For a second order differential equation (many physical systems) in one variable, I know "procedures" to compute the energy. Given q ″ ( t ) = f ( q ( t ) , q ′ ( t ) ) , q ( 0 ) = q 0 , q ′ ( 0 ) = v 0 ,if we're lucky we can read off the related Lagrangian L, introduce p = ∂ L ∂ q , do a Legendre transform and we got the Hamiltonian function H(q,p) for which d d t H ( q ( t ) , p ( t ) ) = 0 for solutions of the differential equation.We can be more exact and give all the conditions for Noethers theorem to hold and the result is that along the flow X ( t ) = ⟨ q ( t ) , p ( t ) ⟩ in phase space given by a solution with initial conditions X(0), the function H(q,p) always takes the same values. It defines surfaces in phase space indexed by initial conditions X(0).I wonder how to view this for a priori first order systems q ′ ( t ) = f ( q ( t ) ) , q ( 0 ) = q 0 ,where I think this must be even simpler. E.g. I thing some functional q ( t ) ↦ q ( t ) − an integral over f ( q ( t ) ) something ,should exists which will be constant for solutions of the equation, i.e. only depend on q 0 . For each f, the functional dependence of this "energy" on " q 0 " will be different.However, I can't seem to find a general relation. What's the theory behind this, is there an energy'ish time integral of motion? What is the functional dependence on the intial condition, for a suitable constant of motion for for first order systems. And what would be the interpretation, given that we speak of a situation with only one initial condition?If it is that case that the system is too restricted so that there is no meaningful geometrical interpretation, then let's think about a system of first order differential equations. This is like the one we generated from phase space, except that it doesn't really come from a second order situation and so the intial conditions aren't really e.g. intial position and velocity. I'm pretty sure there are situation where one considers such a directly generated flow (the equation might be more complicated than exponential flow x ˙ ∝ x), but I don't recall any talk about the time-constant of motion in these systems, or how to interpret it.

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For a second order differential equation (many physical systems) in one variable, I know &quot;procedures&quot; to compute the energy. Given      q    ″    (  t  )  =  f  (  q  (  t  )  ,      q    ′    (  t  )  )  ,        q  (  0  )  =      q    0    ,            q    ′    (  0  )  =      v    0    ,if we&#039;re lucky we can read off the related Lagrangian L, introduce   p  =            ∂      L              ∂      q      , do a Legendre transform and we got the Hamiltonian function H(q,p) for which       d          d      t        H  (  q  (  t  )  ,  p  (  t  )  )  =  0 for solutions of the differential equation.We can be more exact and give all the conditions for Noethers theorem to hold and the result is that along the flow   X  (  t  )  =  ⟨  q  (  t  )  ,  p  (  t  )  ⟩ in phase space given by a solution with initial conditions X(0), the function H(q,p) always takes the same values. It defines surfaces in phase space indexed by initial conditions X(0).I wonder how to view this for a priori first order systems      q    ′    (  t  )  =  f  (  q  (  t  )  )  ,        q  (  0  )  =      q    0    ,where I think this must be even simpler. E.g. I thing some functional  q  (  t  )  ↦  q  (  t  )  −  an integral over     f  (  q  (  t  )  )     something  ,should exists which will be constant for solutions of the equation, i.e. only depend on       q    0  . For each f, the functional dependence of this &quot;energy&quot; on &quot;      q    0  &quot; will be different.However, I can&#039;t seem to find a general relation. What&#039;s the theory behind this, is there an energy&#039;ish time integral of motion? What is the functional dependence on the intial condition, for a suitable constant of motion for for first order systems. And what would be the interpretation, given that we speak of a situation with only one initial condition?If it is that case that the system is too restricted so that there is no meaningful geometrical interpretation, then let&#039;s think about a system of first order differential equations. This is like the one we generated from phase space, except that it doesn&#039;t really come from a second order situation and so the intial conditions aren&#039;t really e.g. intial position and velocity. I&#039;m pretty sure there are situation where one considers such a directly generated flow (the equation might be more complicated than exponential flow             x      ˙        ∝  x), but I don&#039;t recall any talk about the time-constant of motion in these systems, or how to interpret it.

alisonhleel3 · Accepted Answer

Assuming the first order system is Hamiltonian, for one degree of freedom, we may construct the hamiltonian by integrating the pair of ode&#039;s using the relation:      q    ′    =            ∂      H              ∂      p              p    ′    =  −            ∂      H              ∂      q      Higher order systems have the same &quot;symplectic&quot; structure, and the integrals could have no closed forms. So we would have:            ∂      H              ∂      p        =  f  (  q  )  ,        p    ′    =  ?  ?Things seem to get notationally ambiguous. The point is that we must look at variables as conjugated pairs when working in the Hamiltonian formulation of mechanics. As a definitive answer to the OP, if we knew that the system was Hamiltonian, we could in principle integrate the above relations to generate the &quot;Hamiltonian&quot;. Also, energy and time are considered to be conjugated in this framework. May I suggest the Poisson bracket as a starting point.

For a second order differential equation (many physical systems) in one variable, I know "procedures

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