How to derive energy expressions thinking of it as a conserved quantity only?

Question

By now I understand that "energy is a conserved quantity" and that's all we need to know. Then, the idea of work comes as the change in kinectic energy of a system and we realise that having energy is being able to do work, but still remembering that the most basic and important fact about energy is it's conservation.

Now, my doubt is, there are many expressions for the total energy of a system, depending on the situation. For example, in classical mechanics the total energy of a system composed by a particle with trajectory $\gamma : \mathbb{R}\to\mathbb{R}^3$ is:

$$E(t) = K(t)+U(\gamma(t))$$

where $K(t) = m|\gamma\dot(t)|^2/2$ and $U(\gamma(t))$ is the potential energy along the trajectory.

Now, how one might deduce that the conserved quantity is $E(t)$ and even further deduce that $K(t)$ must have this form? I mean, starting just with the assumption that energy is a conserved quantity and using Noether's theorem to grant this quantity does exits, how one can show it has this specific form and find the formulas for it?

Thanks very much in advance.

Answer 1

Energy is conserved by definition. Meaning, we define this quantity called "energy" as $E=\frac{1}{2m}v^2 + V$ where V is the potential energy of a conservative force. The reason we define such a quantity is because it is conserved.

One could also look at this in terms of Lagrangian mechanics where we define the Lagrangian $L = \frac{1}{2m}v^2 - V$ such that it reproduces $F=ma$ when applying the Euler-Lagrange equations. We then note by Noether's theorem that since $L$ is invarient under time-translations, there is a conserved quantity which we can derive from $L$ called "energy".

Both viewpoints follow the same reasoning, that we purposefully define energy, or define the Lagrangian, such that energy is conserved where energy, by definition (or derivation in the case of $L$), is kinetic energy plus potential energy.

Answer 2

It's not having energy that does work, it's potential energy that can do work. Imagine for example a box filled with high pressure air and insulated to the outside world. It can't do work, because there's nowhere else for the energy to go. Punch a hole in the container and air will rush out of the box, doing work in the process.

As for the derivation of $\frac{1}{2}mv^2$, it all starts with Newton's laws. Using $F = ma$, can you calculate the amount of work done accelerating an object from rest to a speed $v$?