RDYNLAB - Revision history

Oborn at 17:54, 18 June 2009

2009-06-18T17:54:53Z

Oborn at 17:53, 18 June 2009

2009-06-18T17:53:53Z

Oborn at 17:42, 18 June 2009

2009-06-18T17:42:04Z

Oborn at 17:31, 18 June 2009

2009-06-18T17:31:38Z

Oborn at 17:29, 18 June 2009

2009-06-18T17:29:57Z

Oborn at 17:25, 18 June 2009

2009-06-18T17:25:22Z

Oborn at 17:17, 18 June 2009

2009-06-18T17:17:27Z

Oborn at 22:35, 17 June 2009

2009-06-17T22:35:02Z

Oborn: New page: This is a library for modeling '''R'''elativistic '''DYN'''amics in the '''LAB'' frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for parti...

2009-06-17T22:34:27Z

New page: This is a library for modeling '''R'''elativistic '''DYN'''amics in the '''LAB'' frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for parti...

New page

This is a library for modeling '''R'''elativistic '''DYN'''amics in the '''LAB'' frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for particle motion, including radiative power losses for charged particles, as described in [[Simulating Particle Trajectories|this page]].

I am unable to post code to the wiki, so I have copied and pasted it all here. You will need all of these files in order to compile the code. GSL must be installed before you can compile this library.

The code can be compiled into a shared object library using the autoconf, automake, and libtool tools. Optionally, you could just compile these source files into other programs.

Return to [[Simulating Particle Trajectories]]

@@ Line 72: / Line 72: @@
 In order to implement radiative power loss into the dynamical equations, we need to go back to the relationship between force and momentum, i.e.
-<math>\vec{F}=\frac{d\vec{p}}{dt}=\frac{d\left(\gamma m_0 \vec{v}}{dt}=m_0\frac{d\gamma}{dt}\vec{v}+\gamma m_0\frac{d\vec{v}}{dt}</math>.
+<math>\vec{F}=\frac{d\vec{p}}{dt}=\frac{d\left(\gamma m_0 \vec{v}\right)}{dt}=m_0\frac{d\gamma}{dt}\vec{v}+\gamma m_0\frac{d\vec{v}}{dt}</math>.
 The power losses come through the term involving <math>\frac{d\gamma}{dt}</math>. Including this term into our previous differential equations yields

@@ Line 59: / Line 59: @@
 '''Power Loss for Accelerated Charges'''
 '''Summary of Relevant Formulae'''
@@ Line 70: / Line 99: @@
 In simple cases, such as when a charged particle travels through uniform electromagnetic fields, the set of dynamical equations can be solved analytically. In more complex cases, the system must be solved numerically.
-The solver uses
+The code solves the system of equations using an embedded 8th order Runge-Kutta Prince-Dormand method with 9th order error estimates with adaptive stepsize control, as implemented in the GNU Scientific Library [http://www.gnu.org/software/gsl].
-I am unable to post code to the wiki, so I have copied and pasted it all here. You will need all of these files in order to compile the code. GSL must be installed before you can compile this library.
+I am unable to post code to the wiki, so I have copied and pasted it all [[RDYNLAB Code|here]]. You will need all of the files in order to compile the code. GSL must be installed before you can compile this library.
 The code can be compiled into a shared object library using the autoconf, automake, and libtool tools. Optionally, you could just compile these source files into other programs.
 Return to [[Simulating Particle Trajectories]]

@@ Line 1: / Line 1: @@
-This is a library for modeling Relativistic DYNamics in the LAB frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for particle motion, including radiative power losses for charged particles, as described in [[Simulating Particle Trajectories|this page]].
+This is a library for modeling Relativistic DYNamics in the LAB frame.
 The general problem of modeling relativistic particle trajectories for a given force boils down to solving the following system of six differential equations:
@@ Line 36: / Line 38: @@
 This now gives us
-<math> \frac{E}{c^2}\frac{d\vec{v}}{dt}=\left(\vec{F}_{\parallel}+\vec{F}_{\perp}\right)-\frac{\vec{v}}{c}\left[\frac{\vec{v}}{c}\cdot\left(\vec{F}_{\parallel}+\vec{F}_{\perp}\right)\right]</math>
+<math> \frac{E}{c^2}\frac{d\vec{v}}{dt}=\left(\vec{F}_{\parallel}+\vec{F}_{\perp}\right)-\frac{\vec{v}}{c}\left[\frac{\vec{v}}{c}\cdot\left(\vec{F}_{\parallel}+\vec{F}_{\perp}\right)\right]</math>.
-, <math>m_0</math> is the rest energy of the particle, and <math>\gamma</math> is the usual relativistic factor. These equations are solved subject to an appropriate set of boundary conditions, usually the initial position and momentum of the particle.
+In simple cases, such as when a charged particle travels through uniform electromagnetic fields, the set of dynamical equations can be solved analytically. In more complex cases, the system must be solved numerically.
-In simple cases, such as when the fields are uniform, this set of equations can be solved analytically. However, for the "silver" permanent magnet we plan on using for the pair spectrometer, the fields are anything but uniform. In such a case, the system must be solved numerically.
+The solver uses
 I am unable to post code to the wiki, so I have copied and pasted it all here. You will need all of these files in order to compile the code. GSL must be installed before you can compile this library.

← Older revision		Revision as of 17:31, 18 June 2009
Line 32:		Line 32:
	where		where

−	<math> \vec{F}_{\parallel}=\left(\vec{F}\cdot\hat{v}\right)\hat(v) </math>.	+	<math> \vec{F}_{\parallel}=\left(\vec{F}\cdot\hat{v}\right)\hat{v} </math>.
		+
		+	This now gives us
		+
		+	<math> \frac{E}{c^2}\frac{d\vec{v}}{dt}=\left(\vec{F}_{\parallel}+\vec{F}_{\perp}\right)-\frac{\vec{v}}{c}\left[\frac{\vec{v}}{c}\cdot\left(\vec{F}_{\parallel}+\vec{F}_{\perp}\right)\right]</math>

@@ Line 10: / Line 10: @@
 <math>\frac{\vec{v}}{c}=\frac{\vec{p}c}{E}</math>
-<math>E^2-c^2\vec{p}\cdot\vec{p}=m_0^2c^4</math>
+<math>E^2-c^2\vec{p}\cdot\vec{p}=m_0^2c^4</math>,
 and taking the time derivatives of both, we find
@@ Line 16: / Line 16: @@
 <math> \frac{1}{c}\frac{d\vec{v}}{dt}=\frac{c}{E}\frac{d\vec{p}}{dt}-c\vec{p}\frac{1}{E^2}\frac{dE}{dt}</math>
-<math> 2E\frac{dE}{dt}-2c^2\vec{p}\cdot\frac{d\vec{p}}{dt}=0</math>
+<math> 2E\frac{dE}{dt}-2c^2\vec{p}\cdot\frac{d\vec{p}}{dt}=0</math>.
 Solving this second equation for <math>dE/dt</math> and plugging into the first, also noting that <math>d\vec{p}/dt=\vec{F}</math>, we obtain
-<math> \frac{1}{c}\frac{d\vec{v}}{dt}=\frac{c}{E}\vec{F}-\frac{c}{E^3}\vec{p}\left(c^2\vec{p}\cdot\vec{F}\right)</math>
+<math> \frac{1}{c}\frac{d\vec{v}}{dt}=\frac{c}{E}\vec{F}-\frac{c}{E^3}\vec{p}\left(c^2\vec{p}\cdot\vec{F}\right)</math>.
-Noting that </math>c\vec{p}=\frac{E}{c}\vec{v}</math>, we find
+Noting that <math>c\vec{p}=\frac{E}{c}\vec{v}</math>, we find
-<math> \frac{E}{c^2}\frac{d\vec{v}}{dt}=\vec{F}-\frac{\vec{v}}{c}\left(\frac{\vec{v}}{c}\cdot\vec{F}\right)</math>
+<math> \frac{E}{c^2}\frac{d\vec{v}}{dt}=\vec{F}-\frac{\vec{v}}{c}\left(\frac{\vec{v}}{c}\cdot\vec{F}\right)</math>.

← Older revision		Revision as of 17:25, 18 June 2009
Line 20:		Line 20:
	Solving this second equation for <math>dE/dt</math> and plugging into the first, also noting that <math>d\vec{p}/dt=\vec{F}</math>, we obtain		Solving this second equation for <math>dE/dt</math> and plugging into the first, also noting that <math>d\vec{p}/dt=\vec{F}</math>, we obtain

		+	<math> \frac{1}{c}\frac{d\vec{v}}{dt}=\frac{c}{E}\vec{F}-\frac{c}{E^3}\vec{p}\left(c^2\vec{p}\cdot\vec{F}\right)</math>

		+	Noting that </math>c\vec{p}=\frac{E}{c}\vec{v}</math>, we find
		+
		+	<math> \frac{E}{c^2}\frac{d\vec{v}}{dt}=\vec{F}-\frac{\vec{v}}{c}\left(\frac{\vec{v}}{c}\cdot\vec{F}\right)</math>

← Older revision		Revision as of 17:17, 18 June 2009
Line 1:		Line 1:
	This is a library for modeling Relativistic DYNamics in the LAB frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for particle motion, including radiative power losses for charged particles, as described in [[Simulating Particle Trajectories\|this page]].		This is a library for modeling Relativistic DYNamics in the LAB frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for particle motion, including radiative power losses for charged particles, as described in [[Simulating Particle Trajectories\|this page]].
		+
		+	The general problem of modeling relativistic particle trajectories for a given force boils down to solving the following system of six differential equations:
		+
		+	<math>\frac{d\vec{x}}{dt}=\vec{v}</math>
		+
		+	<math>\frac{d\vec{v}}{dt}=\vec{a}</math>
		+
		+	where <math>\vec{x}</math> is the position of the particle, <math>\vec{v}</math> is the momentum of the particle, and <math>\vec{a}</math> is the acceleration of the particle which can be derived from the force. Classically, we would just use <math>\vec{a}=\vec{F}/m</math>. Such a description, however, is not consistent with special relativity. Indeed, starting from the relativistic relations
		+
		+	<math>\frac{\vec{v}}{c}=\frac{\vec{p}c}{E}</math>
		+	<math>E^2-c^2\vec{p}\cdot\vec{p}=m_0^2c^4</math>
		+
		+	and taking the time derivatives of both, we find
		+
		+	<math> \frac{1}{c}\frac{d\vec{v}}{dt}=\frac{c}{E}\frac{d\vec{p}}{dt}-c\vec{p}\frac{1}{E^2}\frac{dE}{dt}</math>
		+
		+	<math> 2E\frac{dE}{dt}-2c^2\vec{p}\cdot\frac{d\vec{p}}{dt}=0</math>
		+
		+	Solving this second equation for <math>dE/dt</math> and plugging into the first, also noting that <math>d\vec{p}/dt=\vec{F}</math>, we obtain
		+
		+
		+
		+
		+	, <math>m_0</math> is the rest energy of the particle, and <math>\gamma</math> is the usual relativistic factor. These equations are solved subject to an appropriate set of boundary conditions, usually the initial position and momentum of the particle.
		+
		+	In simple cases, such as when the fields are uniform, this set of equations can be solved analytically. However, for the "silver" permanent magnet we plan on using for the pair spectrometer, the fields are anything but uniform. In such a case, the system must be solved numerically.

	I am unable to post code to the wiki, so I have copied and pasted it all here. You will need all of these files in order to compile the code. GSL must be installed before you can compile this library.		I am unable to post code to the wiki, so I have copied and pasted it all here. You will need all of these files in order to compile the code. GSL must be installed before you can compile this library.

← Older revision		Revision as of 22:35, 17 June 2009
Line 1:		Line 1:
−	This is a library for modeling ~~'''R'''elativistic '''DYN'''amics~~ in the ~~'''~~LAB'' frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for particle motion, including radiative power losses for charged particles, as described in [[Simulating Particle Trajectories\|this page]].	+	This is a library for modeling Relativistic DYNamics in the LAB frame. It uses an eigth order Runge-Kutta algorithm to solve the relativistic dynamical equations for particle motion, including radiative power losses for charged particles, as described in [[Simulating Particle Trajectories\|this page]].

	I am unable to post code to the wiki, so I have copied and pasted it all here. You will need all of these files in order to compile the code. GSL must be installed before you can compile this library.		I am unable to post code to the wiki, so I have copied and pasted it all here. You will need all of these files in order to compile the code. GSL must be installed before you can compile this library.