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Sorry about the endless stream of questions about Lagrangians. I am actually beginning to detest them a bit;p
Anyway, if we have a Lagrangian in three dimensional space:
[tex]L=\frac{1}{2}m\dot{\vec{x}}^{2}+e\vec{A}.\dot{\vec{x}}[/tex]
where [tex]A_{i}=\epsilon_{ijk}B_{j}x_{k}[/tex] and B is just a constant (magnetic field).
The question is to find the equations of motion. To do this we can just use the Euler-Lagrange equations thus:
[tex]\frac{{\partial}L}{{\partial}q^{i}}-\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{q}}^{i}}=0[/tex]
Now,
[tex]\frac{{\partial}L}{{\partial}x^{i}}=0[/tex]
for all i.
So the EOM are given by
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=0[/tex]
Now
[tex]\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\dot{x}^{i}+eA_{i}[/tex]
so d/dt of this should just be:
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\ddot{x}^{i}+e\dot{A}_{i}[/tex]
Since [tex]A_{i}=\epsilon_{ijk}B_{j}x_{k}\Rightarrow\dot{A}_{i}=\epsilon_{ijk}B_{j}\dot{x}_{k}[/tex]
so we should get this:
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\ddot{x}^{i}+e\epsilon_{ijk}B_{j}\dot{x}_{k}[/tex]
but according to the solution to the problem it should be:
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\ddot{x}^{i}+2e\epsilon_{kji}B_{j}\dot{x}_{k}[/tex]
Note the factor of 2.
I posted a similar problem and I was lead to believe that I need to differentiate wrt another superscript - I am not sure if this is at all relavent here since we are differentiating wrt time.
Would be really grateful if someone could offer me some advice...
Thanks..
Anyway, if we have a Lagrangian in three dimensional space:
[tex]L=\frac{1}{2}m\dot{\vec{x}}^{2}+e\vec{A}.\dot{\vec{x}}[/tex]
where [tex]A_{i}=\epsilon_{ijk}B_{j}x_{k}[/tex] and B is just a constant (magnetic field).
The question is to find the equations of motion. To do this we can just use the Euler-Lagrange equations thus:
[tex]\frac{{\partial}L}{{\partial}q^{i}}-\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{q}}^{i}}=0[/tex]
Now,
[tex]\frac{{\partial}L}{{\partial}x^{i}}=0[/tex]
for all i.
So the EOM are given by
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=0[/tex]
Now
[tex]\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\dot{x}^{i}+eA_{i}[/tex]
so d/dt of this should just be:
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\ddot{x}^{i}+e\dot{A}_{i}[/tex]
Since [tex]A_{i}=\epsilon_{ijk}B_{j}x_{k}\Rightarrow\dot{A}_{i}=\epsilon_{ijk}B_{j}\dot{x}_{k}[/tex]
so we should get this:
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\ddot{x}^{i}+e\epsilon_{ijk}B_{j}\dot{x}_{k}[/tex]
but according to the solution to the problem it should be:
[tex]\frac{d}{dt}\frac{{\partial}L}{{\partial}{\dot{x}}^{i}}=m\ddot{x}^{i}+2e\epsilon_{kji}B_{j}\dot{x}_{k}[/tex]
Note the factor of 2.
I posted a similar problem and I was lead to believe that I need to differentiate wrt another superscript - I am not sure if this is at all relavent here since we are differentiating wrt time.
Would be really grateful if someone could offer me some advice...
Thanks..
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