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Basic Hamiltonian Mechanics

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Basic Hamiltonian Mechanics BASIC HAMILTONIAN MECHANICS B.W. Montague 1 . INTRODUCTION At the time of Newton, mechanics was considered mainly in terms of forces, masses and velocities, since these quantities were directly tangible in terms of everyday experience. However, the subsequent evolution of celestial mechanics called for more compact and general methods of handling dynamical systems, and led to the increasing use of potentials as the relevant quantities in the mathematical treatment. Another important development of this period was to free classical mechanics from the constraints of specific co-ordinate systems and to facilitate the transformation from one system to another. This is not only a matter of convenience but a powerful tool for finding invariants of the motion, and a fundamental feature of the Hamiltonian formulation. Joseph—Louis Lagrange was one of the outstanding pioneers of this development;his . Méchanique Analytique laid the foundations of the analytic, in contrast with the geometrical, approach to generalised dynamics. Nevertheless he suggested, apparently as a concession to tradition, that mechanics (with the time dimension included), might be considered as the geometry of a four-dimensional space, which was in a sense a precursor of the structure of special relativity. Generalised classical mechanics has developed considerably since the time of Lagrange and remains not only a broad and fundamental part of classical physics as a whole but has become an important part of the framework on which quantum mechanics has been formulated. The range of topics is so large that even in the restricted field of particle accelerators our coverage here of Lagrangian and Hamiltonian dynamics can only be rather limited. More detailed presentations of the subjects can be found in the Bibliography and are cited in the text wherever the need is specially manifest. The Lagrange equations of motion can be presented in a number of different versions, depending on the nature of the problem and the form of the dynamical constraints. A detailed discussion of these topics is given by Goldstein [1] in Chapter 1; for the application to accelerator physics and the Hamiltonian formulation it is sufficient to consider a restricted set of conditions, namely the motion of charged particles in electromagnetic fields, a domain which is thoroughly covered by Jackson [2] in Chapter 12. Furthermore we shall be dealing with the motion of single particles, taking no account of the forces due to space charge. In the Lagrangian formulation the dynamical behaviour of a system with lc degrees of freedom is characterised by a set of generalised "position" co—ordinates qk, generalised "velocity" co-ordinates qk = dqk / dt and the independent variable t, which is often, but not necessarily, the time variable. The "Lagrangian" or Lagrange function L of the form: L(qkvqk>t) is then a function of 2k dynamical variables. 2 . OUTLINE OF LAGRANGIAN AND HAMILTONIAN FORMALISM In the simplest, non-relativistic case where the forces can be derived from a scalar potential V, independent of velocity, the Lagrangian takes the specific form: OCR Output L(q,¢?»¢) = T(<1.<?J) — V(q,r). where T is the kinetic energy, V is the potential energy and the index k is implicit. In the presence of electromagnetic fields, which can be time-dependent, a generalised potential, and consequently a Lagrangian, can be formulated by combining the Lorentz equation for the force with Maxwell’s equations for the electromagnetic fields. The Lorentz equation for the force F on a particle of charge e moving with velocity v in an electromagnetic field is given by F=e[E+v><B], (3) where E and B are respectively the electric and magnetic fields. Now in the case of static fields the part of the force F arising from E can be derived from a scalar potential tp, but the presence of a magnetic field and time-variation of either, requires the introduction of a vector potential A, which contributes to both electric and magnetic forces. Since we are not taking account here of space-charge forces, only two of Maxwell's equations are required, namely: curlE+dB/dz=O (4) div B = O (5) From Eq. (5) it follows that one can write: B = curl A , (6) where A is the magnetic vector potential. Substituting Eq. (6) into Eq. (4) expanding and rearranging terms, it is then straightforward but somewhat lengthy to show that the non relativistic Lagrangian for tirne—dependent electromagnetic fields is: L(q»¤?,r) = T(¢1»¢?»¢) - U (cmr) (7) where the scalar potential V of Eq. (2) has been replaced by a generalised potential: U = ed) — A · v The relativistic Lagrangian is not just the difference between kinetic and potential energies, and is more complicated to derive formally. A full discussion is given in Chapter 7 of Goldstein [1]; here we simply present the form appropriate to accelerator dynamics, vrz: 2 ]/2 L=—mcc—v·v-e>+eA·v[]q (9) We note that in the non-relativistic limit, v << c, this reduces almost to the form of Eq. (7), apart from a constant- mcz, which vanishes on subsequent differentiations. In the framework of Hamiltonian theory the importance of the Lagrangian lies in the formulation of Hamilton's Principle of Stationary Action (sometimes called "least action" which is usually but not always the case). This principle states that the action integral defined by: OCR Output S= |L(q,q,r)dt (10) is an extremum for the dynamically true path of the time trajectory between tl and tz, i.e. 5S= 5jLdt=O to first order The evaluation of this by the calculus of variations, which is very clearly explained in Vol. II of the Feynmann Lectures [3], results in the Lagrangian equations of motion for a conservative system: .¢i(g)_a dt Qqk Qqk (12, The quantity (QL/ Qqk) is known as the canonical momentum; in the simplest cases where there is no vector potential it is the same as the mechanical momentum, but the presence of an A · v term gives rise to a so—called magnetic momentum. One then recognises that Eq. (12) is Newtor1's Second Law in a more modern guise. Summarising, for a system of k degrees of freedom the Lagrangian is a function of 2k dynamical variables (generalised co-ordinates and velocities) and the "time" t, k of these variables (qk) being the "time" derivatives of the other k variables (qk). The corresponding Lagrange equations consist of a set of k second-order differential equations describing the motion of the system. The Hamiltonian formulation of mechanics describes a system in terms of generalised co ordinates (qk) and generalised momenta (pk the same as the canonical momenta we identified in Eq. (12) above. The change of basis from the set (qk,qk,z) to the set (qk, pk,t) is obtained through a Legendre transformation, defined by the function: H(<1k1¤k¢)=Zpk clk —L(q,r2,r) (13) We can take the differential of H on the left-hand side as QH QH QH (14) dH=—dt+** ———dq q}; + —dp 8g;<9Pk ak gl k or altematively, from the right-hand side of (13) as QL QL . QL . dH=ZPk dqk+`L<1k dP1t‘E‘d<lk’Z"T" dQk‘—’d’ (15) OCR Output Designating the canonical momenta of Eq. (12) as 8L Pk = -7- 1 ( 16) 8% the first and fourth summations cancel and the remaining terms can be identified with the corresponding terms in (14) 8H 8L (17) 8t 8: 8H k <2Pk 8H 3% The function H(q,p,t) is the Hamiltonian and Eqs. (18) and (19) are the Hamilton equations of motion. They are first order, 2k in number for k degrees of freedom, and show a remarkable symmetry of form between the generalised position co-ordinates qk and their conjugate momenta pk. This symmetry leads to very flexible transformation properties between sets of dynamical variables. For non-relativistic motion the Hamiltonian is often, though not necessarily, the sum of potential and kinetic energies H(q,p,t) =T +U (20) A relativistic Hamiltonian for a single particle in an electromagnetic field can be derived from the Lagrangian of Eq. (9). In Cartesian co-ordinates, k = x, y, z, the canonical momenta given by (16) are pk =m¤1’ vt +eAk (21) and differ from the component moyvk of the mechanical momentum by the contribution eAk, the electromagnetic momentum. The resulting Hamiltonian is easily shown to be 2 2 2 1/2 H(q,z>,r)=e¢+¢I(p-6A) +m1>¢ l » (22) where the pk have been expressed in vector form. 3 . SOME PROPERTIES OF THE HAMILTONIAN From the Hamiltonian H (qk,p k,t) the Hamilton equations of motion are obtained by . d 8H qk 2 i 2 _ (18) dt 8pk pk 2 FE1; 2 - EI. (19) OCR Output dt k The Poisson bracket of any two dynamical variables f (qk, pk;) and g(qk,pk,t) is defined by {Lg} (9 3 k aflk 6'Pk 6'Pk aqk (23) One sees that {f,f} E 0 {af} = -{f1s} The time derivative off d . é = %+ Z{%Qk + pk} k qk lc (24) + {,*,11} (from (18), (19) and (23) at In particular, iff = H we have dH BH (25) dt Qt If H does not depend explicitly on time it is a constant of the motion. Any invariant of the motion not containing t explicitly, has a vanishing Poisson bracket with H. 4 . CANONICAL TRANSFORMATIONS Canonical transformations are of considerable utility in simplifying problems by an advantageous choice of co—ordinate system. In particular they can be used to reduce the number of degrees of freedom of a system by exposing invariant quantities, or quantities that are "almost invariant" apart from a small parameter, permitting perturbation theory to be applied.
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