formulas:lorentz_force_law
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formulas:lorentz_force_law [2018/04/14 10:24] aresmarrero [Concrete] |
formulas:lorentz_force_law [2018/04/14 10:26] aresmarrero [Concrete] |
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| \begin{equation} | \begin{equation} | ||
| \label{eq:Lagrangian-relativistic-EM} | \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} | \end{equation} | ||
| so we can work out the Euler--Lagrange equations: | so we can work out the Euler--Lagrange equations: | ||
| \begin{align*} | \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 | &= m v_i + e \,A_i | ||
| \end{align*} | \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*} | \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*} | \end{align*} | ||
| So the [[equations:euler_lagrange_equations|Euler-Lagrange equations]] give us (using $A_i=A_j\Bigl(q(t)\Bigr)$: | So the [[equations:euler_lagrange_equations|Euler-Lagrange equations]] give us (using $A_i=A_j\Bigl(q(t)\Bigr)$: | ||
| \begin{align*} | \begin{align*} | ||
| \dot{p} &= F \\ | \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*} | \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 $\fvect{F}_{ij}=$ the electromagnetic field, $F=dA$. Therefore, the equations of motion are | ||
formulas/lorentz_force_law.txt · Last modified: 2018/05/13 09:18 by jakobadmin
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