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--- formulas:lorentz_force_law [2018/04/14 10:24]
aresmarrero [Concrete]
+++ formulas:lorentz_force_law [2018/05/13 09:18] (current)
jakobadmin ↷ Page moved from equations:lorentz_force_law to formulas:lorentz_force_law
@@ Line 15: / Line 15: @@
 <tabbox Concrete>
-----
 **Derivation**
@@ Line 22: / Line 20: @@
 \begin{equation}
 \label{eq:Lagrangian-relativistic-EM}
- L(q,\dot{q}) = m\norm{\dot{q}} + eA_i\dot{q}^i
+ 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{\pa L}{\pa\dot{q}^i} &= m\frac{\dot{q}_i}{\norm{\dot{q}}} + eA_i\\
+ 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 $\norm{v}=1$.  An important observation is that here momentum is no longer simply mass times velocity! Continuing the analysis, we find the force
+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{\pa L}{\pa q^i} &= \frac{\pa}{\pa q^i}\Bigl(e\,A_j\dot{q}^j\Bigr)\\
+ F_i = \frac{\partial L}{\partial q^i} &= \frac{\partial}{\partial q^i}\Bigl(e\,A_j\dot{q}^j\Bigr)\\
- &= e\frac{\pa A_j}{\pa q^i} \dot{q}^j
+ &= 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{\pa A_j}{\pa q^i}\dot{q}^j\\
+ \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{\pa A_j}{\pa 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{d A_i}{dt}\\
- m\frac{d v_i}{dt} &= e\frac{\pa A_j}{\pa q^i}\dot{q}^j
+ m\frac{d v_i}{dt} &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j
- - e\frac{\pa A_i}{\pa q^j}\dot{q}^j\\
+ - e\frac{\partial A_i}{\partial q^j}\dot{q}^j\\
- &= e\left(\frac{\pa A_j}{\pa q^i} - \frac{\pa A_i}{\pa q^j}\right)\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 $\fvect{F}_{ij}=$ the electromagnetic field, $F=dA$.  Therefore, the equations of motion are
+Here, term in parentheses is $F_{ij}=$ the electromagnetic field, $F=dA$.  Therefore, the equations of motion are
 \begin{equation}
-\boxed{
+ m\frac{d v_i}{dt} = e F_{ij}\dot{q}^j,
- \eqngapabove \gapleft
- m\frac{d v_i}{dt} = e\fvect{F}_{ij}\dot{q}^j, \quad\text{(Lorentz force law)}
- \gapright \eqngapbelow
-}
 \end{equation}
-which we call the Lorentz law.)
+which we call the Lorentz law.
 <tabbox Abstract>

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