Classical Electrodynamics · Problem 12.13

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · darwin-lagrangian-two-particles

Problem

(a) Specialize the Darwin Lagrangian (12.82) to the interaction of two charged particles (m1,q1)(m_1, q_1) and (m2,q2)(m_2, q_2). Introduce reduced particle coordinates, r=x1−x2\mathbf{r} = \mathbf{x}_1 - \mathbf{x}_2, v=v1−v2\mathbf{v} = \mathbf{v}_1 - \mathbf{v}_2 and also center of mass coordinates. Write out the Lagrangian in the reference frame in which the velocity of the center of mass vanishes and evaluate the canonical momentum components, px=∂L/∂vxp_x = \partial L/\partial v_x, etc. (b) Calculate the Hamiltonian to first order in 1/c21/c^2 and show that it is H=p22(1m1+1m2)+q1q2r−p48c2(1m13+1m23)+q1q22m1m2c2(p2+(p⋅r^)2r)H = \frac{p^2}{2}\left(\frac{1}{m_1}+\frac{1}{m_2}\right) + \frac{q_1 q_2}{r} - \frac{p^4}{8c^2}\left(\frac{1}{m_1^3}+\frac{1}{m_2^3}\right) + \frac{q_1 q_2}{2m_1 m_2 c^2}\left(\frac{p^2 + (\mathbf{p}\cdot\hat{\mathbf{r}})^2}{r}\right) Compare with the various terms in (42.1) of Bethe and Salpeter [op. cit. (Section 12.6), p. 193]. Discuss the agreements and disagreements.
Intuitive explanation

The Darwin Lagrangian captures relativistic corrections to the Coulomb interaction between two charged particles at order 1/c21/c^2. By transforming to relative and center-of-mass coordinates and then to the CM frame, you isolate the two-body dynamics; the key is carefully computing canonical momenta from the Lagrangian (which include velocity-dependent electromagnetic contributions), then performing a Legendre transform to the Hamiltonian while systematically expanding to first order in 1/c21/c^2. The resulting Hamiltonian reveals how relativistic effects modify the kinetic energy, introduce velocity-dependent corrections to the Coulomb potential, and split into physically interpretable terms—some matching known fine-structure effects and others revealing subtle differences with alternative formulations.

Intuitive explanation

The Darwin Lagrangian captures relativistic corrections to the Coulomb interaction between two charged particles at order 1/c21/c^2. By transforming to relative and center-of-mass coordinates and then to the CM frame, you isolate the two-body dynamics; the key is carefully computing canonical momenta from the Lagrangian (which include velocity-dependent electromagnetic contributions), then performing a Legendre transform to the Hamiltonian while systematically expanding to first order in 1/c21/c^2. The resulting Hamiltonian reveals how relativistic effects modify the kinetic energy, introduce velocity-dependent corrections to the Coulomb potential, and split into physically interpretable terms—some matching known fine-structure effects and others revealing subtle differences with alternative formulations.

Related theory

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