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

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?

• Possible duplicates: physics.stackexchange.com/q/105912/2451 and links therein. Apr 25, 2018 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. Apr 25, 2018 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" Apr 25, 2018 at 21:43
• that's still a quadratic term, in some sense, which is the point Apr 25, 2018 at 21:44

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.