Aside from Noether's Theorem, how do we know energy is conserved?
Energy is the capacity of a system to do work. It's the number that tells me how much "force" a system can apply over a distance. For long I knew we could create forces. However, the only reason... The physical reason that conservation of energy is making sense to me is that when I apply a force to move matter from point $A$ to $B$, I must apply force $x$ on it over displacement $D$ in order to return it I must apply the same amount of force $x$ over $D$. Also the fact that all our natural forces are conservative (in most cases), if work is done by any of the four forces. You must apply the same work again because the natural force is acting against you. Example: Ball is attracted down to Earth's surface by gravity, $9.8$J of work is done ($mgh = 1 \times 9.8 \times 1$) $9.8$N of force has been applied. When the ball is at rest on the surface, I must apply the same amount of force against gravity to move it upwards back to point $A$ ($1$m above the surface). Does this make any sense?
Why is energy conserved besides Noether's Theorem? What other physical observations show it must be true in all our current observable systems?