Newtonian Dynamics

Sir Isaac Newton published in 1687 in the Philosophiae Naturalis Principia Mathematica three laws.1 In plain English they postulate the following:

1) Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.

2) The rate of change of linear momentum equals the impressed force and is in the direction in which the force acts.

3) To every action there is always opposed an equal reaction.

Newton used the word motion instead of linear momentum to define the second law, but the meaning is the same. Like any researcher he used the scientific method to arrive at this formulation. Observations pointed to hypothesis, and testing con­solidated the theory. After more than 300 years of validation, we certainly are justified to call them natural laws.

However, even laws are constrained by assumptions. Newton’s laws are valid in classical physics, where mass does not exceed that of natural substances and velocities are well below the speed of light. Any transgression requires relativistic expansion that Einstein has provided. Furthermore, as particles assume subatomic size, Heisenberg and Schroedinger2 contributed the framework of quantum physics to explain emerging phenomena. All of these modem extensions however point back to Newton’s classical dynamics, as extreme conditions are reduced to the level of our human experience. For any new dynamic theory to be acceptable, it must contain Newtonian dynamics as a limiting case.

The first law is validated by our experience. We do not notice our own linear momentum unless a wall stops us. The wall exerts the force that kills us (second law). Newton’s third law is important in mechanics because it assures us that internal forces cancel in a collection of particles. However, it is the lesser of the three laws because it fails in classical electrodynamics. For modeling of aerospace vehicle dynamics, the second law is of greatest interest. We therefore refer to it mostly just as Newton’s law.

Interestingly, Newton did not specify a frame of reference in formulating his second law. Others attempted to affix what was called the “luminiferous ether” to his law, until Michelson and Morley2 in 1887 disproved the concept. Thus, wisely, Newton left it to the application to pick the proper reference frame, which we call today the inertial reference frame. From his first law we know that any nonaccelerating frame qualifies equally well; but do they exist? In Chapter З, I suggested using the ecliptic of our sun as inertial reference frame, but we know that our solar system is located in the spiral arms of the Milky Way and therefore accelerating. Other theories suggest that all galaxies are fleeting with increasing speed. Where is the inertial frame? It probably does not exist in absolute terms.

Applications determine the inertial frame. Interplanetary travel requires the he­liocentric frame; Earth satellite trajectories use the ecliptic oriented and Earth – center fixed frame, which we call plainly the inertial frame’, and Earth-bound, low-speed flights can use the Earth frame. Whatever the accuracy requirement of your simulation is will determine the choice of inertial reference frame.