Darwin Lagrangian - AbsoluteAstronomy.com

The Darwin Lagrangian (named after Charles Galton Darwin

Charles Galton Darwin

Sir Charles Galton Darwin, KBE, MC, FRS was an English physicist, the grandson of Charles Darwin. He served as director of the National Physical Laboratory during the Second World War.-Early life:...

, grandson of the biologist

Charles Darwin

Charles Robert Darwin FRS was an English naturalist. He established that all species of life have descended over time from common ancestry, and proposed the scientific theory that this branching pattern of evolution resulted from a process that he called natural selection.He published his theory...

) describes the interaction to order

{v^2\over c^2}

between two charged particles in a vacuum and is given by

L^{ } =
L_f + L_{int}^{ }

where the free particle Lagrangian

Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

L_{f} =
{1\over 2} m_1v_1^2 + {1\over 8c^2}m_1v_1^4 +
{1\over 2} m_2v_2^2 + {1\over 8c^2}m_2v_2^4
  ,

and the interaction Lagrangian is

L_{int} =
L_C + L_{D}^{ }

where the Coulomb interaction is

L_{C} =
  -{q_1q_2 \over  r }

and the Darwin

Charles Galton Darwin

Sir Charles Galton Darwin, KBE, MC, FRS was an English physicist, the grandson of Charles Darwin. He served as director of the National Physical Laboratory during the Second World War.-Early life:...

interaction is

L_{D} =
  {q_1q_2 \over  r }{1\over 2c^2}
\mathbf v_1\cdot
\left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right]
\cdot\mathbf v_2
  .

Here

q_1

and

q_2

are the charges on particles 1 and 2 respectively,

m_1

and

m_2

are the masses of the particles,

\mathbf v_1

and

\mathbf v_2

are the velocities of the particles,

c

is the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

\mathbf r

is the vector between the two particles, and

\hat{\mathbf r}

is the unit vector in the direction of

\mathbf r

. The free Lagrangian is the Taylor expansion 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

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

generated by the other particle. If higher-order terms in v/c 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

Retarded potential

The retarded potential formulae describe the scalar or vector potential for electromagnetic fields of a time-varying current or charge distribution. The retardation of the influence connecting cause and effect is thereby essential; e.g...

effects must be accounted for.

Derivation of the Darwin interaction in a vacuum

The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is

L_{int} = -q\Phi +{q\over c} \mathbf u \cdot \mathbf A

where

\mathbf u

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

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 negative gradient is a given vector field....

in the Coulomb gauge is described by (Gaussian units

Gaussian units

Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...

)

\nabla^2 \mathbf A - {1\over c^2} {\partial^2 \mathbf A \over \partial t^2} = -{4\pi \over c} \mathbf J_t

where the transverse current

\mathbf J_t

is the solenoidal current (see Helmholtz decomposition

Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational vector field and a...

) generated by a second particle. The divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of the transverse current is zero. The current generated by the second particle is

\mathbf J = q_2 \mathbf v_2 \delta \left( \mathbf r - \mathbf r_2 \right)
  ,

which has a Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

\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\left( \mathbf k \right) = q_2 \left[ \mathbf 1 - \mathbf{\hat k} \mathbf{\hat k} \right] \cdot \mathbf v_2
\exp\left( -i\mathbf k \cdot \mathbf r_2 \right)
  .

It is easily verified that

\mathbf k \cdot \mathbf J_t\left( \mathbf k \right) = 0
  ,

which must be true if the divergence of the transverse current is zero. We see that

\mathbf J_t\left( \mathbf k \right)

is the component of the Fourier transformed current perpendicular to

\mathbf k

. From the equation for the vector potential, the Fourier transform of the vector potential is

\mathbf A \left( \mathbf k \right)
 = {4\pi \over c} {q_2\over k^2} \left[ \mathbf 1 - \mathbf{\hat k} \mathbf{\hat k} \right] \cdot \mathbf v_2
\exp\left( -i\mathbf k \cdot \mathbf r_2 \right)

where we have kept only the lowest order term in v/c. The inverse Fourier transform of the vector potential is

\mathbf A \left( \mathbf r \right)
\int { d^3 k \over \left ( 2 \pi \right ) ^3 }
 \; \mathbf A \left( \mathbf k \right) \;
{ \exp \left ( i\mathbf \mathbf k \cdot \mathbf r_1 \right )  }

 {q_2\over 2c} {1 \over  r }
\left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right]
\cdot \mathbf v_2

where

\mathbf r = \mathbf r_1 - \mathbf r_2

(see Common integrals in quantum field theory ). The Darwin interaction term in the Lagrangian is then NEWLINE

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L_{D}

 {q_1 q_2\over r} {1 \over 2c^2  }
\mathbf v_1 \cdot
\left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right]
\cdot \mathbf v_2

|} where again we kept only the lowest order term in v/c.

Lagrangian equations of motion

The equation of motion for one of the particles is

{d \over dt} {\partial  \over \partial \mathbf v_1} L\left( \mathbf r_1 , \mathbf v_1 \right)
\nabla_1 L\left( \mathbf r_1 , \mathbf v_1 \right)

where

\mathbf p_1

is the momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

of the particle.

Free particle

The equation of motion for a free particle neglecting interactions between the two particles is

{d \over dt} \left[ \left( 1   + {1\over 2} { v_1^2\over c^2 } \right)m_1\mathbf v_1 \right]

Interacting particles

For interacting particles, the equation of motion becomes

{d \over dt} \left[ \left( 1   + {1\over 2} { v_1^2\over c^2 } \right)m_1\mathbf v_1 +{q_1\over c}\mathbf A\left( \mathbf r_1 \right) \right]
-\nabla {q_1 q_2 \over r}
+\nabla \left[ {q_1q_2 \over  r }{1\over 2c^2}
\mathbf v_1\cdot
\left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right]
\cdot\mathbf v_2 \right]

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{d \mathbf{p}_1\over dt}
 {q_1 q_2 \over r^2}{\hat{\mathbf r}}
+{q_1 q_2 \over r^2}{1\over 2c^2}
\left\{ \mathbf v_1 \left( { {\hat{\mathbf r}}\cdot \mathbf v_2} \right)
+  \mathbf v_2 \left( { {\hat{\mathbf r}}\cdot \mathbf v_1}\right)
- {\hat{\mathbf r}} \left[ \mathbf v_1 \cdot \left( \mathbf 1 +3 {\hat{\mathbf r}}{\hat{\mathbf r}}\right)\cdot \mathbf v_2\right]
 \right\}

\mathbf p_1
\left( 1   + {1\over 2} { v_1^2\over c^2 } \right)m_1\mathbf v_1
+{q_1\over c}\mathbf A\left( \mathbf r_1 \right)

\mathbf A \left( \mathbf r_1 \right)



 {q_2\over 2c} {1 \over  r }
\left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right]
\cdot \mathbf v_2

\mathbf r

Darwin Hamiltonian for two particles in a vacuum

The Darwin Hamiltonian

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

for two particles in a vacuum is related to the Lagrangian by a Legendre transformation

Legendre transformation

In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another...

H =
\mathbf p_1 \cdot \mathbf v_1 + \mathbf p_2 \cdot \mathbf v_2
- L
  .

The Hamiltonian becomes NEWLINENEWLINENEWLINE

H\left( \mathbf r_1 , \mathbf p_1 ,\mathbf r_2 , \mathbf p_2 \right)
\left( 1   - {1\over 4} { p_1^2\over m_1^2 c^2 } \right){ p_1^2 \over 2 m_1}
\; + \; \left( 1   - {1\over 4} { p_2^2\over m_2^2 c^2 } \right){ p_2^2 \over 2 m_2}
\; + \; {q_1 q_2 \over r }
\; - \; {q_1q_2 \over  r }{1\over 2m_1 m_2 c^2}
\mathbf p_1\cdot
\left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right]
\cdot\mathbf p_2
  .

NEWLINENEWLINE \left( 1- {1\over 2} {p_1^2 \over m_1^2 c^2} \right) {\mathbf p_1 \over m_1} - {q_1 q_2\over 2m_1m_2 c^2} {1 \over r } \left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right] \cdot \mathbf p_2 and NEWLINENEWLINENEWLINE

{d \mathbf p_1\over dt}
 {q_1 q_2 \over r^2}{\hat{\mathbf r}}
\; + \; {q_1 q_2 \over r^2}{1\over 2m_1 m_2 c^2}
\left\{ \mathbf p_1 \left( { {\hat{\mathbf r}}\cdot \mathbf p_2} \right)
+  \mathbf p_2 \left( { {\hat{\mathbf r}}\cdot \mathbf p_1}\right)
- {\hat{\mathbf r}} \left[ \mathbf p_1 \cdot \left( \mathbf 1 +3 {\hat{\mathbf r}}{\hat{\mathbf r}}\right)\cdot \mathbf p_2\right]
 \right\}

NEWLINENEWLINE Note that the quantum mechanical Breit equation

Breit equation

The 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 interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and...

originally used the Darwin Lagrangian with the Darwin Hamiltonian as its classical starting point though the Breit equation would be better vindicated by the Wheeler-Feynman absorber theory and better yet quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

Derivation of the Darwin interaction in a vacuum

Lagrangian equations of motion

Free particle

Interacting particles

Darwin Hamiltonian for two particles in a vacuum

See also