The notes from week 1 of John Baez's course in Lagrangian mechanics give some insight into the motivations for action principles.

The idea is that least action might be considered an extension of the principle of virtual work. When an object is in equilibrium, it takes zero work to make an arbitrary small displacement on it, i. e. the dot product of any small displacement vector and the force is zero (in this case because the force itself is zero).

When an object is accelerating, if we add in an "inertial force" equal to $\,-ma\,$, then a small, arbitrary, time-dependent displacement from the objects true trajectory would again have zero dot product with $\,F-ma,\,$ the true force and inertial force added. This gives

$$(F-ma)\cdot \delta q(t) = 0$$

From there, a few calculations found in the notes lead to the stationary action integral.

Baez discusses D'Alembert more than Hamilton, but either way it's an interesting look into the origins of the idea.

Perhaps a simple example is in order. Consider a simple harmonic oscillator (SHO)

$$ S~=~\int_{t_i}^{t_f} \! dt~L, \qquad
L~=~\frac{m}{2}\dot{x}^2 - \frac{k}{2}x^2, \tag{1}$$

with characteristic frequency

$$ \frac{2\pi}{T}~=~\omega~=~\sqrt{\frac{k}{m}}, \tag{2}$$

and Dirichlet boundary conditions

$$ x(t_i)~=~x_i \quad\text{and}\quad x(t_f)~=~x_f. \tag{3}$$

It can be shown that the classical path is only a minimum for the action (1) if the time period

$$ \Delta t~:=~t_f-t_i~\leq~ \frac{T}{2}\tag{4}$$

is smaller than a characteristic time scale $\frac{T}{2}$ of the problem. (If $\Delta t=\frac{T}{2}$ there is a zeromode.) For $\Delta t>\frac{T}{2}$ the classical path is no longer a minimum for the action (1), but only a saddle point. If we consider bigger and bigger $\Delta t$, a new negative mode/direction develops/appears each time $\Delta t$ crosses a multiple of $\frac{T}{2}$.

It is such examples that Ref. 1. has in mind when saying that the principle of least action is actually a principle of *stationary* action. The above phenomenon is quite general, and related to conjugated points/turning points and Morse theory. In semiclassical expansion of quantum mechanics, this behaviour affects the metaplectic correction/Maslov index. See e.g. Ref. 2 for further details.

A similar phenomenon takes place in geometrical optics, where it is straightforward to construct examples of light paths that do not minimize the time, cf. Fermat's principle of least time.

References:

Landau and Lifshitz, Vol.2, *The Classical Theory of Fields,* p. 24.

W. Dittrich and M. Reuter, *Classical and Quantum Dynamics*, 1992, Chapter 3.

## Best Answer

If you have in mind a mechanical system and if you are really referring to the Hamilton's principle (which tells how to find the actual trajectory the system follows between the

fixedinitial and final times atfixedinitial conditions), then I doubt you can find such an example. The reason is that a mechanical system contains the kinetic term in its lagrangian, which is unbounded from above. That is, you can always construct a trajectory $\{x(t), {\dot x}(t)\}$ with as large action as you wish; so there is no maximum, not even a local one.In his answer above, Mark refers to an optical example that sort of maximizes, not minimizes the action. The example can be reformulated in mechanical form using small balls rolling on a surface instead of light rays, but I don't think it really refers to the Hamilton's principle. In this example you send light rays (or small balls) from a fixed initial point into

differentdirections and search for caustics in balls' trajectories. That is, you vary the initial condition (the direction of initial velocity) and at the same time you assume that each trajectory (for each choice of the velocity) is already known. That's not what the Hamilton's principle is about (there you fix all the initial conditions and solve for the trajectory).