechanics (with the time dimension included), might be considered as theapproach to generalised dynamics. Nevertheless he suggested, apparently as a concession to. M chanique Analytique laid the foundations of the analytic, in contrast with the geometrical,Joseph Louis Lagrange was one of the outstanding pioneers of this development;hisof the Hamiltonian but a powerful tool for finding invariants of the motion, and a fundamental featurefacilitate the transformation from one system to another.

4 This is not only a matter ofwas to free classical mechanics from the constraints of specific co-ordinate systems and torelevant quantities in the mathematical treatment. Another important development of this periodmethods of handling dynamical systems, and led to the increasing use of potentials as theHowever, the subsequent evolution of celestial mechanics called for more compact and generalvelocities, since these quantities were directly tangible in terms of everyday the time of Newton, mechanics was considered mainly in terms of forces, masses and1 . MontagueBASIC Hamiltonian MECHANICSis usually but not always the case). This principle states that the action integral defined by: OCR Outputformulation of Hamilton's Principle of Stationary Action (sometimes called "least action" whichIn the framework of Hamiltonian theory the importance of the Lagrangian lies in theapart from a constant- mcz, which vanishes on subsequent note that in the non-relativistic limit, v << c, this reduces almost to the form of Eq.

5 (7),L= mcc v v-e>+eA v(9)[]q2 ]/2 Goldstein [1]; here we simply present the form appropriate to accelerator dynamics, vrz:energies, and is more complicated to derive formally. A full discussion is given in Chapter 7 ofThe relativistic Lagrangian is not just the difference between kinetic and potentialU = ed) A vwhere the scalar potential V of Eq. (2) has been replaced by a generalised potential:(7)L(q ?,r) = T( 1 ? ) - U (cmr)relativistic Lagrangian for tirne dependent electromagnetic fields is:rearranging terms, it is then straightforward but somewhat lengthy to show that the nonwhere A is the magnetic vector potential. Substituting Eq.

6 (6) into Eq. (4) expanding andB = curl A ,(6)From Eq. (5) it follows that one can write:div B = O(5)curlE+dB/dz=O(4)equations are required, namely:Since we are not taking account here of space-charge forces, only two of Maxwell'swhich contributes to both electric and magnetic a magnetic field and time-variation of either, requires the introduction of a vector potential A,the part of the force F arising from E can be derived from a scalar potential tp, but the presencewhere E and B are respectively the electric and magnetic fields. Now in the case of static fields(3)F=e[E+v> <B],an electromagnetic field is given byThe Lorentz equation for the force F on a particle of charge e moving with velocity v infor the force with Maxwell s equations for the electromagnetic , and consequently a Lagrangian, can be formulated by combining the Lorentz equationIn the presence of electromagnetic fields, which can be time-dependent, a generalisedwhere T is the kinetic energy, V is the potential energy and the index k is (q, ?)

7 = T(<1.<?J) V(q,r).Designating the canonical momenta of Eq. (12) as(15) OCR Output.. dH=ZPk dqk+`L<1k dP1t E d<lk Z"T" dQk d QLQL . QLor altematively, from the right-hand side of (13) as** q}; <9Pk(14)dH= dt+ dq + dp 8g; ak gl kQHQH QHWe can take the differential of H on the left-hand side as(13)H(<1k1 k )=Zpk clk L(q,r2,r)through a Legendre transformation, defined by the function:in Eq. (12) above. The change of basis from the set (qk,qk,z) to the set (qk, pk,t) is obtainedordinates (qk) and generalised momenta (pk the same as the canonical momenta we identifiedThe Hamiltonian formulation of mechanics describes a system in terms of generalised comotion of the equations consist of a set of k second-order differential equations describing thevariables (qk) being the "time" derivatives of the other k variables (qk).)

8 The correspondingdynamical variables (generalised co-ordinates and velocities) and the "time" t, k of theseSummarising, for a system of k degrees of freedom the Lagrangian is a function of 2kNewtor1's Second Law in a more modern v term gives rise to a so called magnetic momentum. One then recognises that Eq. (12) isthere is no vector potential it is the same as the mechanical momentum, but the presence of anThe quantity (QL/ Qqk) is known as the canonical momentum; in the simplest cases where(12,. i(g)_a dt Qqk Qqkconservative system:Vol. II of the Feynmann Lectures [3], results in the Lagrangian equations of motion for aThe evaluation of this by the calculus of variations, which is very clearly explained in5S= 5jLdt=O to first orderis an extremum for the dynamically true path of the time trajectory between tl and tz, (10)S= |L(q,q,r)dtdt k(19) OCR Outputpk 2 FE1; 2 - 8pk(18).)

9 D 8H qk 2 i 2 _From the Hamiltonian H (qk,p k,t) the Hamilton equations of motion are obtained by3 . SOME PROPERTIES OF THE Hamiltonian where the pk have been expressed in vector form.(22)H(q,z>,r)=e + I(p-6A) +m1> l 2 2 2 1/2the electromagnetic momentum. The resulting Hamiltonian is easily shown to beand differ from the component moyvk of the mechanical momentum by the contribution eAk,pk =m 1 vt +eAk(21)by (16) arethe Lagrangian of Eq. (9). In Cartesian co-ordinates, k = x, y, z, the canonical momenta givenA relativistic Hamiltonian for a single particle in an electromagnetic field can be derived from(20)H(q,p,t) =T +Upotential and kinetic energiesFor non-relativistic motion the Hamiltonian is often, though not necessarily, the sum ofdynamical pk.

10 This symmetry leads to very flexible transformation properties between sets ofsymmetry of form between the generalised position co-ordinates qk and their conjugatemotion. They are first order, 2k in number for k degrees of freedom, and show a remarkableThe function H(q,p,t) is the Hamiltonian and Eqs. (18) and (19) are the Hamilton equations of3%8H<2 Pkk8H8t 8:(17)8H 8 Lcorresponding terms in (14)the first and fourth summations cancel and the remaining terms can be identified with the8%Pk = -7- 1( 16)8 Lwhere dG is a total differential. This follows from Hamilton s variational principle(26) OCR OutputPk dQk H1(Qk1 PkJ)df- Ep), dqk H(qk,pk,t)dr | = dGi\[kcanonical is(qk,pk) and (Qk,Pk ).]