Background:

• Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$
• Starting from $F = ma$, integrating with respect to distance, and substituting $a\ ds = v\ dv$ (from calculus), one can derive $\int Fdx = KE_f - KE_i$

(Obviously I care about these results because, when combined with $F_{ab} = -F_{ba}$, they give conservation of momentum and a starting point for conservation of energy, but for this question I'm not going to consider this part of the derivation.)

In 1-D, the two bulleted results make sense to me: the starting point is a 2nd-order ODE, and the two results form a coupled system of two 1st-order ODEs. This is precisely what math says should be possible - the number of mathematical constraints has been conserved.

Question:

A similar (slightly more involved) derivation is possible in 3 dimensions, but it's harder for me to classify the resulting mathematical constraints and reassure myself that the transformation doesn't add or remove constraints:

• Starting Point: $\vec{F}=m\ddot{\vec{s}}\ \rightarrow$ 3 coupled second-order ODES
• Result 1: $\int \vec{F}dt = m(\dot{\vec{s}_f} -\dot{\vec{s}_i})\ \rightarrow$ 3 coupled first-order ODEs (viewing initial state as a boundary condition)
• Result 2: $\int \vec{F}\cdot d\vec{x} = \frac{1}{2}m(\dot{\vec{s}_f} \cdot \dot{\vec{s}_f}) - KE_i\ \rightarrow $ Unsure how to classify this
  • Rewriting this as $\int \vec{F}(x,y,z)\cdot d\vec{x} = \frac{1}{2}m(\dot{x}_f^2 +\dot{y}_f^2 +\dot{z}_f^2 ) - KE_i $ makes it clearer that this is a single non-linear ODE with the first derivatives of 3 different dependent variables in it.

Math says that a system of 3 coupled second-order ODEs can be rewritten as a system of 6 coupled first-order ODEs, but I obviously only have 4 equations. What type of ODE is Result 2? A single first-order ODE? A 'triple' first-order ODE with 3 dependent variables?

I'm ultimately looking to reassure myself that commuting the problem 'solve $F=ma$' to the problem 'solve conservation of momentum and conservation of energy' doesn't add or remove mathematical constraints. If anyone can refer me to a textbook which addresses this idea, I'd appreciate that as well.