Constrain Equations in Newton Laws of Motion?

Answer 1

normally ignored are air and friction forces , which in say vehicle physics adds some complexity, apply a force to a mass it will tend to accelerate uniformly(a=f/m), but rolling resistance and exponential air resistance offer opposing forces , until they balance at terminal velocity, where net force and acceleration are zero

Answer 2

The constraint equation in Newton's Laws of Motion is Conservation of Energy.

Newton did not develop formally the Conservation of energy Constraint Equation. He intuited it. He left out the kinetic or vector energy (mcV) in his Gravity Energy. The complete law of Gravitational Energy is E= -mGM/r + mcV = -mu/r + mcV.

The Law of Force, his laws of Motion come from the derivative of the energy. The real derivative is d/dr and the vector derivative is Del= id/dx + jd/dy + kd/dz=dR/rdr where R is a vector and r=ct is the scalar radius.

F= XE=(d/dr + Del)(-mu/r + mcV)

F=m(v^2/r - cv/r cos(b)) + m(dcV/dr + cv/r sin(b)T + v^2/r R/r)

F=mcv/r(v/c - cos(b) + m(dV/dt + v/tsin(b)T + v/c v/t R/r)

Newton's Constraint Equation is Conservation of Energy

0=XE = F

F=0 then v/c=1 and 0=dV/dt + v/t R/r

1. The Inertial law is F=0 , No force equal no acceleration or no change in motion.

2. 0=dV/dt + v2/r R/r the Action Reaction law

3. F= XE=mcv/r(v/c - cos(b) + m(dV/dt + v/tsin(b)T + v/c v/t R/r) is the general F=ma

Newton left out the vector energy mcV. Thus he missed the centrifugal force mcv/r cos(b) and the redshift z=v/c=cosd(b). Today, physicists have mistaken the centrifugal force for the "dark energy" or "cosmological constant".