Configuring Laws of Motion: Static/Dynamic

In summary, the given equations describe a static and dynamic configuration of a system with components A and J. In the static configuration, the equations are modified to account for the absence of a time component. In both configurations, the equations ultimately lead to Maxwell's equations, with the exception of the term 2βAμ in the first equation. This term presents difficulty in connecting the equations to Maxwell's equations.
  • #1
Maniac_XOX
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TL;DR Summary
Is it possible to derive equations for electric field E and magnetic B from the following equation?
$$\Box A_\alpha +\mu^2 A_\alpha = 2\beta A_\mu \partial_\alpha A^\mu + \frac {4\pi}{c} J_\alpha$$

where ##A=(\Phi, \vec A)## and ##J=(\rho, \vec J)##

using a static configuration first where ##α=0##
and then a dynamic one where ##α=i##

knowing that ##E= - \nabla^2 \Phi - \frac {\partial A}{dt}## and ##B= \nabla \times A## and ##-\nabla^2 A + \nabla (\nabla A) = \nabla \times (\nabla \times A)##

I personally tried but because of the ##2\beta A_\mu## term i cannot connect these

My attempts so far:
For static configuration $$- \frac {\partial \Phi}{c^2 \partial t^2} + \nabla^2 \Phi +\mu^2 \Phi = 2\beta A_\mu \frac {\partial A^\mu}{\partial t} + \frac {4\pi}{c} \rho$$
For dynamic configuration $$- \frac {\partial \vec A}{c^2 \partial t^2} + \nabla^2 \vec A +\mu^2 \vec A = 2\beta A_\mu \nabla \vec A + \frac {4\pi}{c} \vec J$$
 
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  • #2
[CORRECTION] ##E=-\nabla \Phi- \frac {\partial A}{\partial t}##
For static configuration $$\nabla^2 \Phi +\mu^2 \Phi = + \frac {4\pi}{c} \rho$$
since in static configuration ##\frac {\partial}{\partial t}=0##

can someone help me derive Maxwell's equations for these equations if it's possible?
 
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FAQ: Configuring Laws of Motion: Static/Dynamic

What is the difference between static and dynamic laws of motion?

The static laws of motion describe the behavior of objects at rest or in a state of constant motion, while the dynamic laws of motion describe the behavior of objects in changing or accelerating motion.

How do the laws of motion affect everyday life?

The laws of motion play a crucial role in our everyday lives, from the way objects move on a daily basis to the design and functioning of machines and vehicles.

What is the significance of Newton's first law of motion?

Newton's first law of motion, also known as the law of inertia, states that an object will remain at rest or in uniform motion unless acted upon by an external force. This law helps us understand the concept of inertia, which is the tendency of an object to resist changes in its motion.

How can the laws of motion be applied in engineering and technology?

The laws of motion are essential in the field of engineering and technology as they help in designing and building structures, machines, and devices that function efficiently and safely. These laws also aid in predicting the behavior of objects in motion, allowing for the development of new technologies.

What are some real-life examples that demonstrate the laws of motion?

Some common examples of the laws of motion in action include a ball rolling down a hill, a car coming to a stop when the brakes are applied, and a rocket launching into space. These examples illustrate the principles of inertia, acceleration, and action and reaction, respectively.

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