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While solving the problem in this question, I found cases where the numerical optimization failed, suspecting unboundedness of the function being minimized. The function approximates the action of the system in question.

I decided that this result could be explained by an unbounded from below action. But I'm still in doubt because it may be my implementation problem.

So, the question is: do there really exist such physical systems with finite number of degrees of freedom, where the action is unbounded from below, given fixed values for $q(t_1)$ and $q(t_2)$? If yes, how can one decide whether a given system is of such type?

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Ruslan
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1 Answers1

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Example: Consider an action functional

$$\tag{1} S[q]~=~\int_{t_i}^{t_f} \! dt ~L, \qquad L~=~\frac{1}{2}m\dot{q}^2-V(q),$$

with Dirichlet boundary conditions (BC)

$$\tag{2} q(t_i)~=~q_i \qquad \text{and}\qquad q(t_f)~=~q_f,$$

where the potential $V$ has a repulsive pole

$$\tag{3} V(q_0)~=~+\infty$$

at $q=q_0$. Then it is possible to choose a virtual $C^1$-curve $\gamma:[t_i,t_f]\to \mathbb{R}$ that satisfies the BC (2) and sits for a while at the pole $q_0$, so that the action functional $$\tag{4} S[\gamma]~=~-\infty$$ is unbounded from below.

In particular, if the exists a unique stationary path $q_{\rm cl}$ [which satisfies the Euler-Lagrange equation and the BC (2)], it cannot minimize the action functional $S$.

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