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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?

Daniel Teixeira
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  • Possible duplicates: https://physics.stackexchange.com/q/105912/2451 and links therein. – Qmechanic Apr 25 '18 at 20:26
  • @Qmechanic this is not a duplicate, I'm not relating Hamiltonian and Lagrangian mechanics, I'm asking if their relation (the question you mentioned) is the same as that between conservation of energy and the equations of motion. – Daniel Teixeira Apr 25 '18 at 20:30
  • There are so many assumptions here it's hard to know what to answer. In a rotating frame for instance, where energy is still conserved, the kinetic energy does not have the form $\frac{1}{2}m\dot{q}^2$ but will contain a Coriolis-type term proportional to powers of $\dot{q}+\vec\omega\times \vec r$ and thus the $\dot{q}$ term does not "vanish" – ZeroTheHero Apr 25 '18 at 21:43
  • that's still a quadratic term, in some sense, which is the point – Daniel Teixeira Apr 25 '18 at 21:44

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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.

AngusTheMan
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