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I'm looking at Problem 1.6 in Mathematics for physics: A guided tour for graduate students by Stone & Goldbart, which rederives the equation for the catenary by parametrizing the arc by $x(s)$ and $y(s)$, where $s$ is the arc length, and considering the potential energy functional

$$ U = \int_0^L ds \, \rho g y(s)$$

subject to the constraint $$\dot{x}^2 + \dot{y}^2 = 1$$ everywhere along the chain. I can solve the problem just fine, but I'm confused by the fact that we've introduced a very strange non-holonomic constraint here, where I would have thought that the Lagrange multiplier method fails, cf. e.g. this Phys.SE post. I'm generally very confused by when the Lagrange multiplier method works and when it doesn't when it comes to different kinds of non-holonomic constraints, so any help on that would be appreciated.

Qmechanic
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quenderin
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1 Answers1

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  1. From one perspective, the dot notation to denote a derivative (wrt. arc length $s$) is misleading: it would be better to use a prime, since it is a spatial derivative in a static$^1$ problem in continuum mechanics; not a time derivative.

    Hence the infinitely many constraints (parametrized by the arc length $s$) are independent of (generalized) velocities, and therefore holonomic from a dynamical point of view, as spatial derivatives are allowed in the field-theoretic generalization of point mechanics. Therefore it does not lead to contradictions with Newton's laws a la this Phys.SE post.

  2. An alternative perspective is to view the arc length $s$ as an independent "time" parameter in an abstract theory defined with a non-holonomic constraint. Since the theory doesn't have to obey Newton's laws, it is consistent.

$^1$ Btw, it can be generalized to a dynamical problem by introducing kinetic terms.

Qmechanic
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