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Lagrangian and Hamiltonian Formulation of Mechanics.

The action integral for a particle of mass $ m$ and charge $ q$ is

$\displaystyle I$ $\displaystyle =\int^{\tau_2}_{\tau_1}\left\{\frac{m}{2}~\eta_{\alpha\beta} \fra...
...ac{dx^\beta}{d\tau}+ qA_\alpha {(x^\gamma)}\frac{dx^\alpha}{d\tau}\right\}d\tau$    
  $\displaystyle \equiv\int^{\tau_2}_{\tau_1}L(x,\dot{x})d\tau$ (1)

where $ [\eta_{\alpha\beta}]=diag(-1,1,1,1)$ and $ A_\alpha$ is the given electromagnetic vector potential. The integral I is an extremum for those worldlines between $ x^\alpha(\tau_1)$ and $ x^\beta(\tau_2)$ which satisfy the Euler-Lagrange equation

$\displaystyle \frac{d}{d\tau}~\frac{\partial L}{\partial \dot{x}^\alpha}-
\frac{\partial L}{\partial x^\alpha}=0,
$

namely,

$\displaystyle m\eta_{\alpha\beta}~\frac{d^2x^\beta}{d\tau^2}=
q~~\frac{\partial...
...a}{d\tau}-
q~\frac{\partial A_\alpha}{\partial x^\beta}~\frac{dx^\beta}{d\tau}
$

or

$\displaystyle \frac{d^2x^\beta}{d\tau^2}=\frac{q}{m}~F^\beta_{~\alpha}~\frac{dx^\alpha}{d\tau}
~~$with$\displaystyle ~F^\beta_{~\alpha}=
\eta^{\beta \gamma}~\left(\frac{\partial A_\alpha}{\partial x^\gamma}-
\frac{\partial A_\gamma}{\partial x^\alpha}\right),
$

which are the Lorentz equations of motion for the charged particle.

A physically and mathematically more advantageous set of equations is based on the introduction of the momentum variables

$\displaystyle p_\alpha\equiv
\frac{\partial L}{\partial\dot{x}^\alpha}=\eta_{\alpha\beta}~\frac{dx^\beta}{d\tau}m+qA_\alpha
$

and the superhamiltonian

$\displaystyle \mathcal{H}$ $\displaystyle =\dot{x}^\alpha~\frac{\partial L}{\partial\dot{x}^\alpha}-L$    
  $\displaystyle =\frac{1}{2m}~\eta^{\alpha\beta}(p_\alpha-qA_\alpha)(p_\beta-qA_\beta)~.$ (2)

In terms of these one has the two sets of Hamiltonian equations of motion

$\displaystyle \frac{dx^\alpha}{d\tau}$ $\displaystyle =~~\frac{\partial\mathcal{H}}{\partial p_\alpha}= \frac{1}{m}~\eta^{\alpha\beta}(p_\beta-qA_\beta)$ (3)

and


$\displaystyle \frac{dp_\gamma}{d\tau}$ $\displaystyle =-\frac{\partial\mathcal{H}}{\partial x^\gamma}= \frac{q}{m}~\eta^{\alpha\beta}(p_\alpha-qA_\alpha)~\frac{\partial A_\beta}{\partial x^\gamma}.$ (4)

One can readily check that this system of first order equations implies the Euler-Lagrange equations.


next up previous contents
Next: Advantage of the Hamiltonian Up: Laser-driven particle mechanics Previous: Motivational Overview   Contents
Ulrich Gerlach 2005-11-07