Lagrangian of a Force Law, with magnetic monopole

In summary, the covariant Lagrangian for the Lorentz force law must be modified in the presence of a magnetic monopole. This additional term includes the Dual Field Tensor, with the magnetic charge 'g', and is added to the original equation of motion. However, finding the correct relation between the potential A and the dual potential a has proven to be difficult, as it is non-linear and non-local. Various attempts have been made, including adding terms using magnetic charge/current/potential, but the desired Lagrangian has not yet been found.
  • #1
praharmitra
311
1
We know the covariant Lagrangian of the Lorentz force law. However, in presence of a magnetic monopole, one must add another term to the force law. This term, of course is the Dual Field Tensor, along with the magnetic charge 'g' as follows -


[tex]m \frac{d^2 x^{\mu}}{d {\tau}^2} = \frac{e}{c}F^{\mu \nu} \frac{d x_{\nu}}{d \tau} + \frac{g}{c}G^{\mu \nu} \frac{d x_{\nu}}{d \tau}[/tex]

Where G is the dual field tensor

Could anyone help me with writing the Lagrangian that gives rise to this equation?
Here is what I have tried till now.

1. Solve the original Lagrangian, get the equation of motion. Add the additional term, of G, and work backwards. However, I have realized that the required relation between A (potential) and a (dual potential, if I can call it that) is non-linear and non-local.

[tex] G^{ \mu \nu } = \partial_{\mu} a_{\nu} - \partial_{\nu} a_{\mu} [/tex]


2. So i tried hit and trial, wrote down, using covariance, symmetry, and dimensions as my guides, several possible Lagrangians, and got the corresponding equation of motion, hoping to give the one that i want.

However, I have been unlucky so far. Have been working on it for several hours.

Can such a Lagrangian be written down explicitly in terms of A and a??
 
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  • #2
Have you tried adding additional terms to the regular Lagrangian which take the same form as the known terms, but using magnetic charge/current/potential instead of their electric counterparts? e.g. there's a term [tex]J^{\mu}A_{\mu}[/tex], so try adding [tex]j^{\mu}a_{\mu}[/tex] where j is magnetic 4-current. That's the first thing I would try.
 
  • #3
no u misunderstood my question. The lagrangian you are talking about will give the Maxwell's Equations. I am not looking for that. I solved that.

What I am looking for is the Lagrangian which will tell me how a magnetic monopole interacts with electric and magnetic fields. I know from texts that the force law for a magnetic monopole is given by the equation as I have written. I want to arrive at a Lagrangian given the force law.

The non-magnetic monopole Lagrangian ( Action) which I am looking to modify is

Integral of [tex] -mc ds - e/c A_{\mu} d x^{\mu}[/tex]

I cannot add terms like magnetic 4-current, because that tells me how charge produces fields.

Also, [tex] a_{\mu} [/tex] hasn't been defined yet. I am yet to calculate a in terms of A, which as I mentioned is non-linear, and non-local. This is what is causing issues

And yes, I am trying to modify th eregular Lagrangian only, but nothing yet :(
 

1. What is the Lagrangian of a Force Law?

The Lagrangian of a Force Law is a mathematical function that describes the dynamics of a system in terms of its position and velocity. It is used in classical mechanics to derive the equations of motion for a system of particles.

2. How is the Lagrangian related to the magnetic monopole?

The Lagrangian of a Force Law with magnetic monopole is a modified version of the standard Lagrangian. It takes into account the presence of a magnetic monopole, which is a hypothetical particle that has only one magnetic pole. The magnetic monopole affects the dynamics of the system, and therefore, it must be included in the Lagrangian.

3. How is the Lagrangian used to calculate the equations of motion?

The Lagrangian is used in a mathematical procedure called the Euler-Lagrange equation to derive the equations of motion for a system. This equation takes into account the kinetic and potential energies of the system, as well as any external forces, to determine the path that the system will follow.

4. Can the Lagrangian be used for systems with multiple magnetic monopoles?

Yes, the Lagrangian can be extended to include multiple magnetic monopoles in a system. This requires adding additional terms to the Lagrangian that account for the interactions between the monopoles. The resulting equations of motion will depend on the positions and velocities of all the monopoles in the system.

5. How does the Lagrangian approach differ from the Newtonian approach?

The Newtonian approach to mechanics involves using Newton's laws of motion to directly calculate the equations of motion for a system. The Lagrangian approach, on the other hand, involves using a mathematical function to derive the equations of motion. This approach can be more efficient for complex systems and allows for the inclusion of constraints and non-conservative forces.

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