What is the benefit of Hamiltonian formalism to promote ($q,\dot{q},t$) from Lagrangian to ($q,p,t$) despite getting the same EOM finally?

Question

Hamiltonian formalism follows $$H(q,p,t)=\sum_i\dot{q_i}p_i-L(q_i,\dot{q}_i,t) $$ and $$\dot{p}=-\frac{\partial H}{\partial q}, \dot{q}=\frac{\partial H}{\partial p} $$ but finally these will get the same equation of motion as Lagrangian formalism. What is the {point, advantage, benefit, etc.} for using (q,p,t) variables in Hamiltonian formalism while we can get EOM from Lagrangian formalism?, and we have to get the Lagrangian $L$ first to get Hamiltonian $H$.

Answer 1

• First of all, Hamilton's equations are 2n first order differential equations, whereas Euler-Lagrange equations are n second order differential equations, which can make them harder to solve.*

• Secondly, switching from $\{q,\dot{q},t\}$ space to $\{q,p,t\}$ space allows for cannonical transformations which are a coordinate change from $\{q,p,t\}$ to $\{Q(q,p),P(q,p),t\}$ in which the equations can be easier to solve.

• The fact that there are two first order differential equation guaranties that paths in the $\{q,p\}$ space do not intersect which is essential for the Liouville theorem (and thus statistical mechanics).

• The $\{p,q,t\}$ coordinate space allows for the Poisson bracket formalism.

• The Hamiltonian is the generator of time translations, and as such allows for a treatment of conserved quantites. This makes the Hamiltonian more fundamental than the Lagrangian.

• The Hamiltonian is essential for the Hamilton-Jacobi formalism which allows to find conserved quantities (and use them to solve the equations of motion).

*Edit: 2n first order differential equations are not easier to solve.