Abstract
Most of us have heard some version of the story that results in Isaac Newton being plunked on the head by an apple and discovering gravity.
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Notes
- 1.
In October 1665, the Great Plague that beset London spread to the surrounding countryside and forced the closure of Cambridge University. As a result, Newton was required to abandon his formal studies and returned to his family’s home in Woolsthorpe Manor, Lincolnshire. During his two years in relative exile, while still in his early twenties, Newton contemplated the problem of planetary motion and constructed a new form of mathematics that we today call calculus.
- 2.
We now know that the planet Venus has a size and mass that is roughly comparable to the size of the earth. When observed from earth, Venus is often called the evening star; it appears quite point-like in the sky. This is not true for the sun and moon but we shall justify the point assumption eventually.
- 3.
In some cases we may use more explicit notation like r apple and r earth for the specific case of an apple and the earth but indices 1 and 2 are faster to type and have more general applicability.
- 4.
The direction component of the force changes sign if we exchange the indices. The position component \(\mathcal{R}\) cannot change sign because it depends only on the magnitude | r 2 −r 1 | . Thus, the mass component \(\mathcal{M}\) cannot change sign.
- 5.
As we shall see, the law of universal gravitation can be considered to provide a definition of mass. Other mass dependencies like M α, could simply be redefined to a new effective mass without the power-law dependence: M eff ≡ M α.
- 6.
In the case of uniform acceleration a where the mass begins with no velocity, the mass travels a distance d in a time t given by \(d = at^{2}/2\). Thus, the acceleration would be \(a = 2d/t^{2}\). Consequently, equal times of travel imply equal accelerations.
- 7.
Kepler’s Astronomia nova was published in 1609. Kepler managed to determine, through painstaking analysis of decades of detailed observations conducted by Tycho Brahe, that the orbit of Mars was an ellipse, with the sun at one focus.
- 8.
Note that the sign has been chosen to represent the fact that the gravitational interaction is an attractive one.
- 9.
In this instance, the equations of motion actually do relate to motion of objects. The phrase “equations of motion” will often be utilized to describe the time evolution of a system, even when there are no physical objects that move.
- 10.
The fundamental theorem of calculus provides that \(f(x) - f(a) =\int _{ a}^{x}d\tau \,df(\tau )/d\tau\), when f is defined on the interval [a, b] and a ≤ τ ≤ b. The indefinite integrals in Equation 2.10 give rise to constants of integration that can be lumped into the single vector constant e.
- 11.
This point may be an obscure one but Isaac Newton was a master of geometry. It was not at all obscure to him. Equation 2.13 also justifies our use of the somewhat curious choice of the variable e to represent the constant of integration in Equation 2.11. The variable e is used historically to represent the eccentricity of conic sections.
- 12.
Here we use the notation ∫ d x f(x) to mean the same thing as ∫ f(x) dx. This emphasizes the rôle of integration ∫ d x as the inverse operator to the differential operator d∕dx.
- 13.
At this point in the semester, students may be unfamiliar with integration techniques. The principle point to be made here is that the right-hand side now is no longer an explicit function of t but of the scalar | r 2 −r 1 | . The result of the integration can be verified by differentiation, which should be familiar.
- 14.
The need for the multiplicative factor \(M_{1}M_{2}/(M_{1} + M_{2})\) arises from a dimensional convention that will be discussed in later sections.
- 15.
The somewhat curious choice of \(\mathcal{E}\) for the integration constants is justified by the fact that it is the first letter of the word energy. We use a capital letter to distinguish it from the eccentricity and also to emphasize its importance in our subsequent discussions.
- 16.
There is an unfortunate notational ambiguity in Equation 2.19. The term v cm is a constant, not a function of the argument t − t 1. This will not be the last time we shall have to be conscious of what the notation means.
- 17.
The masses are positive quantities, as must be the quantities G 2 and J 2.
- 18.
We have routinely used the integral notation ∫ d x f(x) to mean the same thing as ∫ f(x) dx. This practice is especially useful when we encounter multiple integrals because the integration variable is positioned adjacent to the symbol that defines the limits of integration.
- 19.
Albert Einstein’s pursuit of a general theory of relativity was hindered by his inability to master the intricacies of non-Euclidean geometry. As he wrote to his friend Arnold Sommerfeld in 1912: “I am now working exclusively on the gravitation problem and believe that I can overcome all difficulties with the help of a mathematician friend of mine here [Marcel Grossmann]. But one thing is certain: never before in my life have I toiled any where near as much, and I have gained enormous respect for mathematics, whose more subtle parts I considered until now, in my ignorance, as pure luxury. Compared with this problem, the original theory of relativity is child’s play.”
- 20.
In a very real sense, the mathematics simplifies when we utilize complex numbers, even though the name includes the word complex. Nevertheless, we shall skirt such discussions.
- 21.
Huygens published his Systema Saturnium in 1659, in which he explained that the curious observed behavior of the objects first seen by Galileo was consistent with a flat disk not separate moons.
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Cunningham, M.A. (2015). II On the Motion of Planets. In: Neoclassical Physics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-10647-2_2
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