What is the lagrangian of a free relativistic particle?

[Lior Fa]

Homework Statement

What is the lagrangian of a free reletavistic particle in a electro-magnetic field?
And what are the v(t) equations that come from the Euler-Lagrange equations (given A(x) = B₀/2 crosProduct x)
(B/2 is at z direction)

Homework Equations

The Attempt at a Solution

I've got to: L = mc^2*sqrt(1-(v/c)^2)+q*(A*v-φ)

But don't get to the velocity equations

 

[Let'sthink]

The problem is quite confusing a particle cannot be free if it is in an electromagnetic field because every conceivable particle has electromagnetic attributes.
Kinetic energy, T = m*[1 - {1/(1-(v²/c²))}]*c², where m is rest mass.

 

[kau]

For a free relativistic massive particle ,lagrangian would be $m\int ds$ where ds is the proper time and m is the rest mass..So it is invariant under lorentz transformation. $ds=\sqrt{\eta_{\mu \nu}dx^{\mu}dx^{\nu}}$ So if I parametrize the whole thing with parameter $\tau$ then we have $ds=\sqrt{\eta_{\mu \nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau$ Then we can write $$L=m\int d\tau \sqrt{(\frac{dt}{d\tau})^{2}-(\frac{dx}{d\tau})^{2}}$$ where I have taken mostly negative sign convention...
I think This should be the case with free particle...