For the more mathematically minded..
For those of you with a mathematical bent, here is a brief review of (1D) potential and kinetic energy from the Newtonian viewpoint.
Kinetic energy and the Work-Energy Theorem
Let’s start with Newton’s second law (Key Point 2.2):
F=maUsing the chain rule we rewrite the acceleration:
![\[ a = \frac{d v}{d t} = \frac{d v}{d x} \frac{d x}{d t} = v \frac{d v}{d x} \]](mastermathpng-0.png)
If we differentiate v2 w.r.t x, we get:
![\[ \frac{d (v^2)}{d x} = 2 v \frac{d v}{dx} \]](mastermathpng-1.png)
Therefore we can rewrite Newton’s second law as
![\[ F(x) = m v \frac{d v}{d x} = \frac{d}{dx} \left( \frac{1}{2}mv^2 \right) = \frac{d K}{dx} \]](mastermathpng-2.png)
where K is the kinetic energy.
Integrating both sides, we obtain the work-energy theorem (Key Point 3.7) in one dimension:
![\[ \int^f_i F(x) dx = K_f - K_i \]](mastermathpng-3.png)
Potential Energy
Now let’s assume we can define a function U(x) such that
![\[ F(x) = -\frac{d U(x)}{d x} \]](mastermathpng-4.png)
Inserting this into the work energy theorem
![\[ \int^f_i F(x) dx = - \int^f_i \frac{d U(x)}{d x} dx = -U_f + U_i = K_f - K_i \]](mastermathpng-5.png)
Rearranging, we get that (Key Point 3.16)
Ki+Ui=Kf+UfThis is the law of conservation of energy: total mechanical energy is conserved.
- Naturally this only applies if we can write F=-dU/dx. It is possible to show that this is a necessary and sufficient condition for the force to be conservative.