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For any function $ L(q,\dot{q}) $, the functional $$S = \int_{}{}{L}{\, \mathrm{d}t}$$ is minimised through the Euler-Lagrange equations: $$ \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}} = 0.$$ For the standard Lagrangian $L = \dfrac{1}{2}\dot{q}^2 - V(q)$ this yields the Euler-Lagrange equation: $$\frac{\mathrm{d}^2 q}{\mathrm{d}t^2} = - \frac{\partial V}{\partial q}.$$ In S. Coleman, Fate of the False Vacuum I, it is stated that the above equation also minimises the functional $$\int \sqrt{2(E - V) } \, \mathrm{d}s $$ where $E$ is given as $$ E = \frac{1}{2} \dot{q}^2 + V.$$ Subbing this in, however, I find that the "Lagrangian" reduces to simply $L = |\dot{q}|$ and the Euler-Lagrange equations no longer yield the same equation. Can anyone explain to me why the standard Lagrangian equation can still be used here? I don't quite follow.

EDIT: Here is the passage from the text:

$$\delta \int \mathrm{d}s \sqrt{2(E-V)}=0,\tag{2.8}$$ with fixed end points, are the paths in configuration space traced out by solutions to Euler-Lagrange equations $$\frac{\mathrm{d}^2\vec q}{\mathrm{d}t^2} = - \frac{\partial V}{\partial \vec q},\tag{2.9}$$ with $$\frac{1}{2}\frac{\mathrm{d}\vec q}{\mathrm{d}t}\cdot \frac{\mathrm{d}\vec q}{\mathrm{d}t}+V = E. \tag{2.10}$$

Qmechanic
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Adam P
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1 Answers1

3

Hints:

  1. Eq. (2.8) is the "Jacobi form of the abbreviated action/Maupertuis principle", cf. e.g. Ref. 1.

  2. Note that $\dot{q}$ is hidden inside the measure: $$\mathrm{d}s ~=~\sqrt{(\mathrm{d}q)^2} ~=~\sqrt{\dot{q}^2}\mathrm{d}\lambda, \qquad \dot{q}~:=~\dfrac{\mathrm{d}q}{\mathrm{d}\lambda},$$ where $\lambda$ is the independent variable/parameter.

  3. It is tempting to consider an extended Lagrangian $$ \widetilde{L}~=~\frac{\dot{q}^2}{2e}+(E-V)e, $$ where $e>0$ is an einbein field. If we integrate out/eliminate the $e$ field, we return$^1$ to the square root action (2.8). The stationary action principle (with its Euler-Lagrange equations) corresponds to the gauge $e=1$.

  4. See also this related Phys.SE post.

References:

  1. H. Goldstein, Classical Mechanics; eq. (8.89).

$^1$ This is similar to the relativistic point particle.

Qmechanic
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