Kinetic energy is the integral of momentum: \( \int mv\;=\; \frac 1 2 mv^2 \) Potential energy is the integral of displacement times a constant: \( \int kx\;=\; \frac 1 2 kx^2 \)

Basically, you're right. But, you're missing the link that makes the equation useful at all: Kinetic energy - Work = Constant. I'll just copy-and-paste a response I made to a thread about perpetual energy machines: *** The force of the \(i^{th}\) particle along the \(x\) axis with mass \(m_i\) is given by: \(m_i\(\frac{d^2x_i}{dt^2}\) = X_i\), which is the derivative of the equation for momentum (this is Newton's second law). Now, we integrate it by multiplying both sides by \(\frac{dx_i}{dt}\): \(m_i\(\frac{d^2x_i}{dt^2}\) \(\frac{dx_i}{dt}\) = \frac{1}{2}\(m_i\)\frac{d}{dt}\(\frac{dx_i}{dt}\)^2 = X_i\frac{dx_i}{dt}\) Therefore, \(\frac{1}{2}m_i\(\frac{dx_i}{dt}\)^2 = \int{X_i}{dx_i} + constant\). Sum over all the particles and you have: (1) Kinetic energy - Work = Constant. For force fields like those from magnets, work only depends on the starting and ending paths involved. That means, regardless of the path taken to get from point A to point B, W will be constant. So that means: (2) Work + Potential Energy = Constant. The constant in (2) is arbitrary since it depends on our starting point for the potential energy. So, combining (1) and (2), we have: Kinetic Energy + Potential Energy = Kinetic Energy - Work = Constant = E, the total energy of the system. In other words... the energy in a system is conserved. A perpetual motion machine would contradict this, and hence it would contradict the second law of motion.

Learn some physics beyond high school. What you've written doesn't apply in relativity, quantum mechanics or any system which has potential energy. Another one of your pointless insights.

I thought you got banned. I guess you served your time and then thought really hard to start a thread that will be interesting and not initially considered frivolous. Am I right? I don't thing anyone is disputing that energy is carried by motion though few would allow such a simple statement when the math is so important and is theory specific, various theories will represent the concept differently I suppose. But in general terms motion and energy clearly belong in the same sentence. Do you have a follow up point in response to my conclusion that no one will really dispute the premise? Is it something about the relationship between energy and motion that you would like to understand? Or are you trying to propose something about energy being carried by motion that you would like to discuss? Out with it.