What are the FULL classical electrodynamic equations?

[pellman]

Most (or all) solvable EM problems are either (1) given a fixed static or periodically-varying EM field, find the motions of charged particles, OR (2) given a fixed static or periodically-varying charge distribution, find the resulting EM field.

That is, either (1) solve the Lorentz force equation or (2) Maxwell's equations. But what does the complete problem (1) + (2) look like? Oh, I know, I know, its not solvable. I'd just like to see the differential equations once.

The problem is that all my references give Maxwell's equations in terms of a continuous charge distribution ($$\rho$$ and $$J$$) while the Lorentz force is given for point particles. If we stick to a continuous charge distribution, what I need then, I guess, is the Maxwell equations (got 'em) + the continuity equation (got it) + plus the continuous version of the Lorentz force in terms of $$\rho$$ and $$J$$ and depending on E and B (need it).

If someone could provide that last piece, I'd appreciate it.

Interestingly, in quantum theory this is no big deal. It falls right out of SU(1) gauge invariance, replacing the derivatives of the fields with the "covariant derivative", i.e. the derivative plus a term proportional to the EM vector potential (which really just amounts to the classical p --> p - eA). I am wondering how the classical version compares with the quantum case.

Todd

 

[lalbatros]

Todd,

Consider that a continuous charge distribution is made of individual charges.
A macroscopic averaging of the "microscopic" equations is all that you need, theoretically.
Unfortunately the result may not be so simple than it sounds.
For more details you should read some book on plasma physics.
Plasma physics is precisely the study of fluids of charged particles.

This field is far from being closed. There is a lot of physics behind.

Take for example the -well known- theory of collisions in plasma. The collisions of charged particles is a long-range interaction, which is very different from collisions of "neutral" particles. The theory involves a shielding effect by opposite charges (Debye shielding). Some hypothesis or some "philosophy" are needed to drift from microscopically reversible equations to the irreversible macroscopic equations. The end result is a collision term in the averaged equations (Fokker-Plank for example) responsible for some dissipative behaviour like resistivity, viscosity, ... . This is related to the roots of the 2nd law of thermodynamics, see the difficulty ?

There are also many different possible approximations depending on the situation considered. For example MHD is a kind of equivalent to the Navier-Stokes equation when viscosity and dissipation do not play a significative role, quite an interresting subject (magnetic flux tube keep their topology frozen). Resistive MHD is the same when the effect of resistivity is taken into account (flux tubes can disappear or coalesce). The Vlasov approximation also neglects dissipation but accounts for details of the particle phase-space distribution, it is essential to study some field-particles resonnances, like the famous "Landau damping". See the books for more.

Nothing similar could pop up in quantum mechanics unless -again- a macroscopic system is studied (quantum plasma physics, semiconductors, ...). At the microscopic level, the charge distribution in QM is the density probability, something totally different (an electron cannot collide with itself!).

In summary, the classical charge density is something totally different from the QM probability density.
It is a macroscopic view on a large quantity of individual charged particles.


Michel


Postscriptum:

I translated your question by: "I know how it is in QM thanks to groups theory, but how is it in classical mechanics". But I did not understand your QM approach, could you explain it further?

Reading here on PF, I am daily surprised by how abstract QM (groups theory, QFT, ...) seems to become always more popular than classical physics, not to mention thermodynamics or dynamo and thermic motors !

The advantage may be that old difficulties with quantum physics may simply be forgotten, maybe forever. Because these difficulties were essentially due to teaching classical before quantum.

 

[pmb_phy]

Most (or all) solvable EM problems are either (1) given a fixed static or periodically-varying EM field, find the motions of charged particles, OR (2) given a fixed static or periodically-varying charge distribution, find the resulting EM field.

That is, either (1) solve the Lorentz force equation or (2) Maxwell's equations. But what does the complete problem (1) + (2) look like? Oh, I know, I know, its not solvable. I'd just like to see the differential equations once.

You can see Maxwell's equations at

http://www.geocities.com/physics_world/em/mawell_eq.htm

Notice how these equations are given in terms of densities, i.e. current density and charge density.

In tensor form Maxwell's equations are expressed as

$$\partial^{\alpha}F_{\alpha\beta}= \frac{4\pi}{c}J_{\alpha}$$


The most general form of Maxwell's equations are the Proca equations. In tensor form they are

$$\partial^{\alpha}F_{\alpha\beta} + {\mu}^2A_{\alpha} = \frac{4\pi}{c}J_{\alpha}$$

where $\mu$ is proportional to the photon's proper mass. Setting this to zero gives the equations in that link above.

The problem is that all my references give Maxwell's equations in terms of a continuous charge distribution ($\rho$ and J) while the Lorentz force is given for point particles. If we stick to a continuous charge distribution, what I need then, I guess, is the Maxwell equations (got 'em) + the continuity equation (got it) + plus the continuous version of the Lorentz force in terms of $\rho$ and J and depending on E and B (need it).

The Lorentz force density, in tensor form, relates the 4-force to E, B, $\rho$ and J. They are

$$f^{\alpha} = \frac{1}{c}F^{\beta\lambda}J_{\lambda}$$ = $(\frac{1}{c}$J*E, $\rho$E + $\frac{1}{c}$JxB)

Pete

 

[lalbatros]

Pete,

The equation that you state are well-known, except for the exotic photon proper mass. However, you still need to indicate how the Lorentz force acts on the charge densities and currents.

This link, I think, cannot be established without explicit reference to the charged particles and their law of motion.

For example, a current in an wire cannot be seen as a motion of "the" charge density, but instead it is due to the motion of the electronic density, while the total charge density remains zero. You need to consider (at least) ion and electron charge densities separately. In general you can have many different charged species.

This is to stress that in classical physics (like in QM) there is no evolution equation for total charge density and current.

Beside that, the picture is not very different in QM.
In QM a charge looses its localisation and gets a smooth distribution.
In CM the distribution is a Dirac, that's all. No big deal either.
As an exercice, you might try to replace the equations of motion (on the particles coordinates) by an equation on the density (that will keep it a Dirac, of course).
It may look very familiar. (would it be Vlasov ?)

Michel

 

[pmb_phy]

The equation that you state are well-known, except for the exotic photon proper mass.

The EM equations which include a non-zero proper mass can be found in one of the most popular texts on EM. That text being Classical Electrodynamics, J. D. Jackson. Often here I've seen people post the lower bound on the photon's mass when the topic of "photon mass" arises in this forum. Whether it is well-known will depend on a person's experties as well of the field of physics they are versed in.

However, you still need to indicate how the Lorentz force acts on the charge densities and currents.

The OP wished to know the differential equations for the force/field relations which are the density equivalent of the Lorentz force equation. This is the equation I posted.

This link, I think, cannot be established without explicit reference to the charged particles and their law of motion.

The explicit reference is in the charge density $\rho$ and the current density J. These quantities appear explicitly in the equation I posted.

For example, a current in an wire cannot be seen as a motion of "the" charge density, but instead it is due to the motion of the electronic density, while the total charge density remains zero. You need to consider (at least) ion and electron charge densities separately. In general you can have many different charged species.

The current in a wire is a supperposition of the current due to the positive charges as well as the negative charges. If the charge density is zero in one frame then it need not be zero in another frame (and usually isn't). To obtain these densitities one needs more than the laws of electrodynamics but the law that the equations that govern nature (e.g. laws of electrodyanamics) must be expressible in invariant form, i.e. as tensor equations. To obtain the charge and current densities compared to that in another one simply applies the relavent transformation. The tensors are frame dependant whereas the charge and current densities are frame dependant. The 4-current $J^{\alpha}[/i] = (c\rho, \boldface J)$ is the geometric object which appears in these equations. Only after one chooses a particular observer can one define either charge density or current density. Thus once one has the 4-current expressed in one frame it can be found in all other frames of reference by a mere change in spacetime coordinates.

For an example of where the charge density is zero in one frame but non-zero in another please see the web page I made for this at

http://www.geocities.com/physics_world/em/rotating_magnet.htm

You will notice that while the total charge is zero the charge densities need not be zero.

This is to stress that in classical physics (like in QM) there is no evolution equation for total charge density and current.

I posted the answer to this above, i.e.

$$\partial^{\alpha}F_{\alpha\beta}= \frac{4\pi}{c}J_{\alpha}$$

This may not look like an equation which gives the time evolution of charge and current density, but it is. The equation above is a set of 4 equations. One of them is found by letting $\alpha$ = 0. The partial derivative is then becomes [itex]\partial^{\alpha}[/i] = [itex]\partial^{\0}[/i] which is a time derivative.

Beside that, the picture is not very different in QM.
In QM a charge looses its localisation and gets a smooth distribution.

Actually it is the opposite. In classical EM equations one can deal with either particles or distributions whereas in QM one deals only with particles. In any case I am not versed in quantum electrodynamics and am thus not able to post on that theory so I'm posting only the classical equations.

In CM the distribution is a Dirac, that's all. No big deal either.
As an exercice, you might try to replace the equations of motion (on the particles coordinates) by an equation on the density

I've already done that.

Note: There is another equation but I don't know how to post that using the symbols I know of in Latex.

Pete

 

[lalbatros]

Pete,

I agree with probably everything you said. But I did not understand your comment on the Maxwell's equation :

$$\mathbf{d^*F} = 4\pi\mathbf{J}$$​

This may not look like an equation which gives the time evolution of charge and current density, but it is.

What's the magic I missed?
Do you mean that the Lorentz force on particles can be derived from this equation?
I do not reject the idea, but I would greatly appreciate further explanation or a proof if you confirm, or a reference. For sure, I always tend to identify fields and particles (I mean in electrodynamics!): fields reveal the particles and vice-versa. I find it fascinating that particles could somehow vanish from the fields equations (any comment on that?).

Thanks,

Michel

 

[Rach3]

Pete,

I agree with almost everything you said, except with what I understood from your comment on the Maxwell's equation

$$\mathbf{d^*F} = 4\pi\mathbf{J}$$​

http://en.wikipedia.org/wiki/Hodge_dual

 

[lalbatros]

Rach3,

Because of the way I edit here on the PF, you reacted to my post before I really finished it (maybe it is not yet finished!). My point was not about the algebra or the notations, but about the physics: do the Maxwell's equations imply the Lorentz force?

Regards,

Michel

 

[Stingray]

One generally describes matter by an electromagnetic (4-) current density $J^{a}$ and a stress-energy tensor $T^{ab}$.

The sum of the stress-energy tensors of the matter and electromagnetic field must vanish. One can then show that
[tex]\partial_{b} T^{ab} = - F^{ab} J_{b}[/itex]
This describes how matter couples to an electromagnetic field. The right-hand side could be interpreted as a force density if you like.

The evolution equation for the current is obtained by taking the divergence of the Maxwell equation $\partial_{b} F^{ab}=J^{a}$. The result is just the continuity equation
$$\partial_{a} J^{a} =0$$

This set of equations (plus Maxwell's) isn't fully determinate, but that's just a reflection of the fact that different types of matter move differently. You also need constitutive equations describing the type of matter under consideration to complete the problem (i.e. give an "equation of state").

 

[lalbatros]

Stingray,

Obviously the "motion" of the charge density cannot be derived from the Maxwell's equations. This was quite obvious since the mass of the charged particles (or the mass density) don't appear anywhere.

However there was at some old time this dream to get the mass of the electron from its field energy density. Since then things have been elaborated quite a lot ... and I don't know if the dream has been updated to cover all particles ...

Michel

 

[pmb_phy]

Pete,

I agree with probably everything you said. But I did not understand your comment on the Maxwell's equation :

$$\mathbf{d^*F} = 4\pi\mathbf{J}$$​

What's the magic I missed?

There are three equations which cover EM. Maxwell's equations, the Lorentz force

Equation #1 - the two equations, one for the divergence of E and one for the curl of B are combined into the single covariant equation

$$\partial^{\alpha}F_{\alpha\beta}= \frac{4\pi}{c}J_{\alpha}$$


Equation #2 - the two equations, one for the divergence of B and one for the curl of E are combined into the single covariant equation

$$\partial_{\alpha}G^{\alpha\beta}$$ = 0

where $G^{\alpha\beta}$ is the dual field-strength tensor defined as

$$G^{\alpha\beta} = \frac{1}{2}\epsilon^{\alpha\beta\mu\nu}F_{\mu\nu}$$

Equation #3 - The Lorentz force density, the EM field tensor and the 4-current

$$f^{\alpha} = \frac{1}{c}F^{\beta\lambda}J_{\lambda}$$

To trace the path of a single particle in the resultant EM field one employs the Lorentz force equation. These three equations give you everything that you need to know to solve any problem in EM. That includes the time evolution of source charges and source currents. As far as

do the Maxwell's equations imply the Lorentz force?

the answer is no. The Lorentz force equation must be stated as a separate equation rom Maxwell's equation to find the force on a test particle moving in an EM field. The three equations above give you all you need to know to find the EM field and the given charge and current densities as a function of time and space. The Lorentz force will give you the force on test particle moving in this field. A test charge is defined as that charge which does not significantly alter the field in which the charge is located.

Note: From the questions posted above it appears that one needs to solidify in one's mind what it is we are seeking. As such one needs to distinguish source charges and source currents and the test particles which move in the fields created by these sources.

However there was at some old time this dream to get the mass of the electron from its field energy density.

This is still the case in classical EM. One can calculate the mass of a particle by considering both the bare mass (mass that exists when the charge is zero) and the electromagnetic mass (mass due to charges field due to EM energy). This has been done succesfully in Rorlich's text on EM. This is still the case in Classical EM. Such "particles" have a finite diameter and are referred to as "particles" since one assumes a particle radius which can be neglected.

Pete

 

[pellman]

Thanks, guys. There is a lot for me to digest above.

Pete, I think I almost have everything I need. In your equation $$f^{\alpha} = \frac{1}{c}F^{\beta\lambda}J_{\lambda}$$, what is the left-hand-side? A force density? Relate that to time-derivatives of charge density and current density and I think that's it.

The question of whether we use continuous charge distribution or point charges is almost a matter of taste, since on the scale at which we notice that charges are discrete classical EM fails anyway. I would choose continuous distributions since then we don't run into the infinite self-energies for the field of a point particle.

The Lorentz force equation must be stated as a separate equation rom Maxwell's equation to find the force on a test particle moving in an EM field. ...A test charge is defined as that charge which does not significantly alter the field in which the charge is located.

Hmm. Maybe the Lorentz force density equation by itself is not adequate as I hoped. Since what we are talking about here are the movement of charge distributions which DO significantly alter the field.

lalbatros, I will try put together some LaTex for the quantum version of this for the Klein-Gordon field (the easiest case) soon.

Later edit: btw, I would be happy to have simply a reference that covers this adequately. I don't find it in Jackson's Classical Electrodynamics.

 

[lalbatros]

pellman,

Relate that to time-derivatives of charge density and current density and I think that's it.

That is not possible. This Lorentz force applies to charged particles. Generally, you need to calculate the motion of these charges first, and get currents and densities by summing on the particles. This may eventually lead to an evolution for the current density, but this constitutes then a model for the material considered. Note that you may write the requested equation for each charged species separately. But remember my previous remarks concerning dissipative effect that you need to take also into account and that result from microscopic details. I also like to mention resonant absorption of field energy by particles, the Landau damping, because I find it a wonderful subject and also because it shows you that the full distribution function of the charged particles can also play a role in the evolution of the current densities ... and the fields. This shows clearly that a complete physical picture cannot avoid the details of the density and current "composition", even not only the species but the full distribution function.

As an example, the motion of a thin wire in a magnetic field can indeed be given by an evolution equation for current density. In this case the material model is simply that the magnetic term of the Lorentz force applies to the bulk mass of the wire. The electrons carrying the current are bond to the wire.

An other example I know very well is the magnetized plasma. In this case the individuality of charged species, all sort of ions and electrons cannot be skipped off. The theory of electromatic waves in a magnetized plasma could illustrate that in much detail. Obviously you could get for example some Larmor resonnance on some sort of ions, this would act on the evolution of charge density and currents, in the corresponding frequency domain. There are numerous examples in this field of physics.

You might consider a lot of other examples from physics or chemistry. Consider the hall effect for example or take electrochemistry.

I would be happy to have simply a reference that covers this adequately

Personally, I have two references on this subject, that are very complementary:


Landau & Lifchitz (Fields Theory) is more concise and straight to the point and faster to read, but can be sometimes harder. The "fields" cover electromagnetism as well as gravitation.

Wheeler & al (Gravitation) is long but extremely funny and fascinating to read, specially for the first time. If you are not familiar with modern differential geometry notations, you need to spend some time to learn it, can be funny. It is more difficult to use as a reference book because it is more pedagogy-oriented and for a second reading usually you don't want pedagogy any more. Despite the title "gravitation", it covers electromagnetism too and many other topics, a must have!

The choice depends on the time you make available for reading.

Finally, to repeat myself, plasma physics is essentially dealing with this topic since is deals with free charged particles. My preferred reference is https://www.amazon.com/gp/product/0387126139/?tag=pfamazon01-20. As you can already see from those titles, it is related to your question!


Enjoy!

Michel

 

[pmb_phy]

Thanks, guys. There is a lot for me to digest above.

Pete, I think I almost have everything I need. In your equation $$f^{\alpha} = \frac{1}{c}F^{\beta\lambda}J_{\lambda}$$, what is the left-hand-side? A force density?.

Yes. It is force density.

Relate that to time-derivatives of charge density and current density and I think that's it.

The EM fields are time dependant and this time dependence comes from the first two EM equations I posted. The force density is a function of mass density and 4-acceleration. The 4-acceleration in term tells the charged particles (with finite mass described by the mass density and charge density) how to move. There is no requirement that I know of which demands time derivatives of sources to be given in a differential equation which describes EM. Alot of it is implicit in the equations as I've addressed above and below.

The question of whether we use continuous charge distribution or point charges is almost a matter of taste, since on the scale at which we notice that charges are discrete classical EM fails anyway.

I posted that to describe a test particle moving in a field created by the sources. It was never intended to address charge/current density sources.

I would be happy to have simply a reference that covers this adequately. I don't find it in Jackson's Classical Electrodynamics.

See the section of Jackson which describes Maxwell's equations in covariant form. In Jackson's third edition this section is 11.9 Invariance of Electric Charge; Covariance of Electrodymamics. You'll see how Maxwell's equations are defined. The equations of EM are defined as Maxwell's equations and the Lorentz force equation. Given this set of equations all the information to describe the EM field and the time development of the sources becomes well defined and, along with initial conditions, has a unique solution.

Pete

 

[pmb_phy]

This may eventually lead to an evolution for the current density, but this constitutes then a model for the material considered.

The sources must evolve according to Maxwell's equations and the Lorentz force equation. This can only be omitted when one gives sources which exist independantly of the fields in which they are in.

Pete

 

[lalbatros]

Pete,

Quote:

Originally Posted by lalbatros
This may eventually lead to an evolution for the current density, but this constitutes then a model for the material considered.

The sources must evolve according to Maxwell's equations and the Lorentz force equation. This can only be omitted when one gives sources which exist independantly of the fields in which they are in.

I was thinking to a metal (electronic conductor) or a semiconductor or a resistive plasma far from any resonnance. Take the metal, clearly the current density (and charge) do react directly to the fields. For example by an increase of the current by an emf or even by a displacement of the conductor. Note that these two responses already correspond to different "equation of evolution" for the currents. This was the whole point of my post: current densities are not "autonomous variables", there are usually many more degrees of freedom in materials interacting with em fields and therefore much more physics. I hope I could stimulate some interrest in some of this physics and drift this discussion a little away from the initial question whose answer is simply 'no it works in another way'.

Of course I totally agree that this all reduces ultimately to the Lorentz force and Maxwell's equation.

Michel

 

[pmb_phy]

btw, I would be happy to have simply a reference that covers this adequately. I don't find it in Jackson's Classical Electrodynamics.

I took another look at Jackson and see that section 6.1 Maxwell's Displacement Current; Maxwell Equations. You will note that with regard to Maxwell's equations Jackson states

The set of four equations... known as the Maxwell equationsforms the basics of all classical electromagnetic phenomena. When combined with the Lorentz force equation and Newton's second law of motion, these equations provide the complete desciption of the classical dynamics of interacting charged particles and electrodynamic fields..

This appears to be the textbook statement that you're looking for.

It is useful to know that there must be a mechanism which, in the static case, holds a charge distribution together. E.g. consider a sphere which is uniformly charged. Without something holding the charges together to retain the charge density the charge distribution would not be static. This force is known as the "Poincare stress." An interesting paper on this is

Mass renormalization in classical electrodynamics, Griffiths and Owen, AJP, 51(12), December 1983, pp 1120-1126

..a charged particle requires intervention of some nonmagnetic force to hold it together - as we "turn on" the charge, we must at the same time turn on this so-called "Poincare stress" to keep the particle from falling apart.

Pete

 

[pellman]

lalbatros,

A quantum field theory example looks like the following. Herein h-bar=1 and c=1 and I am not being very careful with the other coeffecients and signs. This is just conceptual.

The free Klein-Gordon field has a Lagrangian (density) like

$$\mathcal{L}_D=\partial^{\mu}\phi^{\dag}\partial_{\mu}\phi-m\phi^{\dag}\phi$$

Actually, this is a Lagrangian for two fields: $$\phi_{1}=\phi+\phi^{\dag}$$ and $$\phi_{2}=\phi-\phi^{\dag}$$. Together they have a conserved "charge" current as if the fields have opposite charge. But without the EM field "turned on", the two fields don't interact.

The free Maxwell Lagrangian is $$\mathcal{L}_{EM}=-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}$$ where $$F^{\alpha\beta}=\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}$$.

To get the interaction we replace $$i\partial_{\mu}\phi$$ with $$\left(i\partial_{\mu}-eA_{\mu}\right)\phi$$ (which is like the familiar p --> p - eA in classical Lagrangians) to get a full Lagrangian

$$\mathcal{L}=\left(\partial_{\mu}-ieA_{\mu}\right)\phi^{\dag} \left(\partial^{\mu}+ieA^{\mu}\right)\phi-m\phi^{\dag}\phi-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}$$

The Euler-Lagrange equations in terms of two fields phi and A give the differential equations describing the evolution of the system.

(The p --> p - eA procedure follows from a deeper consideration of requiring local gauge invariance under SU(1) transformations, btw.)

So now let me ask a more straightforward question, or at least, a question which is more clearly defined.


What is the Lagrangian of a classical system consisting of EM fields and charges? Not a Lagrangian just for the particles in which the potentials are a given but a Lagrangian in which the fields are allowed to vary as well as the charges.

Pete, the trouble I have with the quote from Jackson is that the Lorentz force is for point particles while Maxwell's equations are given in terms of continuous charge distributions. There is a continuous version of the Lorentz force given in a post above but I don't know the continuum analog to Newton's second law.

The Lagrangian method is preferable since it gives the differential equations directly without the intermediary concepts of "force" or "force density".


The Lagrangian was my ultimate goal all along, anyway. I just wanted differential equations so that I could construct a Lagrangian that would yield them. I was curious how it would compare to the quantum Lagrangians.

 

[lalbatros]

pellman,

What is the Lagrangian of a classical system consisting of EM fields and charges? Not a Lagrangian just for the particles in which the potentials are a given but a Lagrangian in which the fields are allowed to vary as well as the charges.

As simple as it looks in Landau-Lifchitz, it is awfull for me to write it in full with this tex stuff.
So I translate the action from L-L to a non-usual style, and I also drop all factors:

S = -path(mc ds) - 4space(A.j dW) - 4space(F.F dW)​

where

path() is an integral over the path of the particle(s)
4space() is a volume integral over 4-dimentional space
ds is a particle path element
dW is a 4-space volume element​

I think you can introduce the current density(ies) and convert the first term to a 4-space integral.
I don't have the result right now.
Note that in Landau-Lifchitz the second integral was already obtained by converting a path integral to a volume integral by introducing the current(s). The initial form of the second term was:

-path(A.dx)

(where dx is the path vector in the 4-space)​

Finally, I guess a 1-1 correspondance with the Lagragian you gave.
Note the second term in your Lagrangian and how it corresponds to the first term here, given the quantum charge density.

Please give us your analysis.


Michel

 

[pmb_phy]

Pete, the trouble I have with the quote from Jackson is that the Lorentz force is for point particles while Maxwell's equations are given in terms of continuous charge distributions. There is a continuous version of the Lorentz force given in a post above but I don't know the continuum analog to Newton's second law.

By Lorentz force Jackson is referring to the force exerted on a charged particle by the EM field. Thiss can be adapted to the case of a continuous charge distribution by a limiting process, the result of which would yield the Lorentz force density.

The Lagrangian method is preferable since it gives the differential equations directly without the intermediary concepts of "force" or "force density".

One can express the Lorentz equation in terms of derivitives as well. One simply replaces the force density by the mass density times the acceleration of the matter which constitutes the charge distribution. By Lagrangian method I assume you're referring to the Lagrangian density function, is that correct? In any case the equations turn out to be identical to each other in the long run. In any case the force is in Lagranges equation anyway. I don't see how you conclude that the Lagrangian method is preferable because the force is not explicitly in Lagrange's equations. They are implicitly in those equations.

Pete

 

[Stingray]

There is a continuous version of the Lorentz force given in a post above but I don't know the continuum analog to Newton's second law.

The analog you want is just what I wrote before: $\partial_{b} T^{ab} = -F^{ab}J_{b}$.

This follows from $\partial_{b} T^{ab}_{(total)}=0$, which may be taken as a fundamental conservation law (it is required by Einstein's equation for example). The stress-energy tensor can also be derived from an action in a standard way, in which case you'll find the same condition on it.

The Lagrangian for a charged fluid is (from Hawking and Ellis p. 70)
$$\mathcal{L} = -\frac{1}{16\pi} F_{ab} F^{ab} -\rho (1+\epsilon) - \frac{1}{2} J^{a} A_{a}$$

 

[pellman]

Cool, Stingray. I will check Hawking and Ellis.

Please give us your analysis.

I will. It will take few days though.

Thanks, all.

 

[lalbatros]

Stingray,

Could you explain the second term from you Lagrangian:

$$\rho (1+\epsilon)$$​

specially the meaning of $$(1+\epsilon)$$ .

For the rest, this also what can be read from the action in Landau-Lifchitz .


Thanks,

Michel

 

[pellman]

What I would like to know is how to get the differential equations from the Lagrangian.

If we apply $$\frac{\partial\mathcal{L}}{\partial\rho}-\partial_{\mu}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\rho)}=0$$ we just get $$(1+\epsilon)+\frac{1}{2}A_0=0$$ which does not make any sense. ($$\rho=J^0$$, right?)

Unless the derivatives of rho are hidden in the 1,2,3 components of J. By the continuity equation, there is a dependence. If $$\partial_{0}\rho+\partial_{k}J^k=0$$ then what is $$\frac{\partial J^k}{\partial(\partial_0\rho)}=$$?

 

[Stingray]

Could you explain the second term from you Lagrangian:

$$\rho (1+\epsilon)$$​

specially the meaning of $$(1+\epsilon)$$ .

$\rho$ is basically the rest mass density. The matter current can be written as $j^{a}:=\rho u^{a}$, which must be conserved: $\partial_{a} j^{a}=0$.

$\epsilon=\epsilon(\rho)$ is a normalized internal potential (or elastic energy density). The pressure is defined to be
$$p:= \rho^{2} \epsilon'(\rho)$$
So specifying the internal potential gives the equation of state for the fluid.

Just to emphasize, $\rho \neq J^{0}$.

Anyway, these are all standard things in continuum mechanics, and I'm not going to be able to explain them very easily in one post. You should probably look at a proper text on the subject for more information. Here are some papers which might also work: http://arxiv.org/abs/gr-qc/0211054"
or
http://arxiv.org/abs/gr-qc/0403073"

 

[samalkhaiat]

what I need then, I guess, is the Maxwell equations (got 'em) + the continuity equation (got it) + plus the continuous version of the Lorentz force in terms of $$\rho$$ and $$J$$ and depending on E and B (need it).

If someone could provide that last piece, I'd appreciate it.

You can write
$$m= \int \mu d^3x$$
$$e= \int \rho d^3x$$

in the Lorentz force

$$\frac{dP_{i}}{dt} = \int d^3x \mu \frac{dv_{i}}{dt}= \int d^3x [ \rho E_{i} + \rho \epsilon_{ijk} v_{j} H_{k}]$$

or

$$\mu \frac{dv_{i}}{dt}= \rho E_{i} +\epsilon_{ijk} J_{j} H_{k}$$

In 4-vector, you have

$$\mu \frac{dV^{\mu}}{ds}= \rho F^{\mu\nu} V_{\nu}$$

or

$$\mu \frac{dV^{\mu}}{dt}= F^{\mu\nu} J_{\nu}$$

This can be turned to conservation statement.

As for the Lagrangian, it is

$$\mathcal{L}= - \mu (1-v^2)^{1/2} -A_{\mu} J^{\mu} -\frac{1}{16\pi} F_{\mu\nu}F^{\mu\nu}$$

regards

sam

 

[pellman]

Thank you very much, Sam. I hope you will be available for further questions in a couple of days after I have digested this much.

 

[pellman]

I have decided to read Landau & Lifchitz Classical Theory of Fields (as someone mentioned above) and hopefully answer the question of this thread while doing so. I hadn't seen that series of books in ten years and had forgotten how good they are.

But in the meantime, what do we do with a Lagrangian like $$\mathcal{L}= - \mu (1-v^2)^{1/2} -A_{\mu} J^{\mu} -\frac{1}{16\pi} F_{\mu\nu}F^{\mu\nu}$$? If we treat each of the $$J^{\mu}$$ as independent, we wind up with

$$\frac{\partial\mathcal{L}}{\partial J^{\alpha}}-\partial_{\mu}\frac{\partial\mathcal{L}}{\partial( \partial_{\mu}J^{\alpha})}=0 \rightarrow A_{\alpha}=0$$

since the the derivatives of J do not appear in the Lagrangian. But that result doesn't make any sense. So we must have $$\frac{\partial J^{\alpha}}{\partial(\partial_{\mu}J^{\beta})}\neq 0$$ for some alpha, beta, mu. How does that work?

 

[lalbatros]

pellman,

It is a great idea to read Landau & Lifchits, it contains an incredible broad range of physics with little-known but important and fundamental subjects all treated in a very rigourous and concise way.

You will not find the answer to your exact problem. However you will quickly remember how these varional principles are developped successively for mechanics, fields, and finally their coupling.

The last part, the coupling of fields and particles is -in fact- what you intend to translate in terms of charge and mass density instead of charge coordinates. Clearly, some tought is needed on how this could be done. But obviously, you need to identify the coordinates that make sense for this development. One simple possibility would be the coordinates of the partciles themselves ... and -naïvely- I could imagine that mapping those coordinates on the coordinates you want to use (densities) would do the job. Honestly, I am very curious to know more about that!

Michel

Poscriptum

Mapping the coordinates of a swarm of particles on the space of density functions does not look as a one-to-one relation: there are probably more possible density functions than possibles swarms of particles (note I am not a mathematician). Maybe this will not be a real problem by assuming some smoothing hypothesis (like going from atoms to fluids mathematically).

Note also that a fluid model might not always be possible. I am thinking for example to a magnetised plasma in a tokamak. At high temperatures, some ions can have cyclotron trajectories that circle largely (not always circles!) and cover regions of different plasma densities and temperature. In this case, it is hard to see how a fluid picture can fit with this physics.

Maybe what you are thinking of is more a classical approach on the border of quantum mechanics: a model where particles have a finite size. If this is your goal, then I think you need to bring additional physics which is basically unknown and that should somehow match with QM on the border of CM and QM. Certainly this is not part of the current classical physics, since this is answered by quantum mechanics. Consider for example that the width of the distribution associated to an electron might be a dynamic variable too, and classical physics has always been silent on that.

Reading L&L could be a source of inspiration. For each variational principle introduced in this book, they explain why the Lagrangian (action) should have the assumed structure. This may guide you to decide about a possible structure for your own model.

 

[lalbatros]

pellman,

When taking the functional derivative of your Lagragian

$$\mathcal{L}= - \mu (1-v^2)^{1/2} -A_{\mu} J^{\mu} -\frac{1}{16\pi} F_{\mu\nu}F^{\mu\nu}$$​

have you tried what happens if you assume the mass and charge $$\mu$$ densities and $$J^{0}$$ are related?
The e/m ratio should provide the relation, unusual in a "fluid theory".

Note also the problem with the speed $$v^{\mu}$$ or $$v^{ }$$. It should also be related to the currents $$J^{\mu}$$ to close the variational principle. Isn't it?

This may go in the right direction, but this may also rise more questions.

Michel

 

[pellman]

Thanks, Michel. I'm going to let this question go for now and keep a lookout for it in my general reading. The surprising thing is that it doesn't have a ready, short answer.

Thanks for the help.

Todd