Force between magnets: Difference between revisions

Revision as of 06:22, 31 January 2014

The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the biologist) describes the interaction to order $\frac{v^{2}}{c^{2}}$ between two charged particles in a vacuum and is given by^[1]

L = L_{f} + L_{int},

where the free particle Lagrangian is

L_{f} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{8 c^{2}} m_{1} v_{1}^{4} + \frac{1}{2} m_{2} v_{2}^{2} + \frac{1}{8 c^{2}} m_{2} v_{2}^{4},

and the interaction Lagrangian is

L_{int} = L_{C} + L_{D},

where the Coulomb interaction is

L_{C} = - \frac{q_{1} q_{2}}{r},

and the Darwin interaction is

L_{D} = \frac{q_{1} q_{2}}{r} \frac{1}{2 c^{2}} v_{1} \cdot [1 + \hat{r} \hat{r}] \cdot v_{2} .

Here q₁ and q₂ are the charges on particles 1 and 2 respectively, m₁ and m₂ are the masses of the particles, v₁ and v₂ are the velocities of the particles, c is the speed of light, r is the vector between the two particles, and $\hat{r}$ is the unit vector in the direction of 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 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 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^[2]

L_{int} = - q Φ + \frac{q}{c} u \cdot A,

where 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 in the Coulomb gauge is described by^[3] (Gaussian units)

\nabla^{2} A - \frac{1}{c^{2}} \frac{\partial^{2} A}{\partial t^{2}} = - \frac{4 π}{c} J_{t}

where the transverse current J_t is the solenoidal current (see Helmholtz decomposition) generated by a second particle. The divergence of the transverse current is zero.

The current generated by the second particle is

J = q_{2} v_{2} δ (r - r_{2}),

which has a Fourier transform

J (k) \equiv \int d^{3} r \exp (- i k \cdot r) J (r) = q_{2} v_{2} \exp (- i k \cdot r_{2}) .

The transverse component of the current is

J_{t} (k) = q_{2} [1 - \hat{k} \hat{k}] \cdot v_{2} \exp (- i k \cdot r_{2}) .

It is easily verified that

k \cdot J_{t} (k) = 0,

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

J_{t} (k)

is the component of the Fourier transformed current perpendicular to k.

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

A (k) = \frac{4 π}{c} \frac{q_{2}}{k^{2}} [1 - \hat{k} \hat{k}] \cdot v_{2} \exp (- i k \cdot r_{2})

where we have kept only the lowest order term in v/c.

The inverse Fourier transform of the vector potential is

A (r) = \int \frac{d^{3} k}{{(2 π)}^{3}} A (k) \exp (i k \cdot r_{1}) = \frac{q_{2}}{2 c} \frac{1}{r} [1 + \hat{r} \hat{r}] \cdot v_{2}

where

r = r_{1} - r_{2}

(see Common integrals in quantum field theory ).

The Darwin interaction term in the Lagrangian is then

$L_{D} = \frac{q_{1} q_{2}}{r} \frac{1}{2 c^{2}} v_{1} \cdot [1 + \hat{r} \hat{r}] \cdot v_{2}$

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

\frac{d}{d t} \frac{\partial}{\partial v_{1}} L (r_{1}, v_{1}) = \nabla_{1} L (r_{1}, v_{1})

\frac{d p_{1}}{d t} = \nabla_{1} L (r_{1}, v_{1})

where p₁ is the momentum of the particle.

Free particle

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

\frac{d}{d t} [(1 + \frac{1}{2} \frac{v_{1}^{2}}{c^{2}}) m_{1} v_{1}] = 0

p_{1} = (1 + \frac{1}{2} \frac{v_{1}^{2}}{c^{2}}) m_{1} v_{1}

Interacting particles

For interacting particles, the equation of motion becomes

\frac{d}{d t} [(1 + \frac{1}{2} \frac{v_{1}^{2}}{c^{2}}) m_{1} v_{1} + \frac{q_{1}}{c} A (r_{1})] = - \nabla \frac{q_{1} q_{2}}{r} + \nabla [\frac{q_{1} q_{2}}{r} \frac{1}{2 c^{2}} v_{1} \cdot [1 + \hat{r} \hat{r}] \cdot v_{2}]

$\frac{d p_{1}}{d t} = \frac{q_{1} q_{2}}{r^{2}} \hat{r} + \frac{q_{1} q_{2}}{r^{2}} \frac{1}{2 c^{2}} {v_{1} (\hat{r} \cdot v_{2}) + v_{2} (\hat{r} \cdot v_{1}) - \hat{r} [v_{1} \cdot (1 + 3 \hat{r} \hat{r}) \cdot v_{2}]}$

p_{1} = (1 + \frac{1}{2} \frac{v_{1}^{2}}{c^{2}}) m_{1} v_{1} + \frac{q_{1}}{c} A (r_{1})

A (r_{1}) = \frac{q_{2}}{2 c} \frac{1}{r} [1 + \hat{r} \hat{r}] \cdot v_{2}

r = r_{1} - r_{2}

Darwin Hamiltonian for two particles in a vacuum

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

H = p_{1} \cdot v_{1} + p_{2} \cdot v_{2} - L .

The Hamiltonian becomes

$H (r_{1}, p_{1}, r_{2}, p_{2}) = (1 - \frac{1}{4} \frac{p_{1}^{2}}{m_{1}^{2} c^{2}}) \frac{p_{1}^{2}}{2 m_{1}} + (1 - \frac{1}{4} \frac{p_{2}^{2}}{m_{2}^{2} c^{2}}) \frac{p_{2}^{2}}{2 m_{2}} + \frac{q_{1} q_{2}}{r} - \frac{q_{1} q_{2}}{r} \frac{1}{2 m_{1} m_{2} c^{2}} p_{1} \cdot [1 + \hat{r} \hat{r}] \cdot p_{2} .$

Hamiltonian equations of motion

The Hamiltonian equations of motion are

v_{1} = \frac{\partial H}{\partial p_{1}}

and

\frac{d p_{1}}{d t} = - \nabla_{1} H

which yield

v_{1} = (1 - \frac{1}{2} \frac{p_{1}^{2}}{m_{1}^{2} c^{2}}) \frac{p_{1}}{m_{1}} - \frac{q_{1} q_{2}}{2 m_{1} m_{2} c^{2}} \frac{1}{r} [1 + \hat{r} \hat{r}] \cdot p_{2}

and

$\frac{d p_{1}}{d t} = \frac{q_{1} q_{2}}{r^{2}} \hat{r} + \frac{q_{1} q_{2}}{r^{2}} \frac{1}{2 m_{1} m_{2} c^{2}} {p_{1} (\hat{r} \cdot p_{2}) + p_{2} (\hat{r} \cdot p_{1}) - \hat{r} [p_{1} \cdot (1 + 3 \hat{r} \hat{r}) \cdot p_{2}]}$

Note that the quantum mechanical Breit equation 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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 pp. 596-598
↑ Jackson, pp. 580-581.
↑ Jackson, p. 242.

[1] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 pp. 596-598

[2] Jackson, pp. 580-581.

[3] Jackson, p. 242.

[1]

[2]

[3]

Force between magnets: Difference between revisions

Revision as of 06:22, 31 January 2014

Contents

Derivation of the Darwin interaction in a vacuum

Lagrangian equations of motion

Free particle

Interacting particles

Darwin Hamiltonian for two particles in a vacuum

Hamiltonian equations of motion

See also

References

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+The '''Darwin Lagrangian''' (named after [[Charles Galton Darwin]], grandson of [[Charles Darwin|the biologist]]) describes the interaction to order <math>\frac{v^2}{c^2}</math> between two charged particles in a vacuum and is given by<ref>{{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 047130932X}} pp. 596-598</ref>
+:<math> L = L_\text{f} + L_\text{int}, </math>
+where the free particle [[Lagrangian]] is
+:<math> L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, </math>
+and the interaction Lagrangian is
+:<math> L_\text{int} = L_\text{C} + L_\text{D}, </math>
+where the [[Coulomb force|Coulomb interaction]] is
+:<math>L_\text{C} = -\frac{q_1 q_2}{r}, </math>
+and the [[Charles Galton Darwin|Darwin]] interaction is
+:<math> L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right] \cdot \mathbf v_2. </math>
+Here ''q''<sub>1</sub> and ''q''<sub>2</sub> are the charges on particles 1 and 2 respectively, ''m''<sub>1</sub> and ''m''<sub>2</sub> are the masses of the particles, '''v'''<sub>1</sub> and '''v'''<sub>2</sub> are the velocities of the particles, ''c'' is the [[speed of light]], '''r''' is the vector between the two particles, and <math>\hat{\mathbf r}</math> is the [[unit vector]] in the direction of '''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]] 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 [[Retarded potential | retardation]] 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<ref> Jackson, pp. 580-581. </ref>
+:<math> L_\text{int} = -q\Phi + {q\over c} \mathbf u \cdot \mathbf A, </math>
+where '''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]] in the [[Coulomb gauge]] is described by<ref> Jackson, p. 242. </ref> ([[Gaussian units]])
+:<math> \nabla^2 \mathbf A - {1\over c^2} {\partial^2 \mathbf A \over \partial t^2} = -{4\pi \over c} \mathbf J_t</math>
+where the transverse current '''J'''<sub>''t''</sub> is the [[solenoidal field | solenoidal current]] (see [[Helmholtz decomposition]]) generated by a second particle. The [[divergence]] of the transverse current is zero.
+The current generated by the second particle is
+:<math>  \mathbf J = q_2 \mathbf v_2 \delta \left( \mathbf r - \mathbf r_2 \right),</math>
+which has a [[Fourier transform]]
+:<math>  \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).</math>
+The transverse component of the current is
+:<math>\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).</math>
+It is easily verified that
+:<math>\mathbf k \cdot \mathbf J_t\left( \mathbf k \right) = 0,</math>
+which must be true if the divergence of the transverse current is zero. We see that
+:<math>\mathbf J_t\left( \mathbf k \right) </math>
+is the component of the Fourier transformed current perpendicular to '''k'''.
+From the equation for the vector potential, the Fourier transform of the vector potential is
+:<math>
+ \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)
+  </math>
+where we have kept only the lowest order term in v/c.
+The inverse Fourier transform of the vector potential is
+:<math>\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 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</math>
+where
+:<math>\mathbf r = \mathbf r_1 - \mathbf r_2 </math>
+(see [[Common integrals in quantum field theory#Transverse potential with mass |Common integrals in quantum field theory ]]).
+The Darwin interaction term in the Lagrangian is then
+::{|cellpadding="2" style="border:2px solid #ccccff"
+|
+<math>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  </math>
+|}
+where again we kept only the lowest order term in v/c.
+==Lagrangian equations of motion==
+The [[equations of motion | equation of motion]] for one of the particles is
+:<math>  {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)  </math>
+:<math>  {d \mathbf p_1 \over dt} = \nabla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) </math>
+where '''p'''<sub>1</sub> is the [[momentum]] of the particle.
+===Free particle===
+The equation of motion for a free particle neglecting interactions between the two particles is
+:<math>{d \over dt} \left[ \left( 1   + {1\over 2} { v_1^2\over c^2 } \right)m_1\mathbf v_1 \right]=0</math>
+:<math> \mathbf p_1 = \left( 1   + {1\over 2} { v_1^2\over c^2 } \right)m_1\mathbf v_1 </math>
+===Interacting particles===
+For interacting particles, the equation of motion becomes
+:<math>  {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]  </math>
+::{|cellpadding="2" style="border:2px solid #ccccff"
+|
+<math> {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\}  </math>
+|}
+:<math> \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)</math>
+:<math>\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  </math>
+:<math>\mathbf r =\mathbf r_1 - \mathbf r_2</math>
+==Darwin Hamiltonian for two particles in a vacuum==
+The Darwin [[Hamiltonian mechanics |Hamiltonian]] for two particles in a vacuum is related to the Lagrangian by a [[Legendre transformation]]
+:<math>  H = \mathbf p_1 \cdot \mathbf v_1 + \mathbf p_2 \cdot \mathbf v_2 - L.</math>
+The Hamiltonian becomes
+{|cellpadding="2" style="border:2px solid #ccccff"
+|
+<math> 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
+  .</math>
+|}
+==Hamiltonian equations of motion==
+The Hamiltonian equations of motion are
+:<math>  \mathbf v_1 = {\partial H \over \partial \mathbf p_1}  </math>
+and
+:<math>  {d \mathbf p_1 \over dt} = -\nabla_1 H </math>
+which yield
+:<math>  \mathbf v_1  =
+\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</math>
+and
+{|cellpadding="2" style="border:2px solid #ccccff"
+|
+<math>  {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\}  </math>
+|}
+Note that the quantum mechanical [[Breit equation]] 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]].
+==See also==
+* [[Static forces and virtual-particle exchange]]
+* [[Breit equation]]
+* [[Wheeler–Feynman absorber theory]]
+==References==
+{{Reflist}}<!--added under references heading by script-assisted edit-->
+[[Category:Magnetostatics]]
+[[Category:Concepts in physics]]

Force between magnets: Difference between revisions

Revision as of 06:22, 31 January 2014

Derivation of the Darwin interaction in a vacuum

Lagrangian equations of motion

Free particle

Interacting particles

Darwin Hamiltonian for two particles in a vacuum

Hamiltonian equations of motion

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