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Advantage of the Hamiltonian Formulation.

One of the chief virtues of the Lagrangian equations of motion is that they remain invariant under an arbitrary point transformation

$\displaystyle \{x^\alpha\}\rightsquigarrow\{Q^\beta=Q^\beta(x)\}$    

Hamilton's equations of motion not only share this virtue but they take it to a higher level: they are invariant under certain more general transformations

$\displaystyle \{x^\alpha,p_\gamma\}\rightsquigarrow \{Q^\beta=Q^\beta(x,p),P_\beta=P_\beta(x,p\},$ (5)

which is to say,

$\displaystyle \frac{dQ^\beta}{d\tau}$ $\displaystyle =\frac{\partial H}{\partial P_\beta} \textrm{~~~and~~~} \frac{dP_\delta}{d\tau}=-\frac{\partial H}{\partial Q^\delta}~.$ (6)

Here $ H$ is the transformed superhamiltonian obtained from $ \mathcal{H}$ with the help of Eq.(5):

$\displaystyle H(Q,P)=\mathcal{H}\left( x(Q,P),p(Q,P)\right)~.

Termed canonical, such transformations have the distinguishing property that they leave invariant the representation of the antisymmetric tensor

$\displaystyle dx^\alpha\wedge dp_\alpha=dQ^\beta\wedge dP_\beta.$    

A sufficient condition for a transformation, Eq.(5), to be canonical is that there exist a scalar function of $ \{x^\alpha\}$ and $ \{P_\beta\}$,

$\displaystyle S=S(x,P).$    

Indeed, letting

$\displaystyle \frac{\partial S(x,P)}{\partial x^\alpha}\equiv~p_\alpha~~$   and$\displaystyle ~\frac{\partial S(x,P)}{\partial P_\beta}\equiv Q^\beta$ (7)

one finds that

$\displaystyle dx^\alpha\wedge dp_\alpha$ $\displaystyle = dx^\alpha\wedge\frac{\partial}{\partial x^\beta}\left(\frac{\pa...
...ial}{\partial P_\beta}\left(\frac{\partial S}{\partial x^\alpha}\right)dP_\beta$    
  $\displaystyle = dx^\alpha\wedge dx^\beta\frac{\partial^2S}{\partial x^\beta\par...
...ial}{\partial x^\alpha}\left(\frac{\partial S}{\partial P_\beta}\right)dP_\beta$    
  $\displaystyle =$   zero$\displaystyle ~+\frac{\partial}{\partial x^\alpha}\left(\frac{\partial S}{\partial P_\beta}\right)dx^\alpha\wedge dP_\beta$    
  $\displaystyle = \frac{\partial}{\partial P_\alpha}\left(\frac{\partial S}{\part...
...\alpha}\left(\frac{\partial S}{\partial P_\beta}\right)dx^\alpha\wedge dP_\beta$    
  $\displaystyle = d\left(\frac{\partial S}{\partial P_\beta}\right)\wedge dP_\beta$    
  $\displaystyle =dQ^\beta\wedge dP_\beta$ (8)

Problem 1: Prove that the representation invariance of Eq.(8) implies the local existence of a scalar $ S(x,P)$ whose gradients yield Eq.(7).

Thus the existence of a scalar $ S$ is both a necessary and a sufficient condition for the invariance expressed by Eq.(8). It also is a sufficient condition for the invariance of Hamilton's equations of motion.

Problem 2: Show that a transformation such as the one given by Eq.(7) transforms the given euations of motion, Eqs.(3)-(4), into the same form, and given by Eq.(6).

Discussion: Taking advantage of the chain rule, let

$\displaystyle \mathbf{\dot{g}}^\tau_{xp}\equiv \frac{dx^\alpha}{d\tau}
\frac{\partial}{\partial P^\beta}=\mathbf{\dot{g}}^\tau_{QP}

be a vector tangent to the phasespace trajectory $ \mathbf{g}^\tau$ relative to the given ( $ x^\alpha;p_\beta$) and the new ( $ Q^\alpha;P_\beta$) coordinates respectively.

a) Show that


$\displaystyle \langle dx^\alpha \wedge dp_\alpha \vert\mathbf{\dot{g}}^\tau_{xp}\rangle$ $\displaystyle = dp_\alpha \frac{dx^\alpha}{d\tau}-dx^\beta \frac{dp_\beta}{d\tau}\langle dQ^\alpha \wedge dP_\alpha \vert\mathbf{\dot{g}}^\tau_{QP}\rangle$ $\displaystyle = dP_\alpha \frac{dQ^\alpha}{d\tau}-dQ^\beta \frac{dP_\beta}{d\tau}$ (9)

b) Point out why

$\displaystyle \langle dx^\alpha \wedge dp_\alpha \vert\mathbf{\dot{g}}^\tau_{xp...
\langle dQ^\alpha \wedge dP_\alpha \vert\mathbf{\dot{g}}^\tau_{QP}\rangle~.

c) Show that

$\displaystyle \langle dx^\alpha \wedge dp_\alpha \vert\mathbf{\dot{g}}^\tau_{xp}\rangle=

implies Hamilton's equations of motion, Eq.(3)-(4).

d) Show that the introduction of the new coordinates $ \{Q^\alpha,P_\alpha\}$ into $ d\mathcal{H}$,

$\displaystyle d\mathcal{H}(x^\alpha(Q^\gamma,P_\delta),p_\beta(Q^\gamma,P_\delta))

yields Eq.(6), Hamilton's equations relative to the new coordinates $ \{Q^\gamma,P_\delta\}$.

next up previous contents
Next: The Dynamical Phase Up: Laser-driven particle mechanics Previous: Lagrangian and Hamiltonian Formulation   Contents
Ulrich Gerlach 2005-11-07