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What is the lagrangian of a free relativistic particle?

  1. Nov 7, 2015 #1
    1. The problem statement, all variables and given/known data
    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)
    2. Relevant equations


    3. 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
     
  2. jcsd
  3. Nov 10, 2015 #2
    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.
     
  4. Nov 10, 2015 #3

    kau

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    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....
     
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