The Dynamical Phase

The most important scalar function for a dynamical system is its dynamical phase. This phase is the (value of) the integral, Eq.(1), evaluated for a worldline $x^\alpha(\tau)$ which starts at $x^\alpha(\tau_1)$, which satisfies the Lagrange equations of motion, and which therefore extremizes this integral. Let us designate this extremal value by

$$S=\int\limits^\tau_{\tau_1}L(x,\dot{x})d\tau ~.$$

Figure 3: Five different world lines having the same starting point and terminating at points where the dynamical phase along the world lines has reached the value $S=17.1$. The locus of such termination points of such world lines with common starting point $\tau =0$ forms the isogram of a scalar function $S(x^\alpha ,\tau )$, the dynamical phase (``Hamilton-Jacobi function'') of the Hamiltonian system.

This integral is a function of the worldline's termination point which we take to be $x^\alpha(\tau)$. If there is only one worldlines between $x^\alpha(\tau_1)$ and $x^\alpha(\tau)$, as is usually the case, the $S$ is a single valued function of the termination point. This is depicted in Figure 3. (If there were several such worldlines, then $S$ would be multivalued.) Thus $S$ is a function of the location $\{x^0(\tau),x^1(\tau)$, $x^2(\tau)$, $x^3(\tau)\}$ of that termination point. It also is a function of the parameter $\tau$ which one uses to parametrize the worldline. Thus one has

$$\int\limits^\tau_{\tau_1}L(x,\dot{x})d\tau=S(x^0,x^1,x^2,x^3,\tau)~.$$