Hamiltonian Constant on integral curves

Question

Let $H \in C^{2}(\mathbb{R}^2)$ and let $(x(t),y(t))$ be a solution to the equations

$$\frac{dx}{dt} = \frac{\partial}{\partial y} H(x(t),y(t))$$ $$\frac{dy}{dt} = -\frac{\partial}{\partial x} H(x(t),y(t))$$ I need to show that $H$ is constant along $(x(t), y(t))$. How would I do so using only the information here?

Moreover, does $$\frac{\partial}{\partial x} \frac{dx}{dt}$$ have a well-defined meaning?

Answer 1

Just apply the chain rule and use the differential equation.

$$ \frac{d}{dt}H(x(t),y(t)) = \partial_x H(x(t),y(t)) \frac{dx}{dt} + \partial_y H(x(t),y(t)) \frac{dy}{dt} $$

Now use the equation to get:

$$ \partial_x H(x(t),y(t)) \partial_y H(x(t),y(t)) - \partial_y H(x(t),y(t)) \partial_x H(x(t),y(t)) = 0 $$

So the function $H$ is constant along the curve $(x(t),y(t))$.