9.3: The Abbreviated Action

Writing the action in the integral form along this constant energy path, we can trivially do the time integral:

\[ S=\int \sum_{i} p_{i} d q_{i}-E\left(t-t_{1}\right)=S_{0}-E\left(t-t_{1}\right) \]

Therefore, from the result \( \delta S=-E \delta t \) it necessarily follows that

\[ \delta S_{0}=\delta \int \sum_{i} p_{i} d q_{i}=0 \]

\( S_{0} \) is called the abbreviated action: this is Maupertuis’ principle.

The abbreviated action for the physical path is the minimum among all paths satisfying energy conservation with total energy E and passing through the designated final point—we don’t care when. Note that not all values of E will work—for example, if we start putting the ball from a low point in the green, we’ll need to give it enough energy to get out of the hollow to begin with. But there will be valid physical paths for a wide range of energy values, since the final arrival time is flexible.

Naturally, since this is a path through configuration space, to evaluate the abbreviated action

\[ S_{0}=\int \sum_{i} p_{i} d q_{i} \]

it must be expressed in terms of the q's. For the usual Lagrangian \( L=T\left(q_{i}, \dot{q}_{i}\right)-V\left(q_{i}\right) \), with \( T=\frac{1}{2} \sum a_{i k}(q) \dot{q}_{i} \dot{q}_{k} \), and momenta

\[ p_{i}==\sum a_{i k}(q) \dot{q}_{k} \]

we find the abbreviated action

\[ S_{0}=\int \sum_{i} p_{i} d q_{i}=\int \sum_{i} a_{i k} d q_{i} d q_{k} / d t \]

This is indeed an integral along a path in configuration space, but we need to get rid of the dt. Physically, we can see how to do this—since we know the total energy E, the kinetic energy at a point is \( E-V\left(q_{i}\right) \) so that determines the speed, hence the time \( d t \text { to move by } d q_{i} \).

That is, (following Landau)

\[ E-V(q)=\frac{1}{2} \sum a_{i k}(q) \dot{q}_{i} \dot{q}_{k}=\frac{\sum a_{i k}(q) d q_{i} d q_{k}}{2(d t)^{2}} \]

from which

\[ d t=\sqrt{\sum \frac{a_{i k} d q_{i} d q_{k}}{2(E-V)}} \]

(This doesn’t look like a very healthy mathematical object, but as you’ll see, it’s fine.)

Hence

\[ S_{0}=\int \sum_{i} p_{i} d q_{i}=\int \sum_{i} a_{i k} d q_{i} d q_{k} / d t=\int \sqrt{\left[2(E-V) \sum a_{i k} d q_{i} d q_{k}\right]} \]

To take a very simple case; if there is no potential, and just a free particle, \( a_{i k}=m \delta_{i k} \) this is nothing but the length of the path multiplied by \( \sqrt{2 m E} \), minimized by a straight line between the two points.

If we have a particle of mass m in a spatially varying potential \( V(\vec{r}) \), the abbreviated action reduces to

\[ S_{0}=\int \sqrt{2 m(E-V)} d \ell \]

where \( d \ell \) is an element of path length. (This is obvious, really—the square root is the absolute value of the momentum, and the momentum vector, of course, points along the path.)

The matrix \(a_{i k}\) sometimes called the mass matrix, is evidently a metric, a measure in the configuration space, by which the “length” of the paths, and particularly the minimum action path, are measured.

Exercise \(\PageIndex{1}\)

Use Maupertuis’ principle to find the path of a cannonball, energy E, fired at a target which is x meters distant horizontally, both cannon and target being at sea level (think ships!).