Newtonian Mechanics

Isaac Newton first pondered space flight in a physically correct manner, but with the wrong technology. He illustrated orbital motion by considering a cannon on a high mountain or perch which shot a ball at a high enough velocity to keep missing the curvature of the Earth. Of course he meant this as a method of illustration, yet at the core he demonstrated how a satellite could be put in Earth orbit. By extension Jules Verne in his novel From Earth to Moon, described a trip to the moon in a large projectile shot from a cannon. Again the cannon was the method of propulsion. The obvious problem here that the crew of this spaceship would be flattened to splatters and bits due to the rapid acceleration to 11 km/sec. The first serious suggestion of rocket flight to space was made by Tsiolkovsky, who also derived the celebrated rocket equation. In the early 20th century Oberth began to lay out the engineering requirements for space rockets. Robert Goddard flew the first successful chemical rocket in 1926. Soon various investigators, von Braun, Winkler, Goddard, Korolev were building larger liquid propelled rockets. It took World War II for the rocket to come into its own. In 1944 Germany used the V-2 rocket to bomb England and other allied positions in a desperate attempt to reverse their diminishing fortunes of war. This was the first clear demonstration that a craft could reach the edge of the Earth's atmosphere by rocketry as the V-2 reached an altitude of 100 km and a speed of over 1000 m/sec.

With the end of the war rocket development in the United States and the Soviet Union proceeded into what then was called the space race. Of course it also lead to the missile race and the existence of Inter-Continental Ballistic Missiles that can deliver a cluster of nuclear bombs to a target and kill millions. Yet pick up a cell phone or make a credit card transaction and one is employing satellites. We get images from the surface of the Saturnian moon Titan, pictures of distant galaxies from the Hubble Space Telescope, watch the martian scenery from robots driving around the martian surface and get our daily weather reports. The rocket makes all of this possible.

To understand how rockets work it is important to understand the physics behind them. This means starting with basic Newtonian mechanics. Before Newton were Copernicus, Kepler and Galileo who respectively laid down the heliocentric solar system, a mathematical description planetary motion, and a description of motion seen on Earth. Galileo furnished some basic equations for the trajectory of an object under an acceleration. This development was stimulated by the introduction of gun powder into Europe, where previous ideas about the flight of a body were found to be inadequate. Gun powder is expensive after all, and a good physics for ballistic flight was needed. Yet even with these early 17th century accomplishments it still remained an open question as to the reason for these principles, and whether the motion of planets had anything to do with the motion of a body in flight here on Earth. Isaac Newton provided the answer in his three laws of motion and his inverse squared law of gravity. It was a grand synthesis of physics that unified the dynamics of bodies here on Earth with the motion of planets. Isaac Newton's Philosophiae Naturalis Principia Mathematica was published in 1687, where after that date it can be said that our views about the nature of the world changed forever [2.1].

Nature and Nature's laws lay hid in night; God said, Let Newton be! And all was light. Alexander Pope

The laws of motion were laid down in the third book of the Principia called De Mundi Systemate (On the system of the world) which give the whole of dynamics. These are codified in the three laws of motion.

Newton's first law of motion is: • A body in a state of motion will remain in that state of motion unless acted upon by a force. This first law introduces the idea of a force by considering its absence. The first law defines an inertial reference frame. On this frame there are no external forces acting upon it and one will observe a body at rest there remaining at rest. Further, this observer may be on another inertial reference with some relative motion to this body, and this observer will see this body move with a constant velocity and remain in this state of linear motion. This first law illustrates something about the nature of inertia as laid down by Galileo in his equation for the linear motion of a body without acceleration x = vt, and constant velocity. It further gives the appropriate reference frame from which the other two laws of physics apply. In other words the laws of mechanics are correct for an observer in a reference frame that is not accelerating.

Newton's second law of motion is:

• The acceleration of a body, or equivalently the change in it momentum, is directly proportional to a force applied to it.

This is the dynamical principle of classical mechanics. The momentum of a body is its mass times its velocity p = mv, mass times velocity, where that velocity is determined by an observer in an inertial reference frame. The time rate of change of the momentum is given by calculus as dp/dt = ma, where the acceleration is a = dv/dt. The first law of motion tells us that the only reliable reference frame from which to observe these dynamics is from an inertial reference frame. From this inertial reference frame the acceleration a is measured. In order for this acceleration to exist there must be a force F exerted on the mass. This leads to the famous expression for the second law of motion

The unit of force kg — m/sec2 is called a Newton N. The first law tells us that only the acceleration is directly measured and not the force. This is even the case for the measurement of a weight. A spring has a force F = —kx where x is the extension of the spring and k is a constant. This equation indicates a spring exerts a force in the opposite direction of its extension, where this force and distance are related by the spring constant k with units of N/m. The measurement of a weight involves the measurement of F = —mg for g = 9.8m/s2 the acceleration of gravity here on the Earth's surface. The second law result F = —kx = —mg infers the weight by the extension of a spring, which indirectly measures the force. Newton's third law of mechanics is:

• Whenever one body exerts a force on a second body, the second body exerts a force of equal magnitude on the first in the opposite direction.

The third law tells us that ultimately momentum is conserved. The action of a force by the second law means that the material body exerting this force will experience an oppositely directed force of the same magnitude. So while there might be an F = ma on one body there is ultimately an —F = —m'a' exerted on another body or mass. The net force is F—F = 0 which conforms to the first law of motion. Further, if one body experiences a change in momentum 5p it does so by inducing an opposite change in momentum —5p on another mass. Fundamentally the third law tells us that the second law operates in a way that is homogeneous and isotropic. In other words how forces acts between bodies is independent of their position and orientation in space. The two forces are always in the opposite direction, and further that the second law of motion is independent of where in space this takes place. This means that Newtonian dynamics is independent of the translation of an inertial frame to another position or its rotational orientation. The first law also indicates that the third law is obeyed for any two masses who's center of mass is travelling at any velocity with respect to an inertial frame. Formally this means that space is isotropic and homogeneous and obeys a set of symmetries given by Galilean translations and orthogonal rotations.

Newton's second law tells us that mechanics is deterministic. Newton's second law of motion is a second order differential equation F = md2x/dt2, which remains the same if the time variables t is replaced by -t. This means that if the dynamics of a body is played backwards in time the dynamics conform to the same principle. Information concerning the initial configuration or state of a system is preserved by dynamics. Newton's laws then imply that the motion of everything can in principle be understood with arbitrary precision and for any time into the future or the past. The second order form of the differential equation for Newton's second law indicates a time reverse invariance. This later was found to imply conservation of energy. Newtonian mechanics was considered in the 18th century to illustrate the clockwork universe. However, it turns out these differential equations for systems involving three or more bodies, or masses, defy integration in a closed form. Classical mechanics is strictly deterministic, but that determinism in most cases is not exactly computable. This is particularly if there are three or more bodies interacting with each other. Isaac Newton in his computations ran into difficulties with modelling the solar system, where he suggested that God might have to intervene to keep it stable [2.2].

Newtonian mechanics unified the mechanics of bodies in motion terrestrially with the motion of the planets. Ultimately this meant that the solar system and beyond lost some of their attributed divine qualities. The motion of Mars is fundamentally no different from the motion of a cannon ball. Prior to Newton there was a general sense that some type of distance rule between bodies governed their motion. Kepler envisioned an inverse linear law for motion, where he lacked the three laws of mechanics and thought the rule should only apply in the plane of planetary motion. Newton's success was in demonstrating that this law of motion. As discussed below the connection with Kepler's law demonstrates that the law of gravity is an inverse square law.

King Oscar II of Sweden, in honor of his 60th birthday, offered a prize for a mathematical demonstration on the stability of the solar system. A year later in 1888 Henri Poincare offered an alternative solution, by demonstrating that this question was not posed properly [2.3]. A system with three or more bodies is not generally integrable, and further any small perturbation on the system will have amplified effects on orbits. This means the system will radically diverge from an expected result at some future time by even the smallest of perturbation. The example in the stability of the solar system is the orbit of a satellite with a small mass in the reduced two-body problem for the two larger masses. It is easy to reduce the dynamics of two mutually interacting bodies to one body. Let this two-body problem be subjected to a perturbation by some distant or small mass. The perturbed two-body orbit is found to be close to an elliptical Keplerian orbit, or the orbit is an approximation to a Keplerian orbit, which is a solution to the reduced two-body problem. Poincare considered a plane transverse to the Kepler orbit for the two-body problem, called a Poincare section. When the actual orbit passes through this plane with each revolution it defines a point on the plane. The unperturbed two-body orbit always pass through the same point. The perturbed two-body orbit might under the right conditions pass through a repeated cycle of points, but in general the orbit hits a different point in this plane with every revolution. To an observer the developing array of these pass through points has no apparent pattern to their occurrence. This pattern in general has no discernable structure and appears completely random.

This was a precursor to the theory of chaos, which in this case is called deterministic chaos. It is a curious matter that a subject centered around F = ma, which is completely deterministic, gives rise to chaos. The term deterministic chaos indicates that in principle the dynamics can be integrated numerically arbitrarily into the future, but the computer would require an infinite floating point capability. Even the smallest truncation of a numerically integration of a three body problem will evolve into a significant contribution that results in a divergence in the actual motion from the numerical simulation. Yet, nature is deterministic and "knows" where it is heading, but we are not able to have that knowledge.

Isaac Newton was a co-founder of calculus, which he laid down in the first book of his Principia De Motu Corporum (On the motion of bodies), along with Gottfried Leibniz, which Newton developed in a proximal way in order to work his laws of mechanics. His approach involved what he called "fluxions," which are infinitesimals. Leibniz introduce the differential, which has a more mathematically well defined concept by limits. Newton's fluxions had some of the content of differentials, but lacked precision and were largely less workable.

It is common to hear that Newtonian mechanics is somehow passe as having been replaced by relativity theory and quantum mechanics so that Newton's classical mechanics is approximate to situations of low velocities and a large scale. While this is true, I urge the reader to again read these three law of motion. It should be apparent that these are truly remarkable principles! There is the central dynamical principle in the second law of motion, which has great utility, but that this law obtains for a certain system of observers on inertial frames within a space that has a mathematical structure of symmetry of translations and rotations. Symmetry structures in more modern physics reveal a crucial connection: conservation principles (such as Newton's third law of motion) is equivalent to a principle of symmetry for the space of consideration. Newton's third law implies conservation of momentum with a symmetry to space. Emmy Noether was the first to clearly codify this connection between symmetry and a conservation law, where her insight was endorsed by Albert Einstein. Isaac Newton was the first to write dynamical principles that made this sort of connection.

The dynamics of F = ma may be used to solve a wide range of problems. A small illustration might be in order here. The spring force is F = — kx, which says that a spring exerts a force proportional to its distention x. This force is equated by the second law to the mass times acceleration of the mass. So Newton's second law indicates that a mass on a spring will in general obey motion according to

Yet the velocity is the time rate of change of position or v = dx/dt. This leads to the differential equation md^x/dt2 = —kx. (2.3)

Without any explicit reference to the theory of differential equations, it is known that functions which satisfy this differential equation are trigonometric functions of sines and cosines. The position of the mass held by this spring is x = Asin{^Jk]rn t) + Bcos{^Jk/m t). The amplitudes A and B depend upon the initial conditions of the problem. Anyone familiar with trigonometric functions knows that this is oscillatory. The spring vibrates up and down in general if it is extended and then released.

For a pendulum in motion the situation is similar. If the angle of swing is small its motion approximates that of a mass on a spring. This results in a dynamical motion that depends only upon the length of the string holding the mass. The mass of the line is regarded as far less than the mass of the mass at the end, and is treated as irrelevant. This was something noted by Galileo, but he was unable to codify in terms of some consistent dynamics. This is a simple problem common in elementary physics texts.

Most elementary classes in physics treat the environment here on Earth as an inertial reference frame. However, this is strictly not the case since the Earth is rotating. However, these effects are small enough to be ignored. Further, local gravity F = -mg for a body sitting on a surface is countered by a normal force N due to the strength of the material under the surface. Hence the net acceleration a = (N/m — g) is zero. A body sitting on a surface is then said to be on an inertial reference frame. Yet this turns out not to be strictly the case for reasons due to general relativity. Yet for basic Newtonian mechanics, which is admitted to be an approximation, these assumptions are made with a good measure of safety.

Isaac Newton introduced another ingredient to his dynamics. This is the universal law of gravitation. Near the surface of the Earth there is a force of gravity F = mg, but this will be different for a region removed from the Earth's surface. Newton realized gravity operated between masses in general. Further, this attractive force is what binds planets in the solar system in their orbits. Newton derived a general rule for gravity. The story of Newton's apple, likely apocryphal, illustrates how he had the insight that when the apple fell to the Earth the apple also pulled the Earth to it, though very slightly. This is consistent with his third law of motion. So for two masses mi and m2 separated by a distance r this force must be F = F(mi, m2, r). This force must also fit within his second law of motion F = ma.

It is necessary to find the acceleration for a particle in circular motion. A circularly moving particle that defines some S^ in some time St traverses an arc of length rSfy. If at one point on the arc it has a velocity v tangent to the circle, then some Sv is needed to change the velocity to that tangent to the second point v ^ v + Sv. The change in velocity Sv points towards the center of the circle. Figure 2.1 illustrates the existence of two similar triangles, one for the small pie slice in the orbital circle and the other for the velocity and its change. By the similarities of these two triangles it is

evident that

Sv v

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