## what are the hamilton equations of motion for homogeneous lagrangians?

For a Lagrangian $L(x^k,\dot{x}^k)$ which is homogeneous in the $\dot{x}^k$ in the first degree, the usual Hamiltonian vanishes identically. Instead an alternative conjugate momenta is defined as

$y_j=L\frac{\partial L}{\partial \dot{x}^j}$

which can then be inverted to give the velocities as a function of the position and momenta

$\dot{x}^i=\phi^{i}(x^k,y_k)$

The Hamiltonian is then equal to the Lagrangian with the velocities replaced with this function

$H(x^k,y_k)=L(x^k,\phi^{k}(x^l,y_l))$

We then find that

$\dot{x}^i=H\frac{\partial H}{\partial y_i}$

which is one half of the Hamilton equations of motion. But what about $\dot{y}_i$?

I am following Hanno Rund The Hamilton-Jacobi equation in the Calculus of Variations. But Rund moves on from this point to the H-J equation, leaving me wondering about this question.
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 I found the answer to this