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The Darwin Lagrangian (named after
Charles Galton Darwin Sir Charles Galton Darwin (19 December 1887 – 31 December 1962) was an English physicist who served as director of the National Physical Laboratory (NPL) during the Second World War. He was a son of the mathematician George Darwin and a gr ...
, grandson of the naturalist) describes the interaction to order / between two charged particles in a vacuum where ''c '' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. It was derived before the advent of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
and resulted from a more detailed investigation of the classical, electromagnetic interactions of the electrons in an atom. From the
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model was a model of the atom that incorporated some early quantum concepts. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear Rutherford model, model, i ...
it was known that they should be moving with velocities approaching the speed of light. C.G. Darwin, ''The Dynamical Motions of Charged Particles'', Philosophical Magazine 39, 537-551 (1920). The full
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for two interacting particles is L = L_\text + L_\text, where the free particle part is L_\text = \frac m_1 v_1^2 + \frac m_1 v_1^4 + \frac m_2 v_2^2 + \frac m_2 v_2^4, The interaction is described by L_\text = L_\text + L_\text, where the
Coulomb interaction Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the ''electrostatic f ...
in
Gaussian units Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
is L_\text = -\frac, while the Darwin interaction is L_\text = \frac \frac \mathbf v_1 \cdot \left mathbf 1 + \hat\mathbf \hat\mathbf\right\cdot \mathbf v_2. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
, is the vector between the two particles, and \hat\mathbf r is the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
in the direction of . The first part is the
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of free Lagrangian of two relativistic particles to second order in ''v''. The Darwin interaction term is due to one particle reacting to the
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
generated by the other particle. If higher-order terms in are retained, then the field degrees of freedom must be taken into account, and the interaction can no longer be taken to be instantaneous between the particles. In that case retardation effects must be accounted for.


Derivation in vacuum

The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is L_\text = -q\Phi + \frac \mathbf u \cdot \mathbf A, where is the relativistic velocity of the particle. The first term on the right generates the Coulomb interaction. The second term generates the Darwin interaction. The
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field \mathbf, a ' ...
in the
Coulomb gauge In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...
is described by \nabla^2 \mathbf A - \frac \frac = -\frac \mathbf J_t where the transverse current is the solenoidal current (see
Helmholtz decomposition In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational ( curl-free) vector field and a sole ...
) generated by a second particle. The
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of the transverse current is zero. The current generated by the second particle is \mathbf J = q_2 \mathbf v_2 \delta, which has a
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
\mathbf J\left( \mathbf k \right) \equiv \int d^3r \exp\left( -i\mathbf k \cdot \mathbf r \right) \mathbf J\left( \mathbf r \right) = q_2 \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right). The transverse component of the current is \mathbf J_t ( \mathbf k) = q_2 \left \mathbf 1 - \hat\mathbf \hat\mathbf \right\cdot \mathbf v_2 e^. It is easily verified that \mathbf k \cdot \mathbf J_t ( \mathbf k) = 0, which must be true if the divergence of the transverse current is zero. We see that \mathbf J_t ( \mathbf k ) is the component of the Fourier transformed current perpendicular to . From the equation for the vector potential, the Fourier transform of the vector potential is \mathbf A \left( \mathbf k \right) = \frac \frac \left \mathbf 1 - \hat\mathbf \hat\mathbf \right\cdot \mathbf v_2 e^ where we have kept only the lowest order term in . The inverse Fourier transform of the vector potential is \mathbf A \left( \mathbf r \right) =\int \frac \; \mathbf A ( \mathbf k ) \; e^ = \frac \frac \left mathbf 1 + \hat\mathbf \hat\mathbf\right\cdot \mathbf v_2 where \mathbf r = \mathbf r_1 - \mathbf r_2 (see '). The Darwin interaction term in the Lagrangian is then L_\text = \frac \frac \mathbf v_1 \cdot \left mathbf 1 + \hat\mathbf r \hat\mathbf r\right\cdot \mathbf v_2 where again we kept only the lowest order term in .


Lagrangian equations of motion

The
equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
for one of the particles is \frac \frac L\left( \mathbf r_1 , \mathbf v_1 \right) = \nabla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) \frac = \nabla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) where is the
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
of the particle.


Free particle

The equation of motion for a free particle neglecting interactions between the two particles is \frac \left \left( 1 + \frac \frac \right)m_1 \mathbf v_1 \right= 0 \mathbf p_1 = \left( 1 + \frac \frac \right)m_1 \mathbf v_1


Interacting particles

For interacting particles, the equation of motion becomes \frac \left \left( 1 + \frac \frac \right)m_1 \mathbf v_1 + \frac \mathbf A\left( \mathbf r_1 \right) \right= - \nabla \frac + \nabla \left[ \frac \frac \mathbf v_1 \cdot \left mathbf 1 + \hat\mathbf r \hat\mathbf r\right\cdot \mathbf v_2 \right] \frac = \frac \hat + \frac \frac \left\ \mathbf p_1 = \left( 1 + \frac \frac \right)m_1\mathbf v_1 + \frac \mathbf A\left( \mathbf r_1 \right) \mathbf A \left( \mathbf r_1 \right) = \frac \frac \left mathbf 1 + \hat\mathbf r \hat\mathbf r\right\cdot \mathbf v_2 \mathbf r = \mathbf r_1 - \mathbf r_2


Hamiltonian for two particles in a vacuum

The Darwin
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
for two particles in a vacuum is related to the Lagrangian by a
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a rea ...
H = \mathbf p_1 \cdot \mathbf v_1 + \mathbf p_2 \cdot \mathbf v_2 - L. The Hamiltonian becomes H\left( \mathbf r_1 , \mathbf p_1 ,\mathbf r_2 , \mathbf p_2 \right)= \left( 1 - \frac \frac \right) \frac \; + \; \left( 1 - \frac \frac \right) \frac \; + \; \frac \; - \; \frac \frac \mathbf p_1\cdot \left mathbf 1 + \mathbf \mathbf\right\cdot\mathbf p_2 . This Hamiltonian gives the interaction energy between the two particles. It has recently been argued that when expressed in terms of particle velocities, one should simply set \mathbf = m\mathbf in the last term and reverse its sign. K.T. McDonald
''Darwin Energy Paradoxes''
Princeton University (2019).


Equations of motion

The Hamiltonian equations of motion are \mathbf v_1 = \frac and \frac = -\nabla_1 H , which yield \mathbf v_1 = \left( 1- \frac \frac \right) \frac - \frac \frac \left mathbf 1 + \hat\mathbf r \hat\mathbf r\right\cdot \mathbf p_2 and \frac = \frac \; + \; \frac \frac \left\


Quantum electrodynamics

The structure of the Darwin interaction can also be clearly seen in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
and due to the exchange of
photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that ...
in lowest order of
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. When the photon has
four-momentum In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum i ...
with wave vector its
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
in the
Coulomb gauge In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...
has two components. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, ''Relativistic Quantum Theory'', Pergamon Press, Oxford (1971). : D_(k) = gives the Coulomb interaction between two charged particles, while : D_(k) = \left( \delta_ - \right) describes the exchange of a transverse photon. It has a polarization vector \mathbf_\lambda and couples to a particle with charge q and three-momentum \mathbf with a strength - q\sqrt\,\mathbf_\lambda\cdot\mathbf/m. Since \mathbf_\lambda\cdot\mathbf = 0 in this gauge, it doesn't matter if one uses the particle momentum before or after the photon couples to it. In the exchange of the photon between the two particles one can ignore the frequency \omega compared with c\mathbf in the propagator working to the accuracy in v^2/c^2 that is needed here. The two parts of the propagator then give together the effective Hamiltonian : H_(\mathbf) = - \mathbf_1\cdot \left( \mathbf 1 - \hat\mathbf \hat\mathbf \right) \cdot \mathbf_2 for their interaction in k-space. This is now identical with the classical result and there is no trace of the quantum effects used in this derivation. A similar calculation can be done when the photon couples to Dirac particles with spin and used for a derivation of the
Breit equation The Breit equation, or Dirac–Coulomb–Breit equation, is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) intera ...
. It gives the same Darwin interaction but also additional terms involving the spin degrees of freedom and depending on the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
.


See also

*
Static forces and virtual-particle exchange Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces ...
*
Breit equation The Breit equation, or Dirac–Coulomb–Breit equation, is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) intera ...
*
Wheeler–Feynman absorber theory The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is a theory of electrodynamics based on a relativistic correct ...


References

{{reflist Magnetostatics Equations of physics