A State–Space Lagrangian Model for Two Interacting Magnetic Dipoles

A State–Space Lagrangian Model for Two Interacting Magnetic Dipoles Parker Emmerson *Yaohushuason December 7, 2025 Abstract We construct a state–space Lagrangian model for the interaction of two magnetic dipoles and connect it explicitly to the standard magnetostatic dipole–dipole force law. Starting from an algebraic invariant I≡α2(l2−h2)=(q−s)2= (γx −θr)2, we interpret the scalar I as a common “state difference” shared by two magnets and build a potential energy U ( I )that reproduces the familiar dipole–dipole interaction energy U ( x ) ∝ − 1 /x3 for coaxial dipoles in one dimension. A Lagrangian L = T−U ( I )is then derived, yielding via the Euler–Lagrange equation the correct F ( x ) ∝ 1 /x4 magnetostatic force. We further compute the jerk ... x in this model and show how the time integral of separation (“absement”) can be incorporated phenomenologically as a history–dependent correction to the dynamics. We extend the framework to include full orientation dynamics and propose a Velocity–Matching Magnetodynamics (VMM) overlay with explicit jerk and absement effects, derive an energy identity for jerk, and present testable predictions including frequency–independent hysteretic loss and a cubic-frequency reactive signature relevant to superconducting systems. This provides a coherent variational framework linking a state– space invariant to realistic magnetostatic forces while enabling higher–order and memory effects. Contents 1 Introduction 2 2 Variables, Dimensions, and State–Space Invariant 3 2.1 Physical configuration and coordinates . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 State variables and the invariant I.......................... 3 3 Potential as a Function of the Invariant and Magnetostatic Limit 4 3.1 Standard dipole–dipole interaction in 1D . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Constructing U(I)fromtheinvariant......................... 4 4 Lagrangian Formulation and Equations of Motion 4 4.1 Kinetic energy and reduced mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2 Lagrangian and force law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 Orientation Dynamics: Angles, Forces, and Torques 5 6 Jerk and Absement in the Model 6 6.1 Jerkasaderivedquantity ............................... 6 6.2 Absement and history–dependent forces . . . . . . . . . . . . . . . . . . . . . . . 6 6.3 Explicit jerk–dependent forces and an energy identity . . . . . . . . . . . . . . . . 6 1 7 Comparison With Standard Magnetostatics and Experiments 6 8 Numerical Integration of the 1D Snap/Push Dynamics 7 9 Toward a State–Space “Theory” of Magnetism 8 10 Velocity–Matching Magnetodynamics (VMM): Definitions and Postulates 9 10.1 Shared state, simultaneity, and rest . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.2 Jerk as a high–order regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.3Absementasmemory.................................. 9 10.4 Governing VMM equations with orientation . . . . . . . . . . . . . . . . . . . . . 10 11 Energy Balance and Stability Notes 10 12 Nondimensionalization and First–Order System 10 13 Remarks, Regimes, and Scaling 11 14 Novel Insights and Testable Predictions 11 14.1Invariantgeometry................................... 11 14.2 Velocity matching as a rest manifold . . . . . . . . . . . . . . . . . . . . . . . . . 11 14.3Jerkenergyidentity .................................. 11 14.4 Absement as hysteretic loss and bias . . . . . . . . . . . . . . . . . . . . . . . . . 11 14.5Timescalecompetition ................................. 11 14.6 New observables and protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 15 Implications for Superconductors 11 15.1 Meissner state: image–dipole mapping . . . . . . . . . . . . . . . . . . . . . . . . 12 15.2 Type-II with pinning: absement as hysteresis surrogate . . . . . . . . . . . . . . . 12 15.3 Reactive electrodynamics: jerk as a cubic-frequency signature . . . . . . . . . . . 12 15.4 Coupling to superconducting loops . . . . . . . . . . . . . . . . . . . . . . . . . . 12 16 Conclusions and Outlook 12 A Mathematica (orientation + translation) 13 1 Introduction The interaction of two magnetic dipoles is well understood in classical electrodynamics through Maxwell’s equations and the dipole approximation. For two coaxial, parallel dipoles with moments m1 and m2 separated by a distance x , the standard magnetostatic result for the interaction energy is Udip(x)=−µ0 2π m1m2 x3,(1) and the corresponding axial force is Fx(x)=−∂Udip ∂x =−3µ0m1m2 2πx4.(2) Here µ0is the vacuum permeability. See, e.g., [1,2]. In parallel, one may wish to describe the interaction of two magnets in terms of an abstract “state space”, where each magnet carries internal variables and the system evolves toward configurations that minimize some mismatch of these states. The goal of this paper is to show 2 that such a state–space framework can be made consistent with known magnetostatic results, and to extend it to include higher–order and memory effects. We begin from an algebraic invariant I≡α2(l2−h2)=(q−s)2= (γx −θr)2,(3) assign physical dimensions to the variables, and construct a potential energy U ( I )that, when expressed in terms of the physical separation x , reproduces U ( x ) = −C/x3 for a magnetic coupling constant C . From this, we derive a Lagrangian L = T−U ( I )and obtain the equations of motion via the Euler–Lagrange equations. We then extend to full orientation dynamics, and develop a Velocity–Matching Magnetodynamics (VMM) overlay that incorporates phenomenological jerk and absement effects, deriving associated energy identities and testable predictions, including implications for superconductor systems. 2 Variables, Dimensions, and State–Space Invariant 2.1 Physical configuration and coordinates We consider two bar magnets approximated as point dipoles with magnetic moments m1 and m2 . We restrict attention to a one–dimensional geometry where the dipoles are: •coaxial, •parallel, and •free to move along a common axis. Let x ( t )denote the center–to–center separation between the two magnets as a function of time t , with [ x ] = L . We also introduce a fixed reference length r with [ r ] = L , which may be interpreted as a characteristic size (length scale) of the magnets. In this paper, r is treated as a constant parameter, not as a dynamical variable. 2.2 State variables and the invariant I Define a dimensionless state difference ∆≡γx −θr, [γ]=[θ] = L−1,(4) so that [∆] = 1. The coefficients γ and θ may be interpreted as geometric or coupling parameters that convert the physical length scales x and r into an abstract, dimensionless state coordinate. Introduce alternative scalar, dimensionless state variables q and s such that q−s = ∆, and dimensionless internal coordinates l, h, β, α, with [l] = [h]=[β]=[α] = 1, h =lsin β. (5) The central state–space invariant is I≡α2(l2−h2) = (q−s)2= (γx −θr)2= ∆2.(6) By construction, I is dimensionless. The equality of these forms expresses the fact that the same scalar I can be parametrized in internal coordinates ( l, h, α ), in abstract state variables ( q, s ), or in the physical coordinate xand the reference length rvia (γx −θr). 3 3 Potential as a Function of the Invariant and Magnetostatic Limit 3.1 Standard dipole–dipole interaction in 1D For two coaxial, parallel dipoles with moments m1 and m2 , the axial dipole–dipole interaction energy and force are given by Eqs. (1) and (2). It is convenient to introduce the constant C≡µ0m1m2 2π,[C] = J·m3,(7) so that Udip(x)=−C x3, Fx(x)=−3C x4.(8) 3.2 Constructing U(I)from the invariant We seek a potential U ( I )which: (i) depends only on the dimensionless invariant I from Eq. (6), and (ii) reproduces the standard dipole–dipole energy Udip(x)when expressed in terms of x. Recall that I= ∆2and ∆=γx −θr, so that √I=|∆|. Observing that ∆+θr = (γx −θr) + θr =γx, (9) we define U(I)≡ −C γ3√I+θr−3.(10) Under the assumption ∆ > 0in the 1D axial configuration of interest (i.e., we take the positive branch √I = ∆), we have √I + θr = γx and therefore U ( I ) = −C/x3 . Thus U ( I )is exactly equivalent to the standard dipole energy when expressed in terms of x: U(I(x))=Udip(x).(11) Importantly, U ( I )is indifferent to the parametrization of I and hence encodes the interaction in a coordinate–free way within state space. 4 Lagrangian Formulation and Equations of Motion 4.1 Kinetic energy and reduced mass Let the inertial masses of the magnets be M1 and M2 (we reserve m1, m2 for the dipole moments). The effective reduced mass for the relative motion is M=M1M2 M1+M2 .(12) The kinetic energy of the relative coordinate is T=1 2M˙x2. 4.2 Lagrangian and force law Define the Lagrangian L=T−U(I) = 1 2M˙x2−U(I),(13) with U(I)as in Eq. (10) and I= (γx −θr)2. Equivalently, using U(I(x))=−C/x3, L=1 2M˙x2+C x3.(14) The Euler–Lagrange equation yields d dt∂L ∂˙x−∂L ∂x = 0 =⇒M¨x=−3C x4,(15) which matches the standard magnetostatic force in Eq. (8). 4 5 Orientation Dynamics: Angles, Forces, and Torques We extend to explicit orientation dynamics of each magnet. Let mi=mini,ni·ni= 1, i = 1,2, be the dipole moments with unit vectors ni . Let the center-to-center displacement be r = xˆ r , with x>0and ˆ ra unit vector. The general dipole–dipole interaction energy is U(n1,n2, x) = µ0 4πx3hm1·m2−3 (m1·ˆ r) (m2·ˆ r)i.(16) In a planar configuration (both dipoles confined to the plane spanned by ˆ r and a fixed transverse direction ˆ t), parametrize ni= sin θiˆ t+ cos θiˆ r, i = 1,2, with θimeasured from ˆ r. Then U(x, θ1, θ2) = µ0m1m2 4πx3hcos(θ1−θ2)−3 cos θ1cos θ2i.(17) Define K(x)≡µ0m1m2 4πx3, D(θ1, θ2)≡cos(θ1−θ2)−3 cos θ1cos θ2, so that U=K(x)D(θ1, θ2). The translational force along the line is Fx≡ −∂U ∂x =3µ0m1m2 4πx4D(θ1, θ2),(18) which reduces to Fx = − 3 C/x4 when θ1 = θ2 = 0 (coaxial, parallel) since D = − 2and C=µ0m1m2/(2π). The torques (about the axis perpendicular to the plane) are τ1=−∂U ∂θ1 =K(x)hsin(θ1−θ2)−3 sin θ1cos θ2i,(19) τ2=−∂U ∂θ2 =−K(x)hsin(θ1−θ2) + 3 cos θ1sin θ2i.(20) Let M be the reduced mass for translation, and I1, I2 the moments of inertia for rotation in the plane. The full conservative Lagrangian is L=1 2M˙x2+1 2I1˙ θ2 1+1 2I2˙ θ2 2−U(x, θ1, θ2).(21) Euler–Lagrange equations yield M¨x=3µ0m1m2 4πx4D(θ1, θ2),(22) I1¨ θ1=τ1(x, θ1, θ2), I2¨ θ2=τ2(x, θ1, θ2).(23) To retain the state–space structure, we generalize Eq. (10) by defining ∆≡γx −θr, I ≡∆2, and write the potential as U(I, θ1, θ2)=−C γ3f(θ1, θ2) (√I+θr)3, f(θ1, θ2)≡ −1 2D(θ1, θ2),(24) so that √I + θr = γx implies U = −C f ( θ1, θ2 ) /x3 and the equations of motion reduce to Eqs. (22) and (23). For θ1=θ2= 0,f= 1 and U=−C/x3. 5 6 Jerk and Absement in the Model 6.1 Jerk as a derived quantity From Eq. (15), a(t) = ¨x(t)=−3C Mx(t)4. The jerk (third derivative of position) is j(t) = ... x(t) = d dt−3C Mx4=12C M ˙x(t) x(t)5.(25) Jerk is an emergent property of the strongly nonlinear force law; it becomes large for small separations xand significant velocities ˙x. 6.2 Absement and history–dependent forces The absement is the time integral of displacement: A(t)≡Zt 0 x(τ) dτ , (26) with dimension [ A ] = L·T . In the conservative Lagrangian Eq. (14), absement does not appear. To incorporate memory, introduce a phenomenological force FA(t)=−kAA(t),(27) where kAhas units of force per absement. Then M¨x(t)=−3C x(t)4−kAZt 0 x(τ) dτ , (28) which is nonlocal in time. 6.3 Explicit jerk–dependent forces and an energy identity If one includes jerk explicitly, posit Fj(t)=kj... x(t),(29) with [kj]=M T. The resulting equation of motion is M¨x+kj... x=−3C x4.(30) The instantaneous power due to jerk is Pj=Fj˙x=kj... x˙x. Using ... x˙x=d dt−(¨x)2, Zt1 t0 Pjdt =kjh¨x˙xit1 t0−kjZt1 t0 (¨x)2dt. (31) Consequently, for kj> 0and bounded boundary term, jerk acts as a net smoother (dissipative in the sense of reducing Ra2dt ), whereas an impetus effect requires either kj< 0or a signconditioned variant (e.g., Fj = kjsgn ( ˙x ) ... x ). A term proportional to ... x typically lies outside standard Lagrangian form (Ostrogradsky); here it is treated phenomenologically. 7 Comparison With Standard Magnetostatics and Experiments In the regime where the magnets can be treated as point dipoles (separations large compared to magnet size), Eqs. (10), (14) and (15) reproduce the standard dipole interaction energy U = −C/x3 and force Fx = − 3 C/x4 (Eq. (8)). The orientation–dependent generalization in Section 5 matches the textbook angular dependence. Deviations at smaller separations reflect finite–size and edge–field effects and can be incorporated by refining the mapping I7→ x or by including higher multipoles. 6 8 Numerical Integration of the 1D Snap/Push Dynamics Consider the 1D conservative equation M¨x=−3C x4, C =µ0m1m2 2π,(32) with initial conditions x(0) = x0>0,˙x(0) = v0. Define κ≡3C M, y ≡x x0 , τ ≡t t0 , t0≡rx5 0 κ. Then Eq. (32) becomes the parameter-free dimensionless ODE d2y dτ2=−1 y4, y(0) = 1,dy dτ (0) = ν0≡v0 t0 x0 .(33) Derived kinematics: v(t) = ˙x=x0 t0 dy dτ , a(t) = ¨x=−κ x4,(34) j(t) = ... x=12C M ˙x x5,(35) A(t) = Zt 0 x(τ)dτ =x0t0Zt/t0 0 y(˜τ)d˜τ. (36) Energy conservation gives E=1 2M˙x2−C x3=Mx2 0 2t2 0 dy dτ 2 −2 3y3!=const.(37) For v0 = 0, y ( τ )decreases to zero in finite time, with |a| and |j| diverging as y→ 0 + ; for v0> 0 sufficiently large, yinitially increases (“push apart”). Numerical recipes. A simple explicit RK4 for Eq. (33): Given y, u = dy/dtau: f1 = u g1 = -1 / y^4 f2 = u + 0.5*h*g1 g2 = -(1 / (y + 0.5*h*f1)^4) f3 = u + 0.5*h*g2 g3 = -(1 / (y + 0.5*h*f2)^4) f4 = u + h*g3 g4 = -(1 / (y + 0.5*h*f3)^4) y_next = y + (h/6)*(f1 + 2*f2 + 2*f3 + f4) u_next = u + (h/6)*(g1 + 2*g2 + 2*g3 + g4) Stop when y <= y_min > 0 (collision cutoff). Mathematica (1D). (* Parameters *) mu0 = 4 Pi*10^-7; m1 = 0.5; m2 = 0.5; (* A m^2: dipole moments *) C = mu0 m1 m2/(2 Pi); 7 M = 0.02; (* kg: reduced mass *) x0 = 0.10; v0 = 0.0; (* ODE with absement A as an auxiliary state if desired *) tmax = 0.2; sol = NDSolve[ {x’’[t] == -3 C/(M x[t]^4), x[0] == x0, x’[0] == v0, A’[t] == x[t], A[0] == 0}, {x, A}, {t, 0, tmax}, Method -> {"EventLocator", "Event" -> x[t] - 0.005, "EventAction" -> "StopIntegration"}][[1]]; xplot = Plot[Evaluate[x[t] /. sol], {t, 0, tmax}, PlotLabel -> "x(t)"]; vplot = Plot[Evaluate[D[x[t], t] /. sol], {t, 0, tmax}, PlotLabel -> "v(t)"]; aplot = Plot[Evaluate[D[x[t], {t, 2}] /. sol], {t, 0, tmax}, PlotLabel -> "a(t)"]; jplot = Plot[Evaluate[D[x[t], {t, 3}] /. sol], {t, 0, tmax}, PlotLabel -> "j(t)"]; Aplot = Plot[Evaluate[A[t] /. sol], {t, 0, tmax}, PlotLabel -> "A(t)"]; 9 Toward a State–Space “Theory” of Magnetism We summarize the framework in a compact set of postulates compatible with Maxwellian magnetostatics and extendable to richer dynamics: •State invariant: There exists a scalar Ibuilt from a dimensionless “state difference” ∆, I≡∆2,∆=γx −θr, with [γ]=[θ]=L−1and [x]=[r] = L. •Orientation factor: Angular dependence of the interaction enters through f(n1,n2,ˆ r) = n1·n2−3(n1·ˆ r)(n2·ˆ r), the standard dipolar angular factor. •Invariant potential: The interaction energy is the composite U(I, n1,n2)=−C γ3f(n1,n2,ˆ r) (√I+θr)3, which reduces to U=−C f/x3when √I+θr =γx. •Lagrangian: L=1 2M˙x2+1 2ω⊤ 1I1ω1+1 2ω⊤ 2I2ω2−U(I, n1,n2), with ωiand Iithe angular velocities and inertia tensors. • Dynamics: Euler–Lagrange equations reproduce the standard dipole force and torque in the magnetostatic limit, and generate nonlinear time evolution that naturally exhibits rapidly changing accelerations (jerk). History dependence (e.g., absement) can be introduced through nonlocal terms while preserving the conservative limit. 8 10 Velocity–Matching Magnetodynamics (VMM): Definitions and Postulates We formalize a state–space dynamical overlay on magnetostatics: simultaneity via velocity matching, jerk as a high-order regulator, and absement as memory–repelling. 10.1 Shared state, simultaneity, and rest Define the shared state difference ∆(t)≡γ x(t)−θ r =q(t)−s(t),(38) so that ˙ ∆=γ˙x=q−s. The simultaneity (velocity–matching) rest manifold is ˙ ∆(t)=0 ⇐⇒ ˙x(t) = 0 ⇐⇒ q(t)=s(t).(39) Enforce relaxation toward ˙ ∆=0with a Rayleigh dissipation Rv=cv 2˙ ∆2=cvγ2 2˙x2,[cv]=M L2/T, (40) which produces the nonconservative force Fv=−∂Rv ∂˙x=−cvγ2˙x. (41) 10.2 Jerk as a high–order regulator In physical space v= ˙x,a= ¨x,j=... x. Introduce Fj=kjj, [kj]=M T, (42) which, by Eq. (31), acts as a smoother for kj> 0(reducing Ra2dt ) and as an impetus for kj< 0 or sign–conditioned variants. 10.3 Absement as memory Define the absement (time integral of separation) A(t)≡Zt 0 x(τ)dτ, [A] = L T. (43) Introduce a memory–repelling force FA= +kAA, [kA]=M/T 3,(44) which grows with the time spent separated, providing a persistent outward bias. For a harmonic x(t)=Xsin ωt,A(t) = −(X/ω) cos ωt +const, so ⟨PA⟩=⟨Fosc Av⟩=−1 2kAX2,(45) independent of ω, a hallmark of hysteretic (Bean-like) loss. 9