# Asymmetry in Hamilton Equations

I noticed that in deriving Hamilton equations from the total deriveative of the Hamiltonian with respect to time, for the first equation $$\frac{dx_k}{dt}=\partial_{p_k}H$$ we do not need Lagrange's equation, which are needed to derive $$\frac{dp_k}{dt}=-\partial_{x_k}H.$$

It is clear to me that only the vector $$(x_k(t),p_k(t))$$ can describe the evolution of the system in its phase space, and hence the first equation alone does not make much sense, but is there any phyisical/mathematical reason behind this asymmetry?

1. The first half $$v=\frac{\partial H}{\partial p}$$ of Hamilton's equations is the Legendre transformed relation of $$p=\frac{\partial L}{\partial v}$$.
2. The second half $$\frac{dp}{dt}=-\frac{\partial H}{\partial x}$$ of Hamilton's equations corresponds to Lagrange equations $$\frac{dp}{dt}=\frac{\partial L}{\partial x}$$.
3. In both Hamiltonian & Lagrangian approaches we identify $$v=\frac{dx}{dt}$$.