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
where the free particle part is
The interaction is described by
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
while the
Darwin interaction is
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
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
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
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
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 ...
The transverse component of the current is
It is easily verified that
which must be true if the divergence of the transverse current is zero. We see that
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
where we have kept only the lowest order term in .
The inverse Fourier transform of the vector potential is
where
(see ').
The Darwin interaction term in the Lagrangian is then
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
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
Interacting particles
For interacting particles, the equation of motion becomes
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 ...
The Hamiltonian becomes
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
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
and
which yield
and
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). ]
:
gives the Coulomb interaction between two charged particles, while
:
describes the exchange of a transverse photon. It has a polarization vector
and couples to a particle with charge
and three-momentum
with a strength
Since
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
compared with
in the propagator working to the accuracy in
that is needed here. The two parts of the propagator then give together the effective Hamiltonian
:
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