Is the relation between Hamilton's and Lagrange's equations the same as that between conservation of energy and the equations of motion?

Question

Conservation of energy is, usually, a $\textbf{first order}$ non linear differential equation, generally written as

$$ \frac{m\dot{q}^2}{2} +V(q) = cte. $$

Taking the derivative yields the usual equation of motion.

$$ m\ddot{q} + V'(q) = 0 $$

(The $\dot{q}$ term vanishes.) Which is a $\textbf{second order}$ ODE. Since the non linear term $\dot{q}^2$ vanishes, this is easier to solve. There are hints for a kind of duality between first order and second order equations.

The same system could be described by Lagrange's equations

$$ \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{x}}\right) = \frac{\partial L}{\partial x} $$

or Hamilton's equations,

$$ \frac{dp}{dt} = -\frac{\partial H}{\partial q}\quad ,\quad \frac{dq}{dt} =\frac{\partial H}{\partial p} $$

Again, there is the same hint about duality. Lagrange's equations are second order, and Hamilton's equations are first order. $\textbf{Is this the same duality as before?}$ Or is it a coincidence?

Answer 1

Conservation of a Hamiltonian is not a universal trait for a system. For this to occur the following must hold $$ \frac{\partial H}{\partial t} = 0 $$ otherwise it is not a constant of the motion. This is a manifestation of Noether's theorem.

The duality observed between Lagrange and Hamilton's equations follows from writing a second order system as a coupled first order system, through a Legendre transform.

Equations of motion must contain time derivatives in order to describe time evolution.