Ufyukyu

5 2 $begingroup$ I've been reading about Hamiltonian mechanics which in its mathematical description uses Poisson manifolds From my limited understanding, on a Poisson manifold $M$ we can look at the Poisson bracket as a gadget that gives a smooth vector field ${f,- }$ for every smooth function on $M$ . This gives a nice way to write Hamilton's equations of motion. My questions are: how should I visualize this vector field ${f,- }$ ? What's its connection to the function $f in C^infty(M)$ ? What's the connection of the flow of ${f,- }$ to the function $f$ ? Am I correct in saying that ${f,g } = 0$ means that $g$ is constant along the flow of ${f,- }$ ? If that helps, my background is primarily in algebra, so I'm asking about a physicist's/geometer's way of thinking about t

0 $begingroup$ Let $a_1, a_2, b, c in mathbb{R}$ such that $ a_1 neq 0,, $ $a_2 neq 0,, $ $ a_1 > a_2,, $ $ b neq 0,,$ $c neq 0,,$ and $ b_1 < b < b_2 < 2(a_1 - a_2)$ , where : $$b_1 = frac{2(a_1-a_2)}{sqrt{1+frac{4a_1}{c}} + 1} qquad b_2 = frac{2(a_1-a_2)}{sqrt{1+frac{4a_2}{c}} + 1}$$ Prove that $k_1k_2 - k_0 > 0,$ where : $$k = frac{a_1a_2c}{b^3(a_1-a_2)}left(b+frac{2(a_1-a_2)}{sqrt{1+frac{4a_1}{c}} - 1}right)left(frac{2(a_1-a_2)}{sqrt{1+frac{4a_1}{c}} + 1}right)$$ $$k_2 = frac{(a_1+a_2)b^2+2c(a_1-a_2)^2}{(a_1-a_2)b}$$ $$k_1 = cleft[frac2b -frac{1}{a_1 - a_2}right][(a_1 + a_2)b + (a_1-a_2)c]$$ $$k_0 = -k(b-b_1)(b-b_2)$$ This was a Bonus question in a test I took last week, I computed $k_1k_2$ and kept doing circular simplifications but I never re