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--- formulas:lorentz_force_law [2018/03/28 13:16]
jakobadmin [Intuitive]
+++ formulas:lorentz_force_law [2018/04/14 10:27]
aresmarrero [Concrete]
@@ Line 15: / Line 15: @@
 <tabbox Concrete>
+**Derivation**
-<note tip>
+The [[formalisms:lagrangian_formalism|Lagrangian]] for a charge $e$ with mass $m$ in an electromagnetic potential $A$ is
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
+\begin{equation}
-</note>
+\label{eq:Lagrangian-relativistic-EM}
+ L(q,\dot{q}) = m|{\dot{q}}| + eA_i\dot{q}^i
+\end{equation}
+so we can work out the Euler--Lagrange equations:
+\begin{align*}
+ p_i = \frac{\partial L}{\partial \dot{q}^i} &= m\frac{\dot{q}_i}{|{\dot{q}}|} + eA_i\\
+ &= m v_i + e \,A_i
+\end{align*}
+where $v$ is the velocity, which we normalize such that $|v|=1$.  An important observation is that here momentum is no longer simply mass times velocity! Continuing the analysis, we find the force
+\begin{align*}
+ F_i = \frac{\partial L}{\partial q^i} &= \frac{\partial}{\partial q^i}\Bigl(e\,A_j\dot{q}^j\Bigr)\\
+ &= e\frac{\partial A_j}{\partial q^i} \dot{q}^j
+\end{align*}
+So the [[equations:euler_lagrange_equations|Euler-Lagrange equations]] give us (using $A_i=A_j\Bigl(q(t)\Bigr)$:
+\begin{align*}
+ \dot{p} &= F \\
+ \frac{d}{dt}\Bigl(mv_i+eA_i\Bigr) &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j\\
+ m\frac{d v_i}{dt} &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j - e\frac{d A_i}{dt}\\
+ m\frac{d v_i}{dt} &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j
+ - e\frac{\partial A_i}{\partial q^j}\dot{q}^j\\
+ &= e\left(\frac{\partial A_j}{\partial q^i} - \frac{\partial A_i}{\partial q^j}\right)\dot{q}^j .
+\end{align*}
+Here, term in parentheses is $F_{ij}=$ the electromagnetic field, $F=dA$.  Therefore, the equations of motion are
+\begin{equation}
+ m\frac{d v_i}{dt} = e F_{ij}\dot{q}^j,
+\end{equation}
+which we call the Lorentz law.
 <tabbox Abstract>

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