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Science Forum Index » Physics - Relativity Forum » Relativistic Dynamics...
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| Dono... |
 Posted: Sat Jul 12, 2008 6:39 pm |
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Sinnce m_0 *d^2x/dt^2=k (constant) is Galilei invariant but not
Lorentz invariant, we have to redefine the impulse in SR as p=
\gamma(v)*m_0*v
We can prove that in the proper frame of the object F :
d(\gamma(v)*v)/dt=\gamma(v)^3*dv/dt
We can also prove (after some computations) that , in a frame F'
moving with constant speed V wrt the frame F:
d(\gamma(v')*v')/dt'=\gamma(v')^3*dv'/dt'
We can further prove that
\gamma(v')^3*dv'/dt'=\gamma(v)^3*dv/dt=k/m_0
This means that by redefining the relativistic impulse as
\gamma(v)*m_0*v , the equations of motion under constant force (F=k)
are Lorentz invariant. So far, so good.
I need some help with the situation when F is NOT constant.
Here are several examples:
1. F= - q*x (common spring)
2. F= -q *sin(theta) (common pendulum)
3. An even nastier case is the case of the torsion pendulum where we
need the relativistic equivalent for the Newtonian p=I*d(theta)/dt
where I is the momentum of inertia . It is not obvious what that
formula would be.
In these particular cases, the fact that the left term of the equation
is invariant (\gamma(v')^3*dv'/dt'=\gamma(v)^3*dv/dt) is of no good,
since the right term is obviously not Lorentz invariant since neither
x, nor theta are Lorentz invariants.
Of course, Hooke law and the pendulum law are laws derived
empirically, so the obvious approach would be to redefine them in
such a fashion that they become Lorentz invariant. Did you see any
literature on this? |
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