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How can differential equations describing a physical object's true spatial paths (orbits) be derived from the time-independent Jacobi-Maupertuis principle of least action? According to this, it is possible.

The time-independent version of Maupertuis' principle yields (Lanczos 1970, Landau and Lifshitz 1969) corresponding differential equations for the true spatial paths (orbits).

I guess I am looking for equations analogous to the Euler-Lagrange conditions, but for true spatial paths (orbits) instead of true trajectories as a function of time.

This PSE post is relevant, but is concerned with the derivation of equations of motion, whereas here I am concerned with the derivation of true spatial paths (orbits).

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    $\begingroup$ Comment to the question (v1): It seems the question formulation is mixing the following non-standard terminology used in the linked Scholarpedia article: 1. a true path = a parametrized on-shell curve, i.e. a function of time; and 2. a true orbit = an un-parametrized on-shell curve, i.e. not a function of time. $\endgroup$ – Qmechanic Dec 15 '15 at 13:59
  • $\begingroup$ @Qmechanic I have tried to address your comment with an edit. $\endgroup$ – ben Dec 15 '15 at 16:42
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Comments to the question (v4):

  1. Recall first of all, that in the principle of stationary action, the initial and final times, $t_i$ and $t_f$, are fixed; while in the abbreviated action principle, the initial and final times, $t_i$ and $t_f$, are free, but the initial and final positions, $q^j_i$ and $q^j_f$, together with the energy $E$, are fixed.

  2. There exist reparametrization-invariant formalisms. However, let us here just outline an elementary pedestrian approach. This consists in picking one of the position coordinates (which possibly after a relabelling we call) $q^1\equiv \lambda$ to play the role of the curve parameter $\lambda$. The other position coordinates $q^2, \ldots, q^n$ are now viewed as functions of $q^1$.

  3. In other words, we view the curve as a graph. The bad news is that this might only work locally, and we may have to patch solutions together globally.

  4. The good news is that $q^1\equiv \lambda$ is naturally fixed at the initial and final position, $q^1_i$ and $q^1_f$.

  5. Now let there be given one of the Maupertuis-like action functional, say $A$. After some straightforward manipulations, $A$ becomes of the form $$A[q^2, \ldots, q^n]~=~ \int_{q^1_i}^{q^1_f} d\lambda~ f\left(E,\lambda, q, \frac{dq}{d\lambda}\right) .$$ It should be straightforward to see which equations describe the stationary points of $A$!

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  • $\begingroup$ Ok, but it sort of seems like you have just trivially relabeled the time parameter as a space parameter. $\endgroup$ – ben Dec 29 '15 at 15:25

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