Hamilton's equations and Euler-Lagrange equation

Question

This paper is about deriving hamilton's equations from Euler-Lagrange equation, what i don't understand is equation 19. In equation 18 the process involved is if we substitute lagrangian $L$ for function $f$ we get one of hamilton's equation but in equation 19 i don't know what process is involved to obtain another hamilton's equation?

Reference: P. Gutierrez, Physics 5153 Classical mechanics Canonical transformation

Answer 1

1. The function $$f(q,\dot{q},p,t)~\equiv~ L_H(q,\dot{q},p,t)~\equiv~ p\dot{q}-H(q,p,t)$$ is the so-called Hamiltonian Lagrangian. Note in particular that its arguments are independent variables.

2. Its EL equations lead to Hamilton's equations (HE): The 1st half of HE is eq. (18). The 2nd half of HE is eq. (19). To derive eq. (19), use the fact that $f$ does not depend on $\dot{p}$, so that the EL eq. (19) simplifies to $\frac{\partial f}{\partial p}=0$.

3. For more details, see e.g. this and this related Phys.SE posts.