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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>
These are probably the preferred pocketknives. The Swiss Army knife and the Handyman are probably the most well known of the multi-goal knives. Boy Scout knives are also multi-purpose knives. Along with the knife blade, multi-purpose knives have can openers, scissors, leather punchers, tweezers, and even screwdrivers. These can come in handy on tenting trips. Nevertheless, if all you’re on the lookout for is a sharp blade, the extra options will probably annoy you. Friction folder knife. That is another methodology of a non-locking blade. [http://Canimpact.org/members/rethaloman/activity/696300/ Friction folder] knives use friction between the blade and the scales to hold the blade in place once opened.<br><br>The Ruger Mark II and Mark III pistols are22-caliber target pistols used primarily for competition shooting, plinking and varmint hunting. The two weapons are based on the identical design, though the Mark III has a magazine security and a slimmer grip handle. The factory Mark III hammer is heavier than the Mark II hammer, which provides [http://www.knife-depot.com emerson knives] the Ruger Mark III a heavier trigger pull. The Volquartsen (VQ) Mark II hammer could be installed instead of the Mark III hammer for a lighter set off pull, resulting in better accuracy, fast magazine release and faster shooting time.<br><br>The most typical lock fastened in all traditional pocket knives is the one that offers method when stress is applied on the blade. Springs are fitted within the pivot to resist pressure that's generated when the knife is used. To shut the blade, all you must [http://www.thebestpocketknifereviews.com/old-timer-knives-pocket-knife-models/ Old Timer Pocket knives] do is press the unsharpened aspect of the blade and the blade will come free and fold into the deal with. This is adequate for most people who make use of such knives Nevertheless, as time goes by, the power of the spring weakens and the knife closes even when the person doesn't need it to.<br><br>Black and Decker introduced the workmate many years ago (my own workmate is round 20 years previous) and in an indication of the longevity of some concepts, continues to be out there in the present day. B&D has expanded the unique line to three models now, with various [http://www.Cartercountymarket.com/ccm-webid/item.php?id=31949&mode=1 capabilities] and prices, but they are all nonetheless the workmate that's liked by so many residence improvement fans. The workmate additionally folds fairly flat, taking over little room within the workshop or storage, and is gentle enough to easily carry to a extra distant job. All but probably the most economical model have folding legs that produce two different work heights.<br><br>The head of the sander known as a cartridge including the 2 wheels rotates around the motor shaft so you may change the place of the belt in relation to the handle. The sharpener has a two position switch with one place being a short lived on and the opposite switching on until you switch it off. Proper now I really [http://www.thebestpocketknifereviews.com/old-timer-knives-pocket-knife-models/ Old timer pocket knife] think the Compression Lock ranks the most effective on all four of my standards, but this being the Golden Age of Gear, now we have a bevy of superb choices–the Tri-Advert lock is great, the Compression lock is something I like, and I think the liner lock is criminally underrated.<br><br>Customized knifemaker Chris Reeve invented the “body lock” to be his up to date model of the liner lock. The body lock functions just like the liner lock nevertheless, instead of utilizing a separate lock bar, it makes use of the actual frame which can also be spring loaded to hold the blade in place. He decided to name in “integral lock”. The mid lock is actually a back lock positioned on the center of the knife spine as a substitute of the back. This gives added energy for the blade lock and allows the blade to face up to more strain than the again lock. Chilly Metal is an avid proponent of the mid lock. Lock Again (Again Lock)<br><br>In a big signal of this rising reputation, Benchmade Knives , an organization not often related to bushcrafting, decided to pick up on the pattern. In March of 2012, they called on their gifted in-house knife designer, outdoor enthusiast Shane Sibert , to assist create Benchmade's first ever bushcraft knife - the 162 Bushcrafter. Determined not to repeat this painful piece of historical past, Benchmade took particular care in constructing the Bushcrafter to make it face up to heavy use. Their main goal was to make sure that the S30V Stainless used on the Bushcrafter would not solely have excellent edge-retention, but additionally be extremely tough, easy to sharpen and chip resistant.<br><br>The SRK comes with a black blade so the very first thing I did was remove the paint. I intended to make use of the SRK for meat slicing and searching, so the painted blade seemed bizarre and Rambo-like. Apart from, I hunt with a number of former navy sorts, and they'd have laughed a "tactical" or survival knife out of camp! The primary fixed that it's best to look for is a fixed blade knife , all good survival knives ought to have a set blade. Most of these knives can deal with way more punishment and hard use than a [http://dallaskoreans.com/index.php?document_srl=404980&mid=Test folding knife] and might be a significantly better device for chopping and cutting.<br><br>The Whiskey Folding KnifeThey’ve taken the basic French Opinel chrome steel blade pocket knife, and added the Son of a Sailor contact with sign [http://wiki.Paconun.eu/index.php/Old_Timer_Knife_Values flag-impressed colorblocking]. This walnut model, the No. 6, is named the Whiskey, and is available in woodland Inexperienced & Neon Coral or Cobalt & Seafoam. Every knife is hand-painted and will differ a bit from the picture. No two are precisely alike, but all are made with care!These knives are made with heavy-duty paint and polish. This can be a software supposed for rugged use, and a rough patina might be earned on the complete finish of the knife, over time. Green & Neon Coral Cobalt & Seafoam
 
:<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]]

Latest revision as of 22:57, 5 January 2015

These are probably the preferred pocketknives. The Swiss Army knife and the Handyman are probably the most well known of the multi-goal knives. Boy Scout knives are also multi-purpose knives. Along with the knife blade, multi-purpose knives have can openers, scissors, leather punchers, tweezers, and even screwdrivers. These can come in handy on tenting trips. Nevertheless, if all you’re on the lookout for is a sharp blade, the extra options will probably annoy you. Friction folder knife. That is another methodology of a non-locking blade. Friction folder knives use friction between the blade and the scales to hold the blade in place once opened.

The Ruger Mark II and Mark III pistols are22-caliber target pistols used primarily for competition shooting, plinking and varmint hunting. The two weapons are based on the identical design, though the Mark III has a magazine security and a slimmer grip handle. The factory Mark III hammer is heavier than the Mark II hammer, which provides emerson knives the Ruger Mark III a heavier trigger pull. The Volquartsen (VQ) Mark II hammer could be installed instead of the Mark III hammer for a lighter set off pull, resulting in better accuracy, fast magazine release and faster shooting time.

The most typical lock fastened in all traditional pocket knives is the one that offers method when stress is applied on the blade. Springs are fitted within the pivot to resist pressure that's generated when the knife is used. To shut the blade, all you must Old Timer Pocket knives do is press the unsharpened aspect of the blade and the blade will come free and fold into the deal with. This is adequate for most people who make use of such knives Nevertheless, as time goes by, the power of the spring weakens and the knife closes even when the person doesn't need it to.

Black and Decker introduced the workmate many years ago (my own workmate is round 20 years previous) and in an indication of the longevity of some concepts, continues to be out there in the present day. B&D has expanded the unique line to three models now, with various capabilities and prices, but they are all nonetheless the workmate that's liked by so many residence improvement fans. The workmate additionally folds fairly flat, taking over little room within the workshop or storage, and is gentle enough to easily carry to a extra distant job. All but probably the most economical model have folding legs that produce two different work heights.

The head of the sander known as a cartridge including the 2 wheels rotates around the motor shaft so you may change the place of the belt in relation to the handle. The sharpener has a two position switch with one place being a short lived on and the opposite switching on until you switch it off. Proper now I really Old timer pocket knife think the Compression Lock ranks the most effective on all four of my standards, but this being the Golden Age of Gear, now we have a bevy of superb choices–the Tri-Advert lock is great, the Compression lock is something I like, and I think the liner lock is criminally underrated.

Customized knifemaker Chris Reeve invented the “body lock” to be his up to date model of the liner lock. The body lock functions just like the liner lock nevertheless, instead of utilizing a separate lock bar, it makes use of the actual frame which can also be spring loaded to hold the blade in place. He decided to name in “integral lock”. The mid lock is actually a back lock positioned on the center of the knife spine as a substitute of the back. This gives added energy for the blade lock and allows the blade to face up to more strain than the again lock. Chilly Metal is an avid proponent of the mid lock. Lock Again (Again Lock)

In a big signal of this rising reputation, Benchmade Knives , an organization not often related to bushcrafting, decided to pick up on the pattern. In March of 2012, they called on their gifted in-house knife designer, outdoor enthusiast Shane Sibert , to assist create Benchmade's first ever bushcraft knife - the 162 Bushcrafter. Determined not to repeat this painful piece of historical past, Benchmade took particular care in constructing the Bushcrafter to make it face up to heavy use. Their main goal was to make sure that the S30V Stainless used on the Bushcrafter would not solely have excellent edge-retention, but additionally be extremely tough, easy to sharpen and chip resistant.

The SRK comes with a black blade so the very first thing I did was remove the paint. I intended to make use of the SRK for meat slicing and searching, so the painted blade seemed bizarre and Rambo-like. Apart from, I hunt with a number of former navy sorts, and they'd have laughed a "tactical" or survival knife out of camp! The primary fixed that it's best to look for is a fixed blade knife , all good survival knives ought to have a set blade. Most of these knives can deal with way more punishment and hard use than a folding knife and might be a significantly better device for chopping and cutting.

The Whiskey Folding KnifeThey’ve taken the basic French Opinel chrome steel blade pocket knife, and added the Son of a Sailor contact with sign flag-impressed colorblocking. This walnut model, the No. 6, is named the Whiskey, and is available in woodland Inexperienced & Neon Coral or Cobalt & Seafoam. Every knife is hand-painted and will differ a bit from the picture. No two are precisely alike, but all are made with care!These knives are made with heavy-duty paint and polish. This can be a software supposed for rugged use, and a rough patina might be earned on the complete finish of the knife, over time. Green & Neon Coral Cobalt & Seafoam

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