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Lagrangian, particle/magnetic need solution check

  1. Jul 2, 2012 #1
    Preparing for classical prelim, just wondering if this solution is correct.

    1. The problem statement, all variables and given/known data
    A particle with mass m and charge q moves in a uniform magnetic field [itex]\boldsymbol{B}=B\boldsymbol{\hat{z}}[/itex]. Write a Lagrangian describing the motion of the particle in the xy plane that gives the correct Lorentz-force equation of motion,
    [itex]m\mathbf{a}=q\mathbf{v}\times\mathbf{B}[/itex]

    2. Relevant equations
    L=T - U

    [itex]\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)=\frac{\partial L}{\partial q}[/itex]

    [itex]\boldsymbol{B}=\boldsymbol{\nabla \times A}[/itex]

    [itex]\boldsymbol{E}=-\boldsymbol{\nabla}\phi +\frac{\partial \boldsymbol{A}}{\partial t}[/itex]

    3. The attempt at a solution
    I'm going to start with the complete lorentz force and remove the electric potential from the lagrangian later.

    [itex]\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})[/itex]

    [itex]\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times (\boldsymbol{\nabla \times A}))[/itex]

    [itex]\mathbf{F}=q(\mathbf{E}+\boldsymbol{\nabla} (\mathbf{v.A})-\boldsymbol{A}(\boldsymbol{\nabla . v}))[/itex]

    [itex]\mathbf{F}=q(-\boldsymbol{\nabla}\phi+\frac{\partial\boldsymbol{A}}{\partial t}+\boldsymbol{\nabla}(\boldsymbol{v.A})-\boldsymbol{A}(\boldsymbol{\nabla v}))[/itex]

    [itex]U=q(\phi -\boldsymbol{v.A})=q\phi-q\boldsymbol{v.A}[/itex]

    We are only interested in the magnetic field so we'll ignore [itex]q\phi[/itex].

    [itex]U=q\boldsymbol{v.A}[/itex]

    [itex]L=\frac{1}{2}mv^{2} + q\boldsymbol{v.A} \hspace{2 mm}\mathbf{:Answer}[/itex]

    As always thanks for the help!
     
  2. jcsd
  3. Jul 5, 2012 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    This is correct, but you should explain how you got it (i.e that you used a specific vector product rule and that 2 of the terms were zero and why) on your exam if you want full marks.

    Your choice of generalized potential is not immediately obvious when looking at this. How did you choose [itex]U[/itex] from this when there are 2 other terms present that depend on your generalized coordinates, momenta and time? There is a step missing in between which will make the choice of [itex]U[/itex] much easier to argue.

    Did you actually show that this choice of Lagrangian gives the correct force law?

    I only see an incomplete motivation for choosing such a Lagrangian, and no calculations to show that it is indeed a correct choice.
     
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