Newtonian laws of motion and conservation principles

Newton’s laws of motion and Newtonian conservation principles such as those for energy and momentum involve the assumption that the vanishing of a certain total time derivative, on integration, yields a fixed constant value as an immediate consequence. While this may ultimately be the case for additional reasons, it is possible to have a properly vanishing total time derivative and yet the individual partial derivates are non-zero. Here, for a particular problem and based only on the requirement that the total time derivative of the quantity vanishes, we investigate the particular mechanism leading to a conventional conservation principle. For the energy and angular momentum totals for planar steady orbiting motion, the partial differential conditions may be formally solved to obtain the general solutions. We determine the general structure for variable energy and angular momentum for which the total time derivatives vanish, and from which it is apparent that the standard expression for constant energy and angular momentum is recovered.