Find the Hamilton's equations

Question

Consider the functional given by $$ \ \large \ J(y)=\int_{a}^{b} \sqrt{(t^2+y^2) (1+\dot y^2) } \ dt .$$

Find the Hamilton's equations .

Answer:

I am unable to find the Hamilton from the functional.

can I get any help or idea?

I think here the Lagrangian is given by

$$ L(t,y, \dot y)=\large \sqrt{(t^2+y^2) (1+\dot y^2) } \ $$

So what would be the Hamilton's equation?

Answer 1

To find the Hamiltonian $H(p,q,t)$ you just define $$p = L_{\dot{q}}(q,\dot{q},t)$$ this will give you a relation between $p$ and $ \dot{q}$, suppose that you can invert the relation (so you can express $\dot{q}$ in terms of $p$ as $\dot{q}(p)$, then the hamiltonian is given by the Legendre transform as $$H(p,q,t) = p \ \dot{q}(p) - L(q,\dot{q}(p),t).$$

Maybe now you can compute the Hamiltonian by your own.

Answer 2

1. Lagrangian: $$L~:=~m\sqrt{1+\dot{q}^2}, \qquad m~ :=~ \sqrt{q^2+t^2}. \tag{1}$$

2. Legendre transformation: Momentum $ \leftrightarrow$ velocity: $$p~:=~\frac{\partial L}{\partial \dot{q}}~=~\frac{m\dot{q}}{\sqrt{1+\dot{q}^2}}\qquad \Leftrightarrow \qquad \dot{q}~=~ \frac{p}{\sqrt{m^2-p^2}}.\tag{2}$$

3. Hamiltonian: $$ H~:=~p\dot{q}-L~=~-\frac{m}{\sqrt{1+\dot{q}^2}}~=~-\sqrt{m^2-p^2}.\tag{3}$$

4. Hamilton's equations: $$ \dot{q}~=~\frac{\partial H}{\partial p}~=~\frac{p}{\sqrt{m^2-p^2}},$$ $$ \dot{p}~=~-\frac{\partial H}{\partial q}~=~\frac{q}{\sqrt{m^2-p^2}}.\tag{4}$$