The principle of least action in classical mechanics (with the Lagrangian formalism) states:

Theorem. A path $\gamma$ between configurations $q_{(1)}$ at time $t_1$ and $q_{(2)}$ at time $t_2$ is a solution to the Euler-Lagrange equations associated to a Lagrangian $\mathcal L$ if and only if it is a stationary point of the action functional $$I_\mathcal L[\gamma] := \int_{t_1}^{t_2}\mathcal L(\gamma,\gamma',t)\ dt$$ associated with the Lagrangian $\mathcal L$.

When illustrating this principle, people tend to make drawings like the following:

$\qquad\qquad\quad$enter image description here

All of these synchronous paths do go from configuration $q_{(1)}$ at time $t_1$ to configuration $q_{(2)}$ at time $t_1$. However, shoulnd't the self-intersecting ones not be allowed?

If a curve passes through configuration $\bar q$ at two different times $t_a$ and $t_b$, it will do so in general at two different generalized velocities $\dot q_a$ and $\dot q_b$. But the principle does not guarantee the existence (nor the uniqueness) of a curve $\gamma$, and instead assumes its existence by hypothesis: to know that the principle may be applied, i.e. that a solution $\gamma$ of the Euler-Lagrange equations exists and is unique, we need to assume that the two configurations $q_{(1)}$ and $q_{(2)}$ are very (infinitesimally) close in time, so that the usual initial-value existence theorems from the theory of ordinary differential equations may be applied.

This means that, in case the self-intersecting trajectory $\gamma$ were to be a solution in this setting, the generalized velocities $\dot q_a$ and $\dot q_b$ would be achieved within an infinitesimally long span of time, that is, the system would be at configuration $\bar q$ and have two different generalized velocities – a very un-physical situation!

Is my reasoning correct?


Yes, self-intersections are allowed, both for stationary and non-stationary paths. E.g. if the configuration space is topological non-trivial. Consider e.g. a circle $\mathbb{S}^1$. Then the EL equation with pertinent Dirichlet boundary conditions (BC) could have infinitely many solutions (often called instantons in physics jargon) because of different winding sectors.


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