What Are the Challenges of Force in Special Relativity?

[sadraj]

Hello

When I had relativity course and I read some books and sources like : Rindler W. Relativity.. special, general, and cosmological (2ed., OUP, 2006)
I found that in these texts authors talk about Newton second law and explain why this law is not Lorentz invarient , but never use this new form of second law in the problems. In example I couldn't find resolving two body problem in those texts.
Clearly I think there is a problem between special relativity and meaning of force which we find in Newton laws. What are these problems ?
Is Lagrangian form of mechanics more elegant when we solve relativistic problems?

Sorry for bad English!

 

[muppet]

Hi sadraj,
The problem might be that SR concerns itself primarily with inertial frames in which Newton's first law holds. For an accelerating object, its own rest frame features "inertial forces", or pseudoforces, that arise as a result of the fact that it itself is accelerating. For example, the coriolis force on Earth has a noticeable effect on things like battleship shells; they appear to experience a lateral force, but in fact the effect is an illusion caused by the Earth's rotation. When you want to address problems such as collisions in SR, the best way is to appeal to conservation of 4-momentum to talk about the initial and final outcomes, and assign all interactions that connect the two into a box marked "there be dragons".
Some people probably think lagrangian mechanics is more elegant full stop The general idea is that everyone must agree on the path which has "least" action, as this dictates the physical motion of the system, so the action is expressed as proportional to the arc length of the path, which is an lorentz invariant quantity. It's a slightly mathsy way of looking at things (the most basic familiarity with the differential geometry of curves helps) but it is quite neat.

 

[sadraj]

Thanks muppet but I couldn't understand you on :

The problem might be that SR concerns itself primarily with inertial frames in which Newton's first law holds.

I think the problem is related to limitation of information propagation's speed. I mean no signal can't travel faster than light.
This causes some problems in Newtonian mechanics principles & they should be changed. For example the meaning of time and simultaneity. Or third law of Newton that is based on simultaneity.
There is another problem. In Newtonian mechanics solving motion equation is equal to solving a second ordered diffrential equation(or some second ordered diffrential eqations less or equal to 3 !) So if you put initial (x,y,z) and initial velocity vector , you can predict what you want about particle at any time. But because of in SR simultaneity depends on frame and is not invarient so F (force) can be a function not just as $$\vec{}r$$ &
$$\vec{}v$$ but function of derivations of $$\vec{}a$$ . This means for predicting next times , initial informations that are needed are not just
$$\vec{}r_0$$ & $$\vec{}v_0$$These facts show that in order to predict nature in high speed levels force is not such a useful tool.

 

[javierR]

Force in special relativity needs to be modified because (1) F=dp/dt and (2) momentum is a frame-dependent object. It is a function of the velocity and acceleration in a given Lorentz frame, so it's still a 2nd order differential equation in position. There isn't any more problem using force than momentum or energy in special relativity (one caveat is that you can't consider gravity as an external force; you have to go to general relativity).
Forces are generally not used often because, as you might have been thinking, the Lagrangian formulation of mechanics does not use forces explicitly like Newton's formulation does. Instead, it's based on energies and Euler-Lagrange equations of motion. This formulation of mechanics is more popular partly because symmetry is such an important concept in modern physics, and the Lagrangian formulation is well-suited for this. Also, quantum mechanics is based on an energy formulation (Hamiltonian mechanics) that is closely related to the Lagrangian formulation. While forces lose their central importance outside of introductory mechanics, you can still frame many results in terms of them.