Action: Forcing Energy to Predict Motion

Abstract

In this paper we use scalar energy, rather than vector force and momentum, to predict how a particle will move. The result is a quantity called action. Action and its relatives undergird Newton's laws and transcend them, also predicting motion in the quantum world and in the curved spacetime of general relativity. An example exhibits action in action.Most introductory physics courses begin with mechanics, as physics itself did historically. We recall everyday experiences with toy wagons, balls, and automobiles and refine descriptions of their motion using vectors: force, momentum, and acceleration. Newton tells us that F = dp/dt. Beyond equations, we learn to represent motion graphically with a worldline, a plot of displacement versus time, which provides a complete description of the path a particle takes through space and time.Energy, which is mathematically simpler than force because it is not a vector, wafts in as a breath of fresh air with forms kinetic K and potential U. But potential energy leads back once again to a vector formulation F = -grad U, telling us that the sharper the incline on which you stand the more difficult it is to resist rolling downhill in the steepest direction.In spite of its awesome power, conservation of energy cannot predict the motion of even a single particle. Why not? Surprise! Because energy is a scalar without direction while displacement of a particle is a vector. Knowing a particle's kinetic energy, we know its speed, and thus the distance ds it will move during the next clock tick dt, but not the direction of that motion, especially in two and three dimensions (Fig. 1).In this paper we force energy to predict how a particle will move. The result is a quantity called action, the invention of a string of geniuses that lived after Newton. To start toward action, think of the simplest possible motion, that of a free particle-a particle subject to no forces. Newton tells us that with respect to an inertial reference frame, a free particle moves in a straight line at constant speed. So choose our space dimension x to lie along the direction of motion of this free particle and plot its worldline (Fig. 2). (Note that the axes are t and x in Fig. 2, not x and y as in Fig. 1.) Constant speed means constant slope of the worldline in the spacetime diagram; that is, a free particle follows a straight worldline. That is what Newton tells us. Predicting Motion Using Kinetic EnergyNow we go over Newton's head and appeal directly to Nature herself, respectfully requesting that she justify the straight worldline of a free particle in terms of our sweet scalar energy. Can she give us an energy reason why the particle follows the straight worldline direct from P to Q in Fig. 2? For example, why doesn't the particle take the alternative worldline PRQ composed of segments A and B? To take this alternative path, the particle would move with higher kinetic energy along the first segment A between P and R, arriving at the final x-value in half the time. Then the particle would relax at rest-with ...

Action physics

“…Background material and further information may be found in the action physics education literature. 6,[9][10][11][12][13][14] As the rationale for action physics includes presenting a unified view of physics, we started our instruction with the foundation, quantum physics. Specifically with the many-paths formulation of quantum physics, introduced by Feynman in 1948.…”

“…A body of literature about implementing action physics instruction has been available for many years, 6,[9][10][11] including in the April 2004 theme issue of this journal, on classical mechanics. [12][13][14][15] This literature was the foundation on which we built our instruction.…”

Developing an action concept inventory

“…One of the first steps is undertaking a literature survey of PER on the inventory's subject. However, as action physics is a relatively new area in PER, this literature is largely confined to content and its delivery [13][14][15][16][17]. There is little discussion of the student misconceptions that might inform the inventory development.…”