Chapter 13 Charged particle in electromagnetic field

A prerequisite for this chapter is the Lagrangian formulation of Maxwell’s equation using gauge potentials.

13.1 Newtonian formulation

Empirical considerations have shown that the force that acts on a charged point particle under the influence of an electromagnetic field is given by the Lorentz force,

\begin{equation} \label {eq:405} \vec F= q\Big (\vec E+\dot {\vec x}\times \vec B\Big ). \end{equation}

For a charged particle that moves at a non relativistic velocity \(v\ll c\), Newton’s second law thus implies

\begin{equation} \label {eq:406} m\ddot {\vec x}=q\Big (\vec E+\dot {\vec x}\times \vec B\Big ). \end{equation}

13.2 Lagrangian formulation

In order to derive the equations of motion (13.2) from a variational principle, one needs to describe the electromagnetic fields in terms of potentials, \(\phi (x^i,t), \vec A(x^i,t)\).

\begin{equation} \label {eq:407} \vec B=\vec \nabla \times \vec A,\quad \vec E=-\partial _t \vec A-\vec \nabla \phi , \end{equation}

the appropriate Lagrangian is

\begin{equation} \label {eq:409} \boxed {L=\frac {1}{2}m \dot {\vec x}^2-q\phi +q\dot {\vec x}\cdot \vec A}. \end{equation}

Indeed, the associated Euler-Lagrange derivatives are

\begin{equation} \begin{split} \label {eq:410} -\frac {\delta L}{\delta x^i}&=\frac {d}{dt}\frac {\partial L}{\partial \dot x^i}-\frac {\partial L}{\partial x^i}=\frac {d}{dt}(m\dot x_i+q A_i)+q\big (\frac {\partial \phi }{\partial x^i}-\dot x^j\frac {\partial A_j}{\partial x^i}\big )\\ &=m\ddot x_i+q\big (\frac {\partial A_i}{\partial x^j}\dot x^j+\frac {\partial A_i}{\partial t}\big )+q\big (\frac {\partial \phi }{\partial x^i}-\dot x^j\frac {\partial A_j}{\partial x^i}\big ). \end {split} \end{equation}

Re-organizing, the associated equations of motion are

\begin{equation} \label {eq:411} m\ddot x_i=q\big (\partial _i A_j-\partial _jA_i\big )\dot x^j-q\big (\partial _i\phi +\partial _tA_i\big )=q\epsilon _{ijk}\epsilon ^{klm}\partial _l A_m\dot x^j-q(\partial _i\phi +\partial _t A_i), \end{equation}

which coincide with (13.2).

13.3 Hamiltonian formulation

The canonical momenta are given by

\begin{equation} \label {eq:412} p_i=\frac {\partial L}{\partial \dot x^i}=m\dot x_i+qA_i \iff \dot x^i=\frac {1}{m}(p^i-q A^i), \end{equation}

while the Hamiltonian is

\begin{equation} \label {eq:413} H=\big [p_i\dot x^i-L\big ]\big |_{\dot x=\dot x(x,p)}=\frac {1}{m}(p^i-q A^i)p_i-\frac {1}{2m}(p^i-q A^i)(p_i-q A_i)+q\phi -q\frac {1}{m}(p^i-qA^i)A_i. \end{equation}

After regrouping terms,

\begin{equation} \label {eq:414} \boxed {H=\frac {1}{2m}(p^i-q A^i)(p_i-q A_i)+q\phi .} \end{equation}

In other words, as compared to the Hamiltonian of a free particle, \(H=\frac {1}{2m}p^i p_i\), the Hamiltonian of a charged particle in an electromagnetic field is obtained by adding the potential \(V=q\phi \) and the so-called rule of “minimal substitution”, \(p^i\to p^i-q A^i\).

Remarks:

(i) In the case of \(N\) particles with coordinates \(x^i_a\), mass \(m_a\) and charge \(q_a\), the Lagrangian becomes

\begin{equation} \label {eq:415} L=\sum _{a=1}^{N}\Big [\frac 12 m_{(a)}\dot x^i_a \dot x_{a i}-q_{(a)} \phi (\vec x_a,t)+q_{(a)}\dot x^i_a A_i(\vec x_a,t)\Big ], \end{equation}

while the Hamiltonian is given by

\begin{equation} \label {eq:416} H=\sum _{a=1}^{N}\Big [\frac {1}{2m_{(a)}}\big [p^i_a -q_{(a)}A^i(\vec x_a,t)\big ]\big [p_{a i} -q_{(a)}A_i(\vec x_a,t)\big ]+q_{(a)}\phi (\vec x_a,t)\Big ]. \end{equation}

One may rewrite the action associated to the Lagrangian (13.10) as \(S=S_P+S_I\), where the action for \(N\) free non-relativistic particles is given in (12.1), while

\begin{equation} \label {eq:interaction} S_I=\sum _{a=1}^{N}\Big [-q_{(a)} \phi (\vec x_a,t)+q_{(a)}\dot x^i_a A_i(\vec x_a,t)\Big ], \end{equation}

which may be re-written as

\begin{equation} \label {eq:418} S_I=\int d^4x A_\mu (x)J^\mu (x),\quad J^0=\sum _{a=1}^N q_{(a)}\delta ^{(3)}(\vec x-\vec x_a(t)),\quad J^i=\sum _{a=1}^N\frac {q_{(a)}}{c}\frac {dx^i}{dt}\delta ^{(3)}(\vec x-\vec x_a(t)), \end{equation}

is the action that describes the interaction of the particles with the electromagnetic field.

(ii) In order to distinguish more clearly the coordinates on which the electromagnetic fields and potentials depend from the position of the particles, it is often useful to change notation and describe the latter by \(z^i_a\) rather than \(x^i_a\).

(iii) In relativistic notation, the current may be written as

\begin{equation} \label {eq:421} J^\mu =\sum _{a=1}^N \int d\tau \frac {q_{(a)}}{c}\frac {dx^\mu _a}{d\tau }\delta ^{(4)}( x^\nu - x^\nu _a(\tau )), \end{equation}

That this expression reduces to that in (13.13) can be seen as follows:

\begin{multline} \label {eq:423} \int d\tau \frac {q_{(a)}}{c}\frac {dx^\mu _a}{d\tau }\delta ^{(4)}( x^\nu - x^\nu _a(\tau ))=\dots =\frac {q_{(a)}}{c}\frac {dx^\mu _a}{d\tau }\delta ^{(3)}(\vec x-\vec x_a) \end{multline} where, in order to show the last equality, one uses the following property of the \(\delta \) function,

\begin{equation} \label {eq:422} \int dx f(x)\delta (g(x))=\frac {f(x_0)}{|g'(x_0)|}, \end{equation}

where \(g(x)\) is a monotonic function that vanishes in \(x_0\), \(g(x_0)=0\).

In this case, one needs to replace the action for the particles \(S_P\) by that appropriate for relativistic ones.

complete by using Felsager exercise 1.4.1 pages 20,28 and equation 1.34 in reverse.