1. Introduction
“When I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, “you look bored; I want to tell you something interesting. “Then he told me something which I found absolutely fascinating, and have, since then, always found fascinating…the principle of least action” [
1].
Analyzing physics problems often requires one to understand how to apply conditions for static equilibrium in those problems. Usually, in school physics courses as well as in introductory physics courses students mainly learn that there are two conditions which one has to apply in such problems: (a) zero resultant force; and (b) zero resultant torque about any axis. Equipped with these two conditions a student usually applies them to deal with most physics situations in those courses. However, dealing with more advanced topics such as advanced mechanics, quantum mechanics, relativity theory, or particle physics demands one considering the system’s energy, usually under equilibrium constrains rather than taking into account the forces acting on the system’s particles and calculate their torque. Students who are not used to consider the particle’s energies have difficulties in understanding and analyzing those situations when they are needed. We therefore suggest exposing students to getting used to consider energies involved in physics situations already in high-school and introductory courses at colleges and universities.
In addition, students’ traditional experiences with the concept of energy leads many to view energy as traveling through machines and wires and changing appearances at different points—what Duit calls a quasi-material conception [
2]. We believe that this view is caused because students are usually applying the concept of energy in dynamic situations. Consider energies under equilibrium situations and applying the least potential energy principle may assist students to get better understanding of the energy concept as an abstract idea rather than a traveling material.
Herein we first introduce the principle of least potential energy and second, offer three examples taking from hydrostatic, electricity and mechanics to demonstrate how to apply the least of potential energy principle. All the examples are elementary problems which students are expected to be able to easily apply the Newtonian approach to solve them. Knowing one approach will enable them to concentrate on the idea behind LPEP approach.
2. Description of the Principle of Least Potential Energy
The least potential energy principle states that a physical system subjected to conservative forces (i.e., gravitational force), forces that are given by a gradient of conservative field will have the lowest potential energy at a stable equilibrium. In other words, according to this principle and under those conditions a system of bodies at rest will adopt a configuration that minimizes total potential energy. This means that in equilibrium the gradient of the total system’s potential energy equals to zero. Mathematically, we may write this as:
where
U(
x,
y,
z) is the potential energy, which its derivatives are continuous. Equation (1) can be obtained from the zero-resultant force condition in equilibrium
. It is known that when conservative forces are involved we may write:
Since the total force
F in equilibrium equals zero we get Equation (1). The principle of least potential energy is actually a private case or a static version of the more general least action principle. According to the least action principle the motion of a physical system from time
t1 to time
t2 is such that the following line integral:
where
T is the kinetic energy, is an extremum relative to all other paths. It was shown that the path with the minimal action is the one satisfying Newton’s second law [
3]. It is clear that for stationary system the principle of least action is reduced to the principle of least potential energy since
T = 0 and the potential energy under equilibrium conditions
U is time independent. The least action principle, which is one of the greatest generalizations in all physical science, was first formulated by Maupertuis in 1746 [
4]. This metaphysical idea was further mathematically developed by the works of Euler, Lagrange, Jacobi and finally formulated by Hamilton and known as Hamilton’s principle [
5,
6,
7].
4. Conclusions
The purpose of this essay was to advocate the use of LPEP in equilibrium situations already on first steps of physics courses. Usually, exposing students to energy consideration at school relates to dynamic situations. Typical question is to find the speed of a ball sliding on a frictionless loop-the-loop track at different points. Students, however, do not learn to consider energies in equilibrium situations. In the current essay we suggested to introduce to students the following two aspects while dealing with equilibrium: (A) The forces acting on the system’s particles, meaning applying the Newtonian rules (zero resultant force and zero resultant torque); and (B) the potential energy of the particles in the systems. Our examples show that such treatment has the following advantages: (1) Deep understanding of equilibrium situations—the two aspects are complementary and each contributes to the comprehension of the equilibrium phenomena. Exposing student to only one aspect the student may get only part of the picture; (2) Variety of presentations—the Newtonian approach is more algebraic in nature and students are thus exposed to mathematic Equations. Potential energy, on the other hand, which is a scalar, may be also described by graphs. For instance, the potential energy in the point charges problem was presented by contour map. Often the most effective way to describe, explore and summarize a set of numbers—even a very large set—is to look at a visual display of those numbers. A good representation of data, both graphically and numerically, may enable powerful and sufficient inferences, obviating the use of delicate and subtle statistical constructs [
16]. Indeed, the point charge problem could be solved visually by looking at the contour graph and identifying the area of the lowest potential energy (the greenest color in the graph). In addition, from the graph a student can relate to other questions that he was not directly asked for. For instance, that the potential is a symmetric and that the region of highest potential energy is near the four-point charges; (3) Overcoming misconceptions—considering energy only in dynamic situations as being usually done in the traditional physics class may explain some misconceptions described in the literature such as described by Gilbert and Pope [
17]: (I) energy is obvious activity; (II) energy is a dormant ingredient within objects, released by a trigger; (III) energy is seen as type of fluid transferred in certain processes. We believe that since in equilibrium there is nothing being “transferred” the students might better understand the idea of energy as an abstract concept which one can use to deal with physical phenomena. In addition, one of the rigorous misconceptions is relating to force as being energy. Equilibrium situations such those that were presented herein presented two solutions one is the Newtonian approach; the other is the energy approach in parallel. This might contribute to the students to grasp the relationship between force and energy.
Stability characterization of the system. The LPEP gives us information about the stability of the system by further classifying the extremum points as minimum, maximum or inflections points. Local minima are stable equilibria, local maxima and inflections points are unstable equilibria while finite regions of constant potential are regions of neutral equilibrium. In the long term we observe only stable and neutral equilibria as the unstable ones are sensitive to small perturbations that drive the system towards stable equilibria.
It is our hope that choosing three simple examples, known to most teachers as a framework for explaining how to use the LPEP in equilibrium as well as the rational provided here will convince teachers to introduce it before their students.
This paper provided four examples of how the Least Potential Energy Principle (LPEP) can be applied. The author suggests that the LPEP should be introduced in high schools, as its application could provide students with the deeper understanding of physical phenomena, as well as provide ground for their dealing with advanced physics afterwards. The examples are the following: ‘Fluid static in a piston of small cross-sectional area’, ‘A system of point charges’, ‘Body connected to a spring on an incline’ and ‘Loaded Flywheel’, where for each of them solutions based on both Newton’s laws and LPEP are provided. All are classic physics problems that students deal with in high school. This paper’s contribution is twofold. At first, it contributes to the discussion of restructuring energy education, and, secondly, provides an example of how to better prepare the next generation of scientists, which is especially required in fields like theoretical Physics and Mathematics that are selected by only few in tertiary education.
Although this subject doesn’t learn in most places, understanding of the least potential energy can leads to improve the other field of physics. By understanding very basic unique phrases and simple calculation the student can exposed to new physics that they didn’t learn before. In order to leads this subject to be universal, this program should have developed into new fields that some of them written these days. Houle et al. [
18] and Hrepic et al. [
19] claim that the right way teaching at the high school can avoid losing student at the colleges and the university. This way has to be universality with known pedagogical methods that encourage student to understand the subject. Voice, as an example, teachers avoid to teach it at the high school level since the complexity of mathematics formula. These students when they come to colleges and university, exposed to the complexity of the mathematics formula, some of them leave, some continue with a real understanding but the major part may understand the math, but not the voice subject as part of waves. So, if you teach the student at the high school the qualities aspects when they are young (at least at the beginning), the student come to the universities with the right tools that can govern on the mathematical level with real understanding. This pedagogical aspect has to suitable for all the student all over the world. Of course, each program has the right synonym. Also, here, the least potential energy should have learned in universality way and with different level. Means, at the high school the level should be with lower math level with a lot of phrases and explain in much more basic way. Furthermore, the Universal Design for Learning Principle roles have to assimilate in the least potential energy in the global learning. First, as I mention the level of high school student and universities are different, so, high school student doesn’t have to finish all the program. Second, the teachers have to learn a glossary of key terms at the beginning of the course, unit, or week. This glossary includes links to online resources where students can find definitions of key terms (e.g., subject encyclopedia through the library). Furthermore, all the lecture should be recorded, so the student from the different level could learned from that. After this recorded class, the student can learn from hard copy note book with different level of exercises. The program should to provide short videos that emphasize or highlight relationships between course concepts, especially when introducing new ideas. Of course, all of that should be in different languages.