1 From Conservation of Energy to the Principle of Least Action: A Story Line 2 3 Jozef Hanca) 4 Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia 5 6 Edwin F. Taylorb) 7 Massachusetts Institute of Technology, Cambridge, MA 02139 8 9 ABSTRACT 10 A schematic story line outlines an introduction to Newtonian mechanics that starts by 11 employing the conservation of energy to predict the motion of a particle in a one-dimensional 12 potential. Incorporating constraints and constants of the motion into the energy expression 13 allows analysis of some more complicated systems. A heuristic transition embeds kinetic and 14 potential energy in the still more powerful principle of least action and Lagrange's equations. 15 16 NOTES TO EDITORS AND REFEREES: 17 1. Preprints of some papers referenced here but not yet published and some papers published 18 elsewhere are available at the website http://www.eftaylor.com/leastaction.html 19 20 2. Comments and suggestions can conveniently be indexed by page and line number in this 21 pdf file. 22 23 I. INTRODUCTION 24 25 It is sometimes claimed that nature has a "purpose", in that it seeks to take the path that produces the 26 minimum action. In fact [according to Feynman's version of quantum mechanics] nature does 27 exactly the opposite. It takes every path, treating them all on [an] equal footing. We simply end up 28 seeing the path with the stationary action, due to the way the quantum mechanical phases add. It 29 would be a harsh requirement, indeed, to demand that nature make a "global" decision (that is, to 30 compare paths that are separated by large distances), and to choose the one with the smallest action. 31 Instead, we see that everything takes place on a "local” scale. Nearby phases simply add and 32 everything works out automatically. 33 1 34 — David Morin 35 2 36 Here is a nine-sentence history of Newtonian mechanics: In the second half of the 1600s 3 37 Newton proposed his three laws of motion, including what has become F = ma. In the mid- 38 1700s Leonhard Euler devised and applied one version of the principle of least action using 39 mostly geometrical methods. On 12 August 1755 the 19-year-old Ludovico de la Grange 40 Tournier of Turin (whom we call Joseph Louis Lagrange) sent Euler a letter with a 41 mathematical attachment that streamlined Euler's methods into algebraic form. "[A]fter 42 seeing Lagrange's work Euler dropped his own method, espoused that of Lagrange, and 4 43 renamed the subject the calculus of variations." Lagrange, in his path-breaking 1788 Analytical 5 44 Mechanics, introduced what we call the Lagrangian function and Lagrange's equations of 6 45 motion. Almost half a century later (1834-35) William Rowan Hamilton developed 1 7 8 1 Hamilton's principle, to which Landau and Lifshitz and Feynman more recently reassigned 9 2 the name principle of least action. Between 1840 and 1860 the conservation of energy was 10 11 3 established in all its generality. In 1918 Emma Noether proved relations between 12 4 symmetries and conserved quantities. Finally, in 1949 Richard Feynman devised the 5 formulation of quantum mechanics that not only explicitly underpins the principle of least 6 action but also demarcates the limits of validity of Newtonian mechanics. 7 13 8 We have proposed that the principle of least action and Lagrange's equations become the 9 basis of introductory Newtonian mechanics. Three recent articles in this Journal demonstrate 10 how to use elementary calculus to derive from the principle of least action (1) Newton's laws 14 15 16 11 of motion, (2) Lagrange's equations, and (3) examples of Noether's theorem. 12 13 How are these concepts and methods to be introduced to students? In the present paper we 14 suggest a reversal of the historical order: Begin with conservation of energy and graduate to 15 the principle of least action followed by Lagrange's equations. 16 17 We start by using the conservation of energy to analyze particle motion in a one-dimensional 18 potential. Much of the power of the principle of least action and its logical offspring 19 Lagrange's equations results from the fact that they are based on energy, a scalar. When we 20 start with the conservation of energy, we not only preview more advanced concepts and 21 procedures but also invoke some of their power. For example, expressions for energy 22 conforming to constraints automatically eliminate corresponding forces of constraint from 23 the analysis of motion. Applying a limited version of Noether's theorem expressed in terms 24 of energy identifies constants of the motion. Using constraints and constants of the motion 25 permits us to reduce to one coordinate the description of some important multi-dimensional 26 systems, whose motion can then be solved completely using the conservation of energy. 27 Equilibrium and statics also derive from the conservation of energy. 28 29 Prerequisites for the proposed introduction include elementary trigonometry, polar 30 coordinates, introductory differential calculus and partial derivatives, and the idea of the 31 integral as a sum of increments. In preparation for later formalism, we symbolize the time 32 derivative with a dot over the variable. 33 34 The story line presented in this article omits most details and is offered for discussion, 35 correction, and elaboration. The authors have no present intent to write an introductory text 36 on Newtonian mechanics that follows the story line of this article. 37 38 II. ENERGY CONSERVATION 39 Begin with a discursive description of the forms of energy and the law of global conservation 40 of energy. Introductory treatments of the conservation of energy and conversion among its 17, 18, 19 41 various forms are available in the literature. How far one goes into engineering 20, 21 42 applications of energy and its environmental consequences is a matter of choice. In 43 addition to expressions for kinetic and potential energy of a particle, this introduction should 44 include angular velocity, moment of inertia, and the kinetic energy of a rigid body rotating 45 about a fixed or parallel-moving axis. 2 1 2 III. ONE-DIMENSIONAL MOTION: ANALYTIC SOLUTIONS 3 Narrow the focus to conservation of mechanical energy of a particle moving in a one- 4 dimensional potential. The complete description of such one-dimensional motion follows 5 from the conservation of energy. Unfortunately, an explicit function of position vs time can be 6 derived for only a small fraction of such systems. In other cases one is encouraged to guess 7 the analytic solution, as illustrated in the following example. Any proposed analytic solution 8 is easily checked by substitution into the energy equation. Heuristic guesses are assisted by 9 the fact that the first time derivative of position, not the second, appears in the energy 10 conservation equation. The following example introduces the potential energy diagram, a 11 central tool in our story line and important in later study of advanced mechanics, quantum 12 mechanics, general relativity, and many other subjects. 13 14 Harmonic oscillator Energy Total energy of particle F A B C DE Potential Kinetic energy energy Position 15 16 Figure 1. The potential energy diagram is central to our treatment. Sample student exercise: For 17 the value of the energy shown, describe the motion of the particle qualitatively but in detail. 18 Compare the magnitudes and directions of each of the following quantities that describe the 19 particle motion at different positions marked A through F: velocity, force, acceleration. In order to 20 describe the motion, what initial condition(s) must be specified in addition to an initial starting 21 point, and are these conditions required for all starting points? How do answers to these 22 questions change for different values of the total energy? 23 24 Qualitative analysis of the parabolic potential energy diagram (or indeed any diagram with 25 motion bounded near a single potential energy minimum) tells us that the motion will be 26 periodic. For the parabolic potential the conservation of energy is written 27 3 1 1 E = mx˙ 2 + kx 2 1 2 2 (1) 2 3 Rearrange Eq. (1) to read: 4 m k x˙ 2 =1− x 2 5 2E 2E (2) 6 7 Recalling the periodicity of motion, we are reminded of trigonometric identity 8 2 2 9 cos θ =1− sin θ (3) 10 11 Set θ = ωt and equate corresponding terms in Eqs. (2) and (3), choosing the positive root, 12 which yields: 2E 1/2 x = sinωt 13 k (4) 14 15 Take the time derivative of this expression and compare its square with the left side of Eq. 22 16 (2). The result tells us that 17 k 1/2 ω = 18 m (5) 19 20 The fact that ω does not depend on the amplitude of the displacement function (4) means that 21 the period is the same whatever the value of the total energy E. The simple harmonic 22 oscillator has wide application because a large number of potential energy curves can be 23 approximated as parabolas near their minima. 24 25 IV. ACCELERATION AND FORCE 26 In the absence of dissipation, force can be defined in terms of energy. In the simple harmonic 27 oscillator example, take the time derivative of both sides of (1) (for constant energy E and 28 mass m). 29 30 0 = mx˙ x˙˙ + kxx˙ (6) 31 32 or 33 k x˙˙ =− x 34 m (7) 35 36 Define force as m times the acceleration.