<s>[INST] Show that all level sets of the Hamiltonian are bounded Consider the system
$$
\dot{x}=y,~~~~~~~~~~\dot{y}=x^2-x^3.
$$
Its Hamiltonian is
$$
H(x,y)=\frac{y^2}{2}-\frac{x^3}{3}+\frac{x^4}{4}.
$$


Show that all level sets of the Hamiltonian are bounded.


As far as I see, it is meant to show that the set
$$
\left\{(x,y)\in\mathbb{R}^2: H(x,y)=c\right\}
$$
is a bounded set.
My computation gives that $H(x,y)=c\Leftrightarrow y=\sqrt{2c+\frac{2}{3}x^3-\frac{1}{2}x^4}$, i.e.
$$
\left\{(x,y)\in\mathbb{R}^2: H(x,y)=c\right\}=\left\{(x,y): y=\sqrt{2c+\frac{2}{3}x^3-\frac{1}{2}x^4}\right\}
$$
How can I see that this is a bounded set?
 [/INST] We could think about different ways to show this, and after a while we might think that it would be nice to show that any level set of $H$ is contained within some ellipse. That would definitely mean that any level set is bounded.
In formulas, this would mean showing that any $x,y$-pair satisfying
$$
\frac{y^2}{2}-\frac{x^3}{3}+\frac{x^4}{4} = c
$$
satisfies
$$
ay^2 + bx^2 \leq d
$$
for some $a,b,d > 0$ which depend on $c$. The first equation already has a $y^2$ term, so we might be able to get away with $a=1/2$. For now let's also take the simplest choice for $b$, namely $b=1$, and hope this doesn't complicate finding $d$.
So now we want to show that if $(x,y)$ satisfies
$$
\frac{y^2}{2}-\frac{x^3}{3}+\frac{x^4}{4} = c \tag{1}
$$
then
$$
\frac{y^2}{2} + x^2 \leq d \tag{2}
$$
for some $d$ depending on $c$.
With our target $(2)$ in mind, let's rearrange $(1)$ like
$$
\frac{y^2}{2} + x^2 = -\frac{x^4}{4} + \frac{x^3}{3} + x^2 + c = p(x) + c. \tag{3}
$$
The polynomial $p(x) = - x^4/4 + x^3/3 + x^2$ is bounded above, and to find this bound we first calculate
$$
p'(x) = -x^3 + x^2 + 2x = -x(x+1)(x-2),
$$
and then check that $p(0) = 0$, $p(-1) = 5/12$, and $p(2) = 8/3$ to conclude that $p(x) \leq 8/3$, and hence, from $(3)$, we conclude that if
$$
\frac{y^2}{2}-\frac{x^3}{3}+\frac{x^4}{4} = c
$$
then
$$
\frac{y^2}{2} + x^2 \leq c + \frac{8}{3},
$$
which completes the proof.
</s>