Kinetic and Potential Energy

Kinetic energy (KE) is the energy an object possesses due to its motion, defined in classical mechanics as e = (1/2)mv².

Potential energy (PE) is the energy an object holds due to its position, among other factors. The PE equation can vary based on the system and the nature of the potential energy. For a falling object, it is represented as PE = mgh, where g is gravitational acceleration and h is the object's height. For a spring that is compressed or extended, we have PE = (1/2)kx², where k signifies the spring's stiffness and x denotes the displacement from its equilibrium position.

These mathematical definitions of KE and PE apply within the context of various mechanical systems, yet they are not universally applicable. For our discussion, it suffices to understand KE and PE as abstract concepts.

Key points to remember include:

1. KE represents the energy linked to an object's motion.
2. PE signifies the energy associated with an object's position (or its potential to create motion).
3. All energy values are determined by the formula: e = mass * velocity².
4. The total energy (KE + PE) in any system is conserved.

Potential energy, as it is utilized, converts into kinetic energy. Conversely, kinetic energy, such as that in a pendulum as it swings past its equilibrium point and is slowed by gravity, transforms back into potential energy. This illustrates the principle of energy conservation: energy neither disappears nor is created; it merely changes form.

In a typical system, constrained by our arbitrary limits, energy is inevitably lost due to factors like surface friction, air resistance, and heat loss, leading to a continual decrease in the total energy of the system.

But what if we envision an ideal oscillating system, one so efficient that no energy can escape? How would we graphically represent this equilibrium of energy?

Euler's formula offers a fascinating solution.

Graphing Energy with Euler’s Formula

When you input Euler’s formula into graphing tools like Wolfram Alpha, the resulting graph often appears as follows:

In this graph, cosine (blue/real) and sine (red/imaginary) values are layered on the same x-y plane. However, this depiction may not be entirely accurate. Multiplying by the imaginary unit i effectively rotates the value 90 degrees from the real axis to the imaginary axis. A more accurate representation would involve illustrating this third axis, the imaginary dimension, with the sine wave depicted perpendicularly to the cosine wave.

Please excuse my rudimentary illustration; I am not a graphic design professional. Nonetheless, the three-dimensional structure and helical nature of the graph should be clear.

The green wave represents our cosine x, situated within the real (x,y) plane, while the purple wave denotes our i*sine x, placed in the imaginary (x,z) plane.

The correlation to an oscillating system is straightforward. Let Cos(x) denote potential energy, while i*Sin(x) represents kinetic energy, with x symbolizing time.

When we compress a perfect spring and release it (x = 0), it possesses maximum potential energy and no kinetic energy. As this potential energy accelerates the spring, it converts into kinetic energy, which peaks at the equilibrium point (x = ?/2) when the potential energy reaches zero. The momentum carries the spring past this point, creating a counteracting tension that depletes the kinetic energy, reaching its apex just as the spring halts at (x = ?). This negative potential energy propels the spring back toward the center, repeating the process in reverse until it reaches the moment of equilibrium again (x = 3?/2), where the force's direction flips, slowing the spring until it comes to a stop at its initial compressed point, x = 0 or x = 2?, at which point the cycle recommences. In an ideal oscillating system, the distinction between these "beginning" and "ending" moments is negligible.

An observant reader may notice that the sine and cosine values do not yield a consistent total at any given x input. For example, sin(0) + cos(0) = 1, while sin(?/4) + cos(?/4) = 1.414. The sum of KE and PE must remain consistent to comply with the law of conservation of energy, presenting a potential issue. However, this discrepancy arises from incorrectly placing both graphs on the same axis and dissipates when we regard i*sin(x) as situated on the imaginary plane. Instead of simple addition of sin(x) + cos(x), we must conduct vector addition, effectively measuring the distance between the two waves for any given x value.

The red lines illustrate our vector addition. Regardless of direction, the line's magnitude remains constant at 1, as the sum calculation mirrors finding the hypotenuse "a" using the Pythagorean theorem, a² = b² + c². At any x value, the squares of sine and cosine always equal one, thus ensuring that, metaphorically, energy is conserved.

This summary of energy is intriguing because it retains a constant magnitude while also depicting the energy's shifting character in terms of its balance between kinetic and potential energy. In the gif below, we visualize the green bar's length as the total energy at any moment, with its direction indicating the character of that energy.

The Universe as an Ideal Oscillating System

The nature of the universe—whether it is infinite or finite, its shape, and the number of dimensions—remains uncertain. On a macro scale, we can confidently state that galaxies are receding from one another, and their velocity is increasing.

This leads us into the realm of hypotheses and thought experiments. We will have to make broad assumptions and observe their implications.

From the observation that galaxies are accelerating apart, physicists largely agree that the universe is expanding. They posit that matter in the universe was once densely packed together in a singularity, which subsequently exploded in a "Big Bang," leading to the current state of the observable universe.

While we cannot assert that all the universe's energy was once confined to a single point, we can reasonably infer that the visible matter was once more condensed than it is now. The Big Bang serves as an ideal limit, representing reality pushed to its logical extreme. Although history may not have unfolded precisely in this manner, it certainly occurred within the parameters of this logical boundary.

In an ideal oscillating system, the moment of the Big Bang corresponds to our maximally compressed spring or a pendulum held at its highest point. At this hypothetical singularity moment, the universe's kinetic energy would be zero, as nothing is in motion. There would be maximum potential energy, an unfathomably high outward pressure resulting from the superposition of all visible matter. This pressure would instantaneously propel the condensed matter outward, converting potential energy into kinetic energy. The system's expansion continues to accelerate as long as positive potential energy exerts an outward force.

At this stage, we must pose fundamental questions regarding the nature of matter, mass, gravity, energy, and space. When considering matter compressed into a single point, what about space? Does the universe extend beyond this point, like an infinite ocean of empty space? Or is space itself compressed within the singularity? Is the outward movement of galaxies also indicative of space's expansion? What constitutes this "outward force"? What drives this separation?

Many physicists attribute the outward force behind the accelerating expansion of the universe to "dark matter," a term coined due to its invisibility, despite its apparent account for up to 85% of the universe's total mass.

Introducing an entirely new form of matter complicates a relatively simple issue. Since Einstein’s General Relativity, we have understood that space possesses physical properties and mediates gravity. If we accept that the space-time field facilitates attraction between distant objects, it is not surprising that it could also account for their repulsion. The perceived "missing mass" phenomenon may vanish if we consider mass as a function of spatial distortion. Rather than merely affecting space-time curvature, as suggested in General Relativity, mass could be more accurately defined as the curvature itself.

I am, however, getting ahead of myself. Redefining mass necessitates a more extensive discussion. In a forthcoming article, I will delve into this new mass definition, its viability, and the myriad implications it could hold for our understanding of the universe.

For now, whether we attribute this unseen pressure to "dark matter" or the nature of space itself, we can conclude two things:

1. It constitutes roughly 85% of the universe's mass.
2. Energy must be expended to generate the outward force.

Given point #1 and applying it to our Euler’s formula graph, we can crudely estimate our temporal position. If 85% corresponds to potential energy, we could assert, “we are at 85% PE and 15% KE,” and through basic calculations—sqrt(0.85), sqrt(0.15)—we might determine we are at approximately 0.921 + 0.387i, or roughly ?/8 along the x-axis.

Of course, this is overly simplistic. We cannot assume dark energy correlates directly to PE, nor can we rely on the 85% estimate's accuracy, especially as we adjust our mass definition. Nevertheless, it is captivating to consider that with Euler’s formula, we might create a model for rough predictions about the future. By estimating our position x, we could project the timeline until the "equilibrium point" at ?/2, the "Big Crunch" at ?, or the next "Big Bang" at 2?. It is also intriguing to envisage time as circular and space as a form of energy—reality itself can compress, expand, and stretch beyond its equilibrium point, much like a spring.