2017-1 A Mathematical Reflection on the Origin of the Laws of Conservation of Energy and Momentum, 050030 Medellin, Colombia ORCID : 0000-0002-7634-7161 doi: 10.13140/RG.2.2.28312.60167 Abstract The purpose of this paper is to demonstrate, from a mathematical point of view, the universal validity of the laws of conservation of energy and momentum. It will also be shown that these conservation laws are a natural consequence of the motion of matter. Finally, the implications of energy and momentum conservation to the collision between two particles are considered, and the validity of the Born-Mayer interaction potential as the reason for collision is discussed. Keywords Conservation of Energy; Conservation of Momentum; Equations of Motion; Particles Collision; Born-Mayer Repulsion 1. The concept of Energy and the Conservation of Mechanical Energy The Merriam-Webster Dictionary [1] defines Energy as a fundamental entity of nature that is transferred between parts of a system in the production of physical change within the system and usually regarded as the capacity for doing work. The term Energy comes from the Greek energeia (meaning action or activity ), which was used by Aristotle to express actuality, reality, existence [2]. In modern physics, Energy is defined as the capacity that an object has for performing work [3]. Since work is related to the motion caused by a force, Energy can basically be considered as a property of material objects related to their capacity to cause motion, a property that can be transferred between different objects. 11/02/2017 Reports 2017-1 (1 / 17)
Even though the concept of Energy is widely known and used in science and in our daily life, it is actually a rather abstract and intangible concept. For gaining a better understanding of the nature of Energy, let us first study the motion of a particle. Let us consider a particle (or a molecule) of mass in space. At time, the particle will be located at position. The position of this particle as a function of time is described by the general nonlinear differentiable function. Although the mathematical form of such general function is unknown, it is possible to use a power series expansion around as follows: Let us know give some names to the derivatives of the position with respect to time: Instantaneous velocity: (1.2) (1.1) Instantaneous acceleration: (1.3) Instantaneous jolt: (1.4) Instantaneous snap: (1.5) Instantaneous crackle: (1.6) Instantaneous pop: (1.7) For all following derivatives of position the name hyperchange will be used 1. The instantaneous -th hyperchange is then: 1 For the seventh and eighth derivatives of position the names lock and drop have been suggested, respectively. 11/02/2017 Reports 2017-1 (2 / 17)
Using these definitions, the position of the particle is now described by: (1.8) Let us now understand some basic properties of this expression. If the position of the particle is described relative to a certain static reference position in space, then the new relative position will be: (1.9) (1.10) Given that the initial position can also be described relative to the static reference position: (1.11) Now, let us assume that the reference position is also in motion ( position as a function of time can also be approximated as: ). Then, the reference (1.12) where the star (*) denotes that it refers to the reference position. 11/02/2017 Reports 2017-1 (3 / 17)
The relative position of the particle, considering a moving reference position becomes: (1.13) The first conclusion here is that, independently of the reference (static or dynamic), the position of the particle can be described by the same mathematical functional form, as long as all derivatives of the position of the particle are relative to the reference position. In the following discussion, the term and all its derivatives refers to the position of the particle under any arbitrary frame of reference. Now, differentiation of the position (Eq. 1.9) with respect to time results in: (1.14) And the following time derivatives can be calculated in the same way: (1.15) 11/02/2017 Reports 2017-1 (4 / 17)
(1.16) (1.17) (1.18) (1.19) This is the full description of the arbitrary motion of the particle as a function of time. (1.20) Given that the position and all its time-derivatives are simultaneously changing, it is possible to find a relation for the behavior of the time-derivatives in terms of the position. This can be done by calculating the position derivative for each time-derivative. Let us consider the case of velocity. Differentiating Eq. (1.14) with respect to position yields: Using the definition of velocity (Eq. 1.2) and rearranging, it is found that: (1.21) (1.22) 11/02/2017 Reports 2017-1 (5 / 17)
and now integrating with respect to : (1.23) (1.24) If both sides of the equality are multiplied by the mass of the particle, then (1.25) The last expression corresponds to the law of mechanical energy conservation.[4] Mass plays a key role in this equation because energy is defined as an extensive property. The term is called kinetic energy of the particle, and the term is called work or potential energy difference, where is the net force acting on the particle. Thus, the change in kinetic energy of the particle is equal to the work exerted on the particle (or to the change in potential energy): or equivalently, (1.26) Similar expressions can be obtained for higher order derivatives of position: (1.27) (1.28) 11/02/2017 Reports 2017-1 (6 / 17)
(1.29) (1.30) (1.31) (1.32) and in general, (1.33) Up to this point, it has been shown that the conservation of mechanical energy is a natural mathematical manifestation of the motion of bodies, as it is the conservation of similar higher order expressions (Eq. 1.28 1.33). Energy can be interpreted thus, as a mathematical abstraction more than as a real tangible entity of nature. And, as a mathematical tautology, the experimental validity of the conservation of energy is thus not surprising. 2. Newton s Third Law of Motion and the Conservation of Momentum Up to this point, we have only considered an individual particle in motion, not interacting with any other particle. Let us now assume that this particle will approach the trajectory of a second particle at some future time. In order to understand the collision, it is important to understand the interaction between the two particles. It is assumed that the work of the interaction between the particles can be described by a function of the distance between the particles at any given moment: 11/02/2017 Reports 2017-1 (7 / 17)
where and are the positions of the particles 1 and 2, respectively, under any arbitrary reference. The instantaneous force acting on particle can be described by (from Eq. 1.26): (2.1) Thus, for each particle we will have: (2.2) (2.3) and therefore: (2.4) which is the mathematical description of Newton s third law of motion. [5] (2.5) Since, then (2.6) or equivalently, (2.7) Integrating Eq. (2.7) with respect to time: 11/02/2017 Reports 2017-1 (8 / 17)
(2.8) where is the integration constant. Now, defining the momentum as the product of the mass and the velocity of a particle, [6] then This means that the total momentum of the system (particles 1 and 2, in this case) remains constant during the collision, based on the only assumption that the interaction potential is a function of the distance between the particles. (2.9) 3. Particles collision: A practical use of Energy and Momentum Conservation Let us consider again the collision between two hard particles with masses and, which at time (before the collision) have velocities and, respectively. During the collision, both momentum and mechanical energy are conserved, as it was shown in the previous sections. Therefore: where and are the corresponding velocities after collision. And (3.1) If the particles are initially moving at constant velocities, and after the collision they reach constant velocities too, then the work of collision can be neglected (conservative systems) and the energy conservation equation becomes: (3.2) From the conservation of momentum (Eq. 3.1) it can be found that: (3.3) 11/02/2017 Reports 2017-1 (9 / 17)
(3.4) or equivalently, (3.5) Replacing this result in the energy conservation equation (Eq. 3.3): (3.6) After some algebra, it is found that: ( ) [ ] Solving the equation, two possible solutions are found: (3.7) ( ) [ ] And after some additional algebra: (3.8) (3.9) One of these solutions ( ) is the initial condition, and it would be the case if no collision occurs. On the other hand, the second solution (when collision occurs) is: 11/02/2017 Reports 2017-1 (10 / 17)
(3.10) Similarly, the velocity of the first particle has two solutions, one is the case of no collision (when ), and the second is: And the relative velocity of particle 1 with respect to particle 2, after the collision is: (3.11) (3.12) That is, the relative velocity of particle 1 with respect to particle 2 after the collision has exactly the same magnitude but in the opposite direction of the relative velocity of particle 1 with respect to particle 2 before the collision, as long as the initial and final velocities of the particles correspond to constant velocities. When both particles have the same mass, are:, then the velocities after the collision (3.13) and (3.14) that is, the velocities between the particles are exchanged. The relation between the relative velocities before and after the collision still holds true: (3.15) Now, if one the particles has a negligible mass compared to the other ( ), then (3.16) and (3.17) 11/02/2017 Reports 2017-1 (11 / 17)
In this case, the massive particle is not affected by the collision. However, the relation between relative velocities still holds true: (3.18) Let us consider now that the initial velocity of the massive particle is. Then, (3.19) (3.20) So in this case, the velocity of the particle with negligible mass after collision is exactly the opposite of its velocity before the collision. The results obtained in this section are of practical importance for understanding the thermal behavior of matter as well as certain thermodynamic laws, as it will be shown in a future work. 4. The reason for collision: The Born-Mayer repulsion Assuming that both particles have a net electric charge of zero, that their dipolar moment is zero and that they are not polarizable, then the only relevant interaction is repulsion. Experimental results have shown that the repulsion potential between two atoms can be represented by the Born-Mayer function [7]: where and are constants, and is the unit direction vector pointing from atom 2 to atom 1, and is given by: (4.1) [ ] corresponds to the interaction energy between the particles when, and is related to the reciprocal of the range of the interaction. It is now assumed that the particles considered present an exponential repulsion potential following the Born-Mayer function. For simplicity, let us now consider the dynamic position of particle 2 as the frame of reference. In this way, the repulsion potential becomes: (4.2) 11/02/2017 Reports 2017-1 (12 / 17)
(4.3) where, and [ ] (4.4) In the following derivation, velocity and acceleration are also relative to particle 2: (4.5) (4.6) Therefore: Considering the Born-Mayer repulsion function, then the acceleration function of particle 1 relative to particle 2 will be: (4.7) (4.8) or equivalently, (4.9) Multiplying both sides of the equation by results in: (4.10) Integrating: 11/02/2017 Reports 2017-1 (13 / 17)
Assuming that at a position the velocity is, then (4.11) Therefore, (4.12) But also notice that from Eq. (4.9): (4.13) So, Eq. (4.13) becomes (4.14) or equivalently, (4.15) and integrating: (4.16) 11/02/2017 Reports 2017-1 (14 / 17)
( ) (4.17) Considering that at time particle 1 was, then (close to the time of the collision), the relative velocity of ( ) (4.18) Resulting in: ( ) ( ) (4.19) Using the exponential function: ( ) (4.20) 11/02/2017 Reports 2017-1 (15 / 17)
And after some algebra: ( ) ( ) (4.21) The resulting expression (Eq. 4.21) corresponds to a sigmoid function. Considering the more general case, then for a long time before the collision, (4.22) and for a long time after the collision, (4.23) Therefore, (4.24) Given that, it can be confirmed again that Eq. (3.15) is valid: (4.25) confirming from a mathematical point of view, that the Born-Mayer repulsion expression is a valid approach for representing the collision between particles. Conclusion It has been shown that the conservation of mechanical energy is a natural mathematical consequence of the motion of bodies (Eq. 1.25). In addition, assuming that the interaction 11/02/2017 Reports 2017-1 (16 / 17)
potential between two particles is a function of their distance, then the momentum is also naturally conserved (Eq. 2.9). During a collision between particles, if the work caused by the interaction can be neglected, then the result of the collision can be easily described as a function of the relative velocities of the particles before the collision (Eq. 3.15). Finally, it has been shown that the Born-Mayer expression (Eq. 4.1) for describing the repulsion interaction potential satisfies the mathematical requirements for a conservative collision between particles. Acknowledgments The author gratefully acknowledges the helpful discussions with Prof. Jaime Aguirre (Universidad Nacional de Colombia). This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Merriam-Webster Online Dictionary, www.merriam-webster.com/dictionary/energy. Accessed on 01/02/2017. [2] Douglas Harper s Online Etymology Dictionary, www.etymonline.com/index.php?term= energy. Accessed on 01/02/2017. [3] Halliday, D., Walker, J., & Resnick, R. (2010). Fundamentals of Physics, Chapter 7. John Wiley & Sons. [4] Halliday, D., Walker, J., & Resnick, R. (2010). Fundamentals of Physics, Chapter 8. John Wiley & Sons. [5] Halliday, D., Walker, J., & Resnick, R. (2010). Fundamentals of Physics, Chapter 3. John Wiley & Sons. [6] Halliday, D., Walker, J., & Resnick, R. (2010). Fundamentals of Physics, Chapter 9. John Wiley & Sons. [7] Abrahamson, A. A. (1969). Born-Mayer-type interatomic potential for neutral ground-state atoms with Z= 2 to Z= 105. Physical Review, 178(1), 76. 11/02/2017 Reports 2017-1 (17 / 17)