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5521 results in History, Philosophy and Foundations of Physics

Page 30 of 277

  • 2 - Prehistory of Variational Principles

  • Alberto Rojo, Oakland University, Michigan, Anthony Bloch, University of Michigan, Ann Arbor
  • Book:
    The Principle of Least Action
    Published online:
    26 March 2018
    Print publication:
    29 March 2018, pp 6-37

  • Summary

    Queen Dido and the Isoperimetric Problem

    Consider a loop of thread lying on a table. How can we distort the loop, without stretching the thread, so that it encloses the maximum area?

    The problem appears in a story told by the Roman poet, Virgil, in his epic poem, The Aeneid (Virgil, 19 BC):

    They sailed to the place where today you'll see

    Stone walls going higher and the citadel

    Of Carthage, the new town. They bought the land,

    Called Drumskin [Byrsa] from the bargain made, a tract

    They could enclose with one bull's hide.

    These verses refer to the legend of Queen Dido who fled her home because her brother, Pygmalion, had killed her husband and was plotting to steal all her money. She ended up on the north coast of Africa, where she was given permission to rule over whatever area of land she was able to enclose using the hide of only one bull. She cut the hide into thin strips, tying them together to form the longest loop she could make, in order to enclose the largest possible kingdom. Queen Dido seems to have discovered how to use this loop to maximize the area of her kingdom: using straight coastline as her side border, she enclosed the largest area of land possible by placing the loop in the shape of a semi-circle.

    Queen Dido's story is now the emblem of the so-called isoperimetric problem: for a fixed perimeter, determine the shape of the closed, planar curve that encloses the maximum area. The answer is the circle. Aristotle, in De caelo, while discussing the motion of the heavens, displays some knowledge or intuition of this result (Aristotle, 350 BC/1922, Book II):

    Again, if the motion of the heavens is the measure of all movement … and the minimum movement is the swiftest, then, clearly, the movement of the heavens must be the swiftest of all movements. Now of the lines which return upon themselves the line which bounds the circle is the shortest; and that movement is the swiftest which follows the shortest line.

  • 1 - Introduction

  • Alberto Rojo, Oakland University, Michigan, Anthony Bloch, University of Michigan, Ann Arbor
  • Book:
    The Principle of Least Action
    Published online:
    26 March 2018
    Print publication:
    29 March 2018, pp 1-5

  • Summary

    The idea of writing a book on the principle of least action came to us after many conversations over coffee, while we pondered ways of communicating to students the ideas of mechanics with an historical flavor. We chose the principle of least action because we think that its importance and aesthetic value as a unifying idea in physics is not sufficiently emphasized in regular courses. To the general public, even to those interested in science at a popular level, the beautiful notion that the fundamental laws of physics can be expressed as the minimum (or an extremum) of something often seems foreign. Nature loves extremes. Soap films seek to minimize their surface area, and adopt a spherical shape; a large piece of matter tends to maximize the gravitational attraction between its parts, and as a result the planets are also spherical; light rays refracting in a glass window bend and follow the path of least time; the orbits of the planets are those that minimize something called the “action;” and the path that a relativistic particle chooses to follow between two events in space-time is the one that maximizes the time measured by a clock on the particle.

    Our initial intention was to write a popular book, but the project morphed into a more technical presentation. Nevertheless we have tried to keep sophisticated mathematics to a minimum: nothing more than freshman calculus is needed for most of the book, and a good part of the book requires only high school algebra. Some familiarity with differential equations would be useful in certain sections. While the different sections have various levels of difficulty, the book does not need to be read in a linear fashion. It is quite feasible to browse through this book, as most of the chapters and many of the sections are relatively self-contained. Sections and subsections that are a bit more technical and that can easily be omitted on a first reading include 1.1, 2.5, 3.2.2, 3.2.5, 4.3, 5.7, 6.2 to 6.6, 7.7 and 8.8. These are marked in the text with an asterisk.

  • 4 - The Optical-Mechanical Analogy, Part I

  • Alberto Rojo, Oakland University, Michigan, Anthony Bloch, University of Michigan, Ann Arbor
  • Book:
    The Principle of Least Action
    Published online:
    26 March 2018
    Print publication:
    29 March 2018, pp 51-78

  • Summary

    Bernoulli's Challenge and the Brachistochrone

    At the end of the seventeenth century, prominent mathematicians gravitated towards problems formulated in terms of maxima and minima. The interest germinated from a combination (of uncertain proportions) of aesthetic preferences, theological reasons, and the sheer allure of mathematical challenges. The most famous of such problems appears in a letter of June 9, 1696, from John Bernoulli to his friend Gottfried Leibniz: “Given two points A and B in a vertical plane, find the path AMB down which a moveable point M must, by virtue of its weight, fall from A to B in the shortest possible time” (Leibniz, 1962; Orio, 2009). In just a week, on June 16, Leibniz wrote back with the solution, adding that he solved the problem against his will, but that he was attracted to its beauty like Eve before the apple. He expressed his solution in the form of a differential equation, and proposed to name the curve tachystoptota (curve of quickest descent). Bernoulli responded a few days later taking up the biblical reference. He was “very happy about this comparison provided that he was not regarded as the snake that had offered the apple” (Knobloch, 2012). Bernoulli points out that Leibniz's solution corresponds to the cycloid (the curve traced out by a point on the rim of a circle rolling on a flat plane) and proposed to name the curve brachistochrone. The cycloid was admired by Galileo “as a very gracious curve to be adapted to the arches of a bridge” (Drake, 1978, p. 406) and was proven by Huygens (1673) to correspond to the isochronous pendulum. In the meantime, Bernoulli had already communicated the problem as a challenge to Rudolf Christian von Bodenhausen in Florence, in Switzerland to his brother Jacob Bernoulli, and in France to Pierre Varignon. In his response to Leibniz, John Bernoulli mentions two solutions. The first is similar to Leibniz's. The second uses a clever map from the swiftest path to the problem of finding the trajectory of a light ray propagating in a medium of continuously varying index of refraction. This mapping is a hallmark of the optical – mechanical analogy, which had a profound influence in later formulations of mechanics.

  • 7 - Relativity and Least Action

  • Alberto Rojo, Oakland University, Michigan, Anthony Bloch, University of Michigan, Ann Arbor
  • Book:
    The Principle of Least Action
    Published online:
    26 March 2018
    Print publication:
    29 March 2018, pp 162-188

  • Summary

    In 1905, Albert Einstein published a series of papers that constitute an unprecedented display of creativity. The paper Einstein (1905) published in June of this annus mirabilis deals with the theory of relativity, a work that eventually brought Einstein rock-star fame. The first sentence of the essay makes an aesthetic observation: “It is known that Maxwell's electrodynamics, as usually understood at the present time, when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena” (italics added). Einstein discusses the meaning of this asymmetry with an example from the theory of electromagnetism. He observes that a magnet in motion relative to a wire loop produces an electrical current in the wire. According to Maxwell's theory, different equations apply when the magnet moves and the wire is stationary and vice versa. In one case, the magnet is moving with respect to the ether (a universal static substance that acts as the medium for transmission of light) and in the other, the magnet is at rest with respect to the ether. This asymmetry was unacceptable to Einstein. If the current is the same in both cases, then one is looking at the same phenomenon from different perspectives, or from different reference frames, thus making the idea of an ether superfluous. McCullagh's prediction (see the quotation on page 110) of an unexpected, simple and beautiful ether is perhaps fulfilled; non-existence is the ultimate simplicity. The new theory is based on two simple postulates:

  • The laws of physics take the same form for “all reference frames for which the equations of mechanics hold good” (inertial frames).

  • Light always propagates in empty space with a velocity c which is independent of the state of motion of the emitting body.

    • From this starting point, as simple as it is audacious, Einstein leads us through a path of impeccable logic that culminates in the notion that time, represented by the ticking of a wristwatch, is not an absolute phenomenon.

    It is remarkable that the equations that Einstein derives existed before his work. In 1895, the Dutch physicist Hendrik A. Lorentz, in order to explain some experiments by Michelson and Morley wrote a set of equations (identical to Einstein's) in which time appeared as a mathematical variable that depended on velocity and position.

  • 3 - An Excursion to Newton's Principia

  • Alberto Rojo, Oakland University, Michigan, Anthony Bloch, University of Michigan, Ann Arbor
  • Book:
    The Principle of Least Action
    Published online:
    26 March 2018
    Print publication:
    29 March 2018, pp 38-50

  • Summary

    The Principia is one of the greatest books of all time. In it Newton formulates a “System of the World,” incorporating his own new mathematical ideas as well as displaying a masterful command of geometry. “Newton was the greatest genius that ever existed,” the mathematician Joseph-Louis Lagrange is alleged to have said, adding, with a grain of humor, “and the most fortunate, for we cannot find more than once a system of the world to establish.”

    We include a chapter on Newton's laws of motion and the Principia even though this work is not directly related to variational principles. We do so for two reasons: the monumental importance of his work on mechanics and the fact that his ideas are crucial in the development of the principle of least action. Moreover, Newton himself was a proponent of the Aristotelian simplicity and economy (Lyssy, 2015). In his first rule for the study of Natural Philosophy we read: “As the philosophers say: Nature does nothing in vain, and more causes are in vain when fewer suffice. For nature is simple and does not indulge in the luxury of superfluous causes” (Cohen and Whitman, 1999, p. 794).

    Newton's Propositions on the Laws of Motion

    Book I of the Principia starts with three axioms, the proverbial Newton's laws of motion. After some definitions, Newton states (Motte, 1729, p. 83):

    LAW I

    Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

    LAW II

    The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed

    LAW III

    To every action there is always an opposed and equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

    In many derivations of the Principia, Newton uses his second law following a method of impulses. The body moves on straight lines that are broken by the actions of impulses acting at regular intervals of time.

  • 5 - D'Alembert, Lagrange, and the Statics-Dynamics Analogy

  • Alberto Rojo, Oakland University, Michigan, Anthony Bloch, University of Michigan, Ann Arbor
  • Book:
    The Principle of Least Action
    Published online:
    26 March 2018
    Print publication:
    29 March 2018, pp 79-111

  • Summary

    In this chapter we visit mechanics in the Age of Enlightenment. In that period, Newton's ideas, which allowed only the study of motion of bodies free in space, were extended to incorporate constraints in mechanical systems. The key figures are James Bernoulli, Jean le Rond d'Alembert and Joseph-Louis Lagrange. The central concept is the principle of virtual work, which establishes the conditions of static equilibrium and its extension to dynamics.

    The Principle of Virtual Work

    The idea of treating a static equilibrium problem using ideas from dynamics goes back to Aristotle's text “Mechanical Problems.” Although his authorship is disputed, it is probably the product of his contemporaries of the Peripatetic School. In the discussion of the lever – although the word equilibrium is never used – we read “the ratio of the weight moved to the weight moving it is the inverse ratio of the distances from the center” (Aristotle, 350 BC/1955, p. 353). This statement is regarded by many as a precursor to the so-called method of virtual velocities, or virtual displacements (Capecchi, 2012). In “On Mechanics,” one of his early works, (Galileo, 1600/1960) borrows Aristotle's idea and treats equilibrium on an inclined plane as an invariance under hypothetical displacements. In the “Discourses,” he uses notions of statics and dynamics in the same sentence: “when equilibrium (that is, rest) is to prevail between two moveables, their [overall] speeds or their propensities to motion – that is, the spaces they would pass in the same time – must be inverse to their weights [gravità]”(Galilei, 1638/1974, p. 173). Since Galileo talks about the “propensity” to move, and the system is at rest, the velocity refers to a hypothetical motion in a time different from the time of our universe. Galileo realized that, for weights on an inclined plane, the determining factor for equilibrium is their motion away from or “removal from the center of the earth”(Galileo, 1600/1960, p. 177). For the inclined plane with two masses of Figure 5.1, the ratio of the masses is 2. In order for the system to be at equilibrium, the ratio of the vertical velocities (if they were displaced in the same amount of time) is 1/2.

Page 30 of 277