From least action in electrodynamics to magnetomechanical energy – a review 9 Hanno Ess´en 0 Department of Mechanics 0 2 Royal Institute of Technology n SE-100 44 Stockholm, Sweden a J January, 2009 1 2 Abstract ] h Theequationsofmotionforelectromechanicalsystemsaretracedback p - tothefundamentalLagrangianofparticlesandelectromagneticfields,via s the Darwin Lagrangian. When dissipative forces can be neglected the s a systems are conservative and one can study them in a Hamiltonian for- l malism. The central concepts of generalized capacitance and inductance c coefficients are introduced and explained. The problem of gauge inde- . s pendenceofself-inductanceisconsidered. Ourmaininterestisinmagne- c tomechanics, i.e. the study of systems where there is exchange between i s mechanicalandmagneticenergy. Thisthrowslightontheconceptofmag- y neticenergy,whichaccordingtotheliteraturehasconfusingandpeculiar h properties. Weapply thetheory toa few simple examples: the extension p of a circular current loop, the force between parallel wires, interacting [ circular current loops, and the rail gun. These show that the Hamilto- 1 nian, phase space, form of magnetic energy has the usual property that v an equilibrium configuration corresponds toan energy minimum. 7 1 2 3 . 1 0 9 0 : v i X r a 1 1 Introduction Electromagnetism is usually taught at the undergraduate level without men- tion of Lagrangians, Hamiltonians, or the principle of least action. In modern theoretical physics of gauge field theory, however, the concept of an invariant Lagrangian density has become the standard starting point. The Lagrangian formalismofanalyticalmechanicswasintroducedintoelectromagnetismalready by Maxwell in his Treatise [1] who, using this approach, derives equations for electric circuits and for electromechanical systems. Since then its importance haskeptgrowing. Onecanthereforearguethatthissetoftoolsshouldbebetter known and become accessible at an earlier stage in the physics curricula. This review attempts to be an aid in such efforts. OurstartingpointisthebasicLagrangiandensityofclassicalelectrodynam- ics as set down early in the last century by Larmor and Schwarzschild. From there we proceed to neglect radiationwhich leads us to the Darwin Lagrangian [2]. Then the path to classical linear circuit theory is traced. It is pointed out thatthe similaritybetweentheLagrangianformulationofmechanicsandofcir- cuit theory has deep physical reasons and is not just a formal similarity. In a longAppendixthegeneralizedcapacitancecoefficientsandthecoefficientsofself and mutual induction of circuit theory are derived, investigated and explained. Lagrangians of electromechanical systems are also seen to arise from the Darwin Lagrangianby introducing suitable constraints, or assumptions, on the possible movements of both the charged particles and the neutral matter in the system. We concentrate on magnetomechanical problems, i.e. problems where there is a magnetic interaction energy involving macroscopic matter. As examplesofsuchproblemsweconsidertheextensionofacircularloopofcurrent, the attractionofparallelcurrents,the interactionbetweentwo circularloops of current, and the rail gun. Finally we discuss the properties of the concept of magnetic energy and clarify some tricky points. 2 Lagrangian electrodynamics Inmodernphysicsonehasfoundthatthemostreliableandfundamentalstarting point in theoretical investigations is the principle of least action. The action is a scalar quantity constructed from a Lagrange density which is a function of the relevant particle and field variables and their (normally first) derivatives. The action for classical electrodynamics is the time integral of the Lagrangian which has three parts, L = + + . (1) m i f L L L L The first part is the Lagrangianfor free non-interacting particles, N N = = m c2 1 v2/c2. (2) Lm Lma − a − a a=1 a=1 X X p Inthenon-relativisticapproximation,whichwemostlyassumevalid,itissimply the kinetic energy. The second is the the interaction Lagrangian, 1 = j A ̺φ dV. (3) i L c · − Z (cid:18) (cid:19) 2 It was published in 1903by KarlSchwarzschild(1873- 1916)and describes the interaction of the charge and current density of the particles with the electro- magnetic potentials. The third and final part is the field Lagrangian, 1 = (E2 B2)dV, (4) f L 8π − Z originally suggested by Joseph Larmor (1857 - 1942) in 1900. The connection with (3) is via the identifications, 1∂A E = φ , and B = A. (5) −∇ − c ∂t ∇× Maxwell’s homogeneous equations are identities obtained by taking the curl of the first of the equations (5), and the divergence of the second. Maxwell’s remaining, inhomogeneous equations, and the equations of motion for the par- ticles under the Lorentz force,are all obtained from the variationof the action, S = dt,with from(1). Itisthisjoining ofboththe equationsdetermining L L thefieldsfromthesources,andtheequationsofmotionofthesourcesduetothe R fields, into a single formalism,thatis the strengthandbeauty of this approach. The variational approach to electromagnetism outlined above can be found in many of the more advanced textbooks on electrodynamics [3, 4, 5, 6, 7, 8]. More specialized works are Yourgrau and Mandlestam [9], Doughty [10], and Kosyakov[11]. 2.1 The Darwin Lagrangian In many types of problems one can neglect the radiation of electromagnetic waves from the system under study, since this phenomenon is proportional to c−3. In those circumstances the field Lagrangian can be rewritten and one f L finds that = 1 . Inserting this in (1) we get, Lf −2Li 1 = + , (6) m i L L 2L for the relevant Lagrangian in the non-radiative case. When the motion of a chargedparticleisknownonecanfindthepotentials,φ,A,thatitproduces,the so called retarded, or Li´enard-Wiechert potentials. Expanding these to order (v/c)2, one finds that acceleration vanishes from the Lagrangian (since it only contributes a total time derivative to this order). The result is a Lagrangian that contains only particle positions andvelocities. There are then no indepen- dent electromagnetic field degrees-of-freedom. Everything is determined by the positionsandvelocitiesofthechargedparticles,andtheresultingLagrangianis the DarwinLagrangian[2], as derivedby CharlesGaltonDarwin(1887- 1962), a grandson of the great naturalist, in 1920. The Darwin Lagrangiancan be written, 1 1 = + j A ̺φ dV, (7) D m L L 2 c · − Z (cid:18) (cid:19) i.e. Eq. (6), where, ̺(r′)dV′ φ(r)= , (8) r r′ Z | − | 3 which is exact in the Coulomb gauge, and where, A(r)= 1 j(r′)+[j(r′)·er′r]er′r dV′. (9) 2c r r′ Z | − | Hereer′r =(r r′)/r r′ . ThisspecificformoftheDarwinvectorpotentialcan − | − | be traced back to a fairly large retardationeffect in the Lorenz gauge Coulomb potential. Its effect is included in the Darwin approximation which, however, uses a Coulomb gauge, A=0. ∇· A more familiar form of the Darwin Lagrangian, for N point particles, is obtained by introducing, N N ̺(r)= e δ(r r (t)), and j(r)= e v (t)δ(r r (t)), (10) a a a a a − − a=1 a=1 X X in the expressions (7) - (9) given above. After skipping self interactions one obtains, N 1 1 = + e v A (r ) e φ (r ) , (11) D m a a a a a a a L L 2 c · − a=1(cid:18) (cid:19) X where, N e φ (r)= b , (12) a r r b bX(6=a)| − | and, N e [v +(v e )e ] A (r)= b b b· rbr rbr . (13) a 2cr r b bX(6=a) | − | Herer andv areparticlepositionandvelocityvectorsrespectively,m ande a a a a their restmassesandchargesrespectively,whilee =(r r )/r r . There rbr − b | − b| arenoindependentfielddegrees-of-freedomandhencenogaugeinvarianceinthe Darwin formalism, which entails action-at-a-distance. Retardation is included to order (v/c)2, a fact which is often missed in the literature. TheDarwinapproachtoelectromagnetismisonlybrieflymentionedinsome advanced textbooks [3, 4, 6]. A book by Podolsky and Kunz [12] is a bit more thorough. Several good fundamental and pedagogical studies can, however, be found in the literature [13, 14, 15, 16, 17, 18, 19, 20]. 2.2 The kinetic energy of currents In the free particle Lagrangian of Eq. (2) the approximation, m L m m m c2+ v2 + av4, (14) Lma ≈− a 2 a 8c2 a is usually done, because of the validity of the Darwinapproachto order (v/c)2. Here we will be concerned with systems in which there are macroscopic charge andcurrentdensitiesconfinedtoelectricallyconductingmatter. In1936Darwin [21]foundthatthemagneticenergycontributiontotheinertiaoftheconduction electrons is roughly 108 greater that the contribution from their rest mass. 4 This means that, so called, inductive inertia dominates. For the dynamics of macroscopic currents and charges in fixed conductors (electric circuit theory) one can consequently also neglect the free particle Lagrangian . In plasma m L physicsthe neglectofparticleinertia iscalledthe forcefree approximation[22]. Skipping , m L 1 1 = j A ̺φ dV, (15) LC L 2 c · − Z (cid:18) (cid:19) is all that then remains of (7). This Lagrangian, together with A and φ given by (8) and (9) respectively, describes electromagnetic systems with inductive and capacitive phenomena. For electromechanical systems, on the other hand, the non-relativistic form of kinetic energy must be retained for the mechanical degrees-of-freedom. Potential energy contributions due to elasticity or gravita- tion may also have to be included. 3 Linear electric circuits The equations governing linear electric circuits are presented in almost every textbook on electromagnetism, and their similarity with those for oscillating mechanical systems is often pointed out. A smaller number of more advanced textsevengoasfaraspresentingaLagrangianformalismunderlyingthecircuit equations [23, 24, 25]. Here we will derive and discuss some standard results in for linear elec- tric circuits starting directly from (15). These are alternatively called current circuits, or networks, in the literature. Assume that all current flows in con- ducting thin (filamentary) wires and that there are n such wires with currents i = e˙ , (k = 1,...,n). It is then easy to show that the magnetic part of (15) k k can be written, n n 1 1 1 = j AdV = L e˙ e˙ . (16) L kl k l L 2 c · 2 Z k=1l=1 XX In a similar way for a fixed arrangement of m (extended) conductors, with charges e (i=1,...,m) on them, the electric part of can be written, i LC L m m 1 1 = ̺φdV = Γ e e . (17) C ij i j L −2 −2 Z i=1j=1 XX The inductance coefficients L and the generalizedcapacitance coefficients Γ kl ij only depend on the geometry of the arrangement. These are derived and ex- plainedinthe Appendices. We havethus foundthatthe Lagrangian(15)under the above assumptions can be written, (e,e˙)= (e˙)+ (e). (18) LC L C L L L This is a validtotalLagrangianfora non-radiatingarrangementofcurrentcar- ryingthinwiresandextendedchargedconductors. This separationofmagnetic and electrostatic effects comes from the central idea that there will be no net charge density on thin wires, and that currents in extended conductors have negligible magnetic effects. 5 3.1 The conductor pair condenser In practice all the chargese on all the m different conductors are not indepen- i dent. Often a circuit is arranged so that the conductors come in pairs that are very close, so called condensers. If each such pair is connected by a wire while being electrically isolated otherwise, the total charge on that subsystem must be a constant which we take to be zero. The number of wires is then half the number of conductors, n = m, and the charges e come in pairs that are equal 2 i and opposite e = e ,(k = 1,...,n = m), while the current in the wire k − k+n 2 connecting them is i = e˙ , see Fig. 1. There is then only n = m degrees-of- k k 2 freedom of the problem. We now assume, without loss of generality, that the coefficients Γ are symmetric in the indices ij, and define the new symmetric ij matrix, Figure 1: A single LC-circuit with an inductance L and a capacitance C. The Lagrangian (20) gives thedynamicsfor n interacting circuits of this type. C−1 =C−1 Γ +Γ Γ Γ , (19) kl lk ≡ kl k+nl+n− kl+n− lk+n where k,l=1,...,n. Using this our Lagrangian(18) can be written, n n 1 (e,e˙)= L e˙ e˙ C−1e e . (20) LLC 2 kl k l− kl k l k=1l=1 XX(cid:0) (cid:1) Here the n charges e and currents e˙ are independent and a diagonal element k k of the L matrix, L , is a self-inductance, while the off diagonal elements cor- kk respond to mutual inductances. A diagonal element, C−1 =Γ +Γ kk kk k+nk+n− 2Γ , of the C−1-matrix represents the inverse capacitance, C , of the cor- kk+n kk responding conductor pair condenser. 3.2 Equivalence of electric and mechanical oscillators The Lagrangian (20) is completely equivalent to that of a mechanical system of coupled oscillators, the L-matrix corresponding to the mass matrix and the C−1-matrix corresponding to the stiffness matrix (of spring constants). This is oftenregardedasapurelyformalcorrespondence,ameremathematicalmapping of one problem on another physically completely different one. This is wrong, 6 however. If we denote the linear density of conducting charge in wire k by λ , k andthe arclengthalongthis wirebys , we findthat the currentinthe wireis, k i =e˙ =λ s˙ . Here, of course, s˙ is the speed of the conducting linear charge k k k k k density. Clearly the charges on the condensers have to be, e = λ s (with a k k k suitable choice of origin and orientation for the arc length). If this is inserted in we find that, LC L n n 1 (e,e˙)= λ λ L s˙ s˙ C−1s s LLC 2 k l kl k l− kl k l ≡ k=1l=1 XX (cid:0) (cid:1) (21) n n 1 (M s˙ s˙ K s s )=T(s˙) V(s)= (s,s˙). kl k l kl k l 2 − − L k=1l=1 XX The generalized coordinates s now have dimension length so we have an or- k dinary mechanical coupled oscillator Lagrangian on the right hand side. The difference is that the mass matrix, M = λ λ L , does not come from rest kl k l kl massbutentirelyfromthe inertiacontainedintheenergyofthemagneticfield. By means of the technique ofsimultaneous diagonalizationoftwo quadratic formsonecanfindalineartransformationto,socalled,normalmodecoordinates and thus decouple the equations of motion, which are, n L e¨ +C−1e =0, k =1,...,n, (22) kl l kl l l=1 X(cid:0) (cid:1) assuming that no further generalized forces enter the problem. In terms of the normal modes q the equations of motion become, k 1 q¨ + q =0, k =1,...,n, (23) k k L C k k so these oscillate independently with angular frequencies ω = 1/√L C . For k k k the corresponding mechanical problem one finds ω = K /M . The expres- k k k sion ω = 1/√LC for the angular frequency of a single LC-circuit, as shown in p Fig. 1, is sometimes referred to as Thomson’s formula. 3.3 Introduction of resistance and external voltage OurLagrangian(20)correspondstoacoupledsystemofundampedelectromag- neticoscillators. Inmostcasesofpracticalinteresttheconnectingwireswillnot beperfectlyconducting. Therewillberesistanceinthesystem. Theenergywill then dissipate and the equations of motion require that there are generalized forces that describe this. Assume that the ohmic resistance in wire k is R . kk This can be achieved with a Rayleigh dissipation function. In a more general case there may also be off diagonal elements R , and, kl n n 1 (e˙)= R e˙ e˙ , (24) kl k l R 2 k=1k=l XX is the most general form of this function for linear circuits. 7 If there is resistance currents e˙ eventually dissipate to zero and arbitrary k initialconditionsonlyleadtotransientdynamics. Inmostapplicationsofcircuit theory one is therefore mainly interestedin systems with added externale.m.f.. This can be done by the additional term, n (t)= e (t), (25) emf k k L V k=1 X added to the Lagrangian . Here (t) is an applied external voltage. A LC k L V constant e.m.f. will not drive a stationary current through a condenser so to get a direct current some of the C−1-matrix eigenvalues must be zero. With harmonically oscillating e.m.f., (t) = sin(ωt), capacitors are no problem k k V V and one is dealing with alternating current circuits. ThegeneralLagrangianequationsofmotionforasystemofcircuitsarethen, d ∂ ∂ ∂ cc cc L L = R, (26) dt ∂e˙ − ∂e −∂e˙ k k k where, (e,e˙,t)= (e,e˙)+ (e,t), (27) cc LC emf L L L is the circuit Lagrangian[24]. The system (22) of equations of motion are then modified so that, n L e¨ +R e˙ +C−1e = (t), k =1,...,n, (28) kl l kl l kl l Vk l=1 X(cid:0) (cid:1) is their new form. This is thus the type of system investigated in linear circuit theory(seee.g.Guillemin[23]orJosephs[26]). Inmechanicalsystemstheohmic resistance terms correspond to dampers (dashpots) and the external e.m.f. to applied external force. 3.4 Energy and Hamiltonian for conservative systems ForLagrangians (q,q˙)with no explicittime dependence, suchas thoseofEqs. L (11) and (20), the quantity, n ∂ (q,q˙)= Lq˙ , (29) k E ∂q˙ −L k k=1 X is known to be a constant of the motion, the energy. For example the Darwin Lagrangian(11) corresponds to the conserved energy, N 1 1 = + e v A (r )+e φ (r ) . (30) D m a a a a a a a E E 2 c · a=1(cid:18) (cid:19) X This expression for the energy goes up if currents are parallel since the vector potential is proportional to terms like e v . This may seem odd since we find b b in Sec. 4.3that parallelcurrentsattract. We will returnto this in Sec.5 below. 8 Returning to circuits we find that, when there is no time dependent forcing andnoohmicresistance,theLagrangianissuchthatthereisaconservedenergy. Using (20) and (29) gives the expression, n n 1 (e,e˙)= L e˙ e˙ +C−1e e , (31) ELC 2 kl k l kl k l k=1l=1 XX(cid:0) (cid:1) for this energy. Recall that if = T V, then = T +V. The effect of a L − E constant forcing, due to permanent constant charge on condensers, is only to shift the equilibrium from e = e˙ = 0. Ignoring this (20) is the most general k k circuit Lagrangianthat conserves the energy (31). The generalized momenta obtained from the Lagrangian (20) are by defini- tion, n ∂ LC p L = L e˙ . (32) k kl l ≡ ∂e˙ k l=1 X The Hamiltonian is obtained by eliminating the generalized velocities in the Lagrangian energy (31) in favor of the generalized momenta. Since, e˙ = k n L−1p , the Hamiltonianwill depend onthe inverseofthe L-matrix. For a l=1 kl l system of coupled LC-circuits we find the Hamiltonian, P n n 1 (e,p)= L−1p p +C−1e e , (33) HLC 2 kl k l kl k l k=1l=1 XX(cid:0) (cid:1) representing its conserved energy as a function of phase space variables. The LagrangianandHamiltonianabove,aswellastheinterpretationofthecanonical momenta as magnetic fluxes discussed below, can be found in an article by Meixner [27], discussing thermodynamic issues. 3.5 Generalized momenta and magnetic flux To find the meaning of the generalized, or canonical, momenta in this case we return to the definition of the magnetic Lagrangian, 1 1 = j AdV. (34) L L 2 c · Z Thevolumeintegrationisonlyoverthefilamentarywiresthatcarrythecurrents i =e˙ , so, using jdV =i dr, we get, k k k n 1 = i A dr, (35) L k L 2c · k=1 Ik X where the line integral is around the loop of wire k. Now, however, A dr = ( A) ds= B ds, (36) · ∇× · · Ik Zk Zk according to Stokes’ theorem. By definition this is the magnetic flux, Φ , k through the loop k. We thus find that, n 1 Φ k = i . (37) L k L 2 c k=1 X 9 Comparing with (16) this gives us that, n Φ k = L i =p , (38) kl l k c l=1 X where the individual terms on in the sum represent contributions to the flux through k from the loops l of the system. Finally then, we have found that the generalized (canonical) momenta p , k of Eq. (32), conjugate to the charges e on the condensers are the magnetic k fluxesthroughthe currentsloops(dividedbyc): p =Φ /c. Thismeansthatif k k one e does not appear in the Lagrangian(because there is no condenser in the l corresponding loop) then that generalized momentum p (or flux) is a constant l of the motion. 4 Electromechanical systems Few texts derive of equations of motion for electromechanicalsystems from the fundamental Lagrangian for particles and fields, only Ne˘imark and Fufaev [28] come close. As should be clear from the above developments electromechanical systems,asopposedtoelectriccircuits,requirethatweretreatfrom(15)backto the Darwin Lagrangian in the form (7), which we had before we neglected rest massinertia. Fromtherethe equationsofmotionforelectromechanicalsystems canbe found by adding constraints,or assumptions,aboutthe motion, thereby reducing the number of degrees-of-freedom,in the way familiar from analytical mechanics. If macroscopic matter moves one must, of course, add the Lagrangian cor- responding to that motion. Further, the induction and capacitance coefficients may now depend on the mechanical degrees-of-freedom corresponding to the motion of thin wires and extended conductors of the system, since this changes its geometry. For an energy conserving system one then typically arrives at a Lagrangianof the form, n n 1 (q,e,q˙,e˙)=T(q,q˙) V(q)+ L (q)e˙ e˙ C−1(q)e e , (39) L − 2 kl k l− kl k l k=1l=1 XX(cid:2) (cid:3) and this will be general enough for our purposes. One notes that if charged conductors movethis produces magnetic effects which may have to be handled. Inmanycases,however,thespeedofthismotionwillbesuchthatthemagnetic effect is negligible. In moregeneralcasesone may,ofcourse,alsohavecoupling terms between q˙ and e˙. In case of doubt the safe method is to start with the DarwinLagrangian(11)andintroducerelevantconstraintsandidealizations. A couple of examples of this procedure can be found in Ess´en [29, 30]. Electromechanical systems are treated e.g. in the books by Ne˘imark and Fufaev [28], Wells [31], and Gossick [32]. Articles discussing various aspects of these systems are [33, 34, 35, 36]. We now proceed to some concrete examples of magnetomechanicalsystems. 4.1 Extension of current carrying ring When a current i flows in a conducting circular loop, or ring, its radius will increase somewhat. This is due to the reaction forces to the forces needed to 10