^{1,a)}

### Abstract

Why is the integrated difference of the kinetic and potential energies the quantity to be minimized in Hamilton’s principle? I use simple arguments to convert the problem of finding the path of a particle connecting two points to that of finding the minimum potential energy of a string. The mapping implies that the configuration of a nonstretchable string of variable tension corresponds to the spatial path dictated by the principle of least action; that of a stretchable string in space–time is the one dictated by Hamilton’s principle. This correspondence provides the answer to the question.

I wish to thank Anthony Bloch, Roberto Rojo, and Arturo López Dávalos for conversations, Alejandro García, Paul Berman, and an anonymous referee for valuable corrections, and Estela Asís for her help with the translation from the Latin of Ref. 14. This work is partially supported by Research Corporation, Cotrell College Science Award.

I. INTRODUCTION

II. THE LEAST ACTION PRINCIPLE AND NONSTRETCHABLE STRINGS

III. HAMILTON’S PRINCIPLE AND STRETCHABLE STRINGS

IV. HAMILTON’S PRINCIPLE USING ELEMENTARY CALCULUS

V. COULD HAMILTON HAVE DISCOVERED QUANTUM MECHANICS?

### Key Topics

- Wave equations
- 7.0
- Bernoulli's principle
- 5.0
- Geometrical optics
- 3.0
- Particle velocity
- 3.0
- Calculus
- 2.0

## Figures

(John Bernoulli’s proof of Snel’s law using the mechanical equilibrium of a string under tension.) Two weights and hang from frictionless pulleys at and , meaning that the tension on the portions and is and . The point of contact, , slides without friction along . The potential energy of the system, , is minimized at equilibrium, where the horizontal components of the tensions and cancel, giving Snel’s law: . (Figure reproduced from Ref. 14.)

(John Bernoulli’s proof of Snel’s law using the mechanical equilibrium of a string under tension.) Two weights and hang from frictionless pulleys at and , meaning that the tension on the portions and is and . The point of contact, , slides without friction along . The potential energy of the system, , is minimized at equilibrium, where the horizontal components of the tensions and cancel, giving Snel’s law: . (Figure reproduced from Ref. 14.)

An alternative version of Bernoulli’s setup: (a) Two weights hanging from frictionless points and are assigned zero potential energy. (b) The two weights are then lifted, and the ends of the strings are joined at point along the line . The work done is equal to the increase in the potential energy: .

An alternative version of Bernoulli’s setup: (a) Two weights hanging from frictionless points and are assigned zero potential energy. (b) The two weights are then lifted, and the ends of the strings are joined at point along the line . The work done is equal to the increase in the potential energy: .

Frictionless pulleys that can slide in horizontal lines with a string passing through them a sufficient number of times gives the trajectory of the particle if is identified with at each segment. Because the string can only pass through each pulley an integer number of times, the ratios of the velocities are approximated by the ratio of the times the rope passes through each segment.

Frictionless pulleys that can slide in horizontal lines with a string passing through them a sufficient number of times gives the trajectory of the particle if is identified with at each segment. Because the string can only pass through each pulley an integer number of times, the ratios of the velocities are approximated by the ratio of the times the rope passes through each segment.

(a) Space–time trajectory of an otherwise free one-dimensional particle acted on by an impulsive force at . (b) Equivalent equilibrium configuration of two segments of a stretchable string with spring constant and an external force .

(a) Space–time trajectory of an otherwise free one-dimensional particle acted on by an impulsive force at . (b) Equivalent equilibrium configuration of two segments of a stretchable string with spring constant and an external force .

## Tables

Analogies used in the principle of least action between mechanics, geometric optics, and the equilibrium of a nonstretchable string.

Analogies used in the principle of least action between mechanics, geometric optics, and the equilibrium of a nonstretchable string.

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