Intuitive

• https://gravityandlevity.wordpress.com/2009/04/13/force-and-energy-which-is-more-real/
• https://gravityandlevity.wordpress.com/2010/01/30/the-universe-is-a-giant-energy-minimization-machine/
• https://gravityandlevity.wordpress.com/2009/04/14/the-where-did-the-energy-come-from-game/
• https://gravityandlevity.wordpress.com/2009/05/16/the-equivalence-of-mass-and-energy-the-center-of-energy/

Concrete

Energy is the conserved quantity that we derive using Noether's theorem if our system is symmetric under temporal translations.

In addition, energy is responsible for temporal translations. We say energy generates temporal translations.

The total energy is defined as \begin{equation} E(t) \equiv K(t) + V(q(t)), \end{equation}

where $K$ denotes the kinetic energy and $V$ the potential energy.

Proof the the total energy is conserved

For a system with a conservative force the relationship between force and potential energy is given by $ \nabla V \equiv - F$.

In addition, Newton's second law $F = ma$ implies \[ \begin{split} \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) + \nabla V(q(t))\cdot v(t) \\ &= 0, \qquad\text{(because $F=-\nabla V$)}. \end{split} \]

Kinetic Energy

Kinetic energy is defined as \begin{equation} K(t) \equiv \frac{1}{2}m\,v(t)\cdot v(t). \end{equation} This quantity is useful because \[ \begin{split} \frac{d}{dt}K(t) &= m\,v(t)\cdot a(t) \\ &= F(q(t))\cdot v(t). \end{split} \] We can see here that the kinetic energy goes up whenever we push an object in the direction of its velocity. Moreover, it goes down whenever we push it in the opposite direction.

In addition, we have \[ \begin{split} K(t_1)-K(t_0) &= \int_{t_0}^{t_1} F(q(t))\cdot v(t)\,dt \\ &= \int_{t_0}^{t_1} F(q(t))\cdot \dot{q}(t)\, dt. \end{split} \] This tells us that the change of kinetic energy is equal to the work done by the force. The work is defined as the integral of $F$ along the trajectory.

Potential Energy

$ \nabla V \equiv - F,$

where $F$ denotes the force.

Abstract

Why is it interesting?

Energy is the most important concept in physicsBrian Skinner