Alright, so recently I've been attempting to prove physics equations and it's really highlighted my lack of understanding in math. So what I went out and tried to prove was the simple law of kinetic energy using the law of momentum. The first scenario I thought of was a simple ball sliding down a ramp (forget angular momentum). Now, I was able to take F=ma (using gravity) along with the law of momentum and convert the equation into mgh = 1/2mv^2 fairly easily. At first, I thought I had done everything I wanted, but then I started thinking of problems where say, the ramp was curved but the total height had remained the same but the total distance varied, and I wanted to prove that as well.
So here's the problem. I could (try) to prove this for a bunch of different equations, but the physics law states that mass' velocity is ONLY dependent on height and force. How exactly can I create (and furthermore KNOW) that the kinetic energy equation holds up for any curve or path (assuming no loss as heat etc)? I could find it with a basic ramp, but how exactly can I prove it to be a state function? Thanks.
One can show that Newton's laws of motion in the particular case $F=-\nabla U(x)$ ($\nabla$ denotes the gradient of a scalar field) imply Hamilton's equations of motion where the Hamiltonian is $H(x,v)=U(x)+\frac{1}{2} m |v|^2$ (where the bars denote the length of a vector). Hamilton's equations of motion make it very easy to check that the Hamiltonian is preserved by just directly computing its derivative with respect to time. This should be derived in your textbook (possibly much later than the section you are in now).