A simple derivation of Kepler's laws without solving differential equations

Abstract

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to an explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by beginners at university (or even before) can be considered as a first application of mechanical concepts to a physical problem of great historical and pedagogical interest.

Re-establishing Kepler’s first two laws for planets in a concise way through the non-stationary Earth

“…The position of Mars M1 on May 13, 1950, is now selected as a reference point ( figure 7; table 1). In △ SEj1M1, the angle ∠SEj1M1 = μj1 is observable, and ∠SM1Ej1 = αj1 is calculable, which can be achieved from (6) and is listed in…”

“…(1) The Law of Equal Areas Referring to figures 1 and 3 and combining (3) and ( 6), the Jupiter-Sun distance d can also be represented by the Earth-Sun distance rj, where the Earth-Sun distance is set to be rj1 = r0 = 100000 on August 6, 1962. For the other 4 randomly selected observation dates, the calculated Jupiter-Sun distances d from ( 3) and (6) are listed in table 6. Applying the relationships as shown in figure 4, one may obtain the angular speed ω of Jupiter at the different dates shown in table 6.…”

Had the planet Mars not existed: Kepler's equant model and its physical consequences

“…This may be one reason why he considered that the Sun was responsible for the velocity of the planets and that he took already in [5] this proportionality to be a general law of planet motion (the "velocity law"). 9 Kepler was also aware, as he noticed in his later comments on [5], that this law implied the (incorrect) proportionality of the period T to 2 R . We discuss this "velocity law" further in section 7.…”

“…It is easy to verify it in the equant model, using respectively equation (6) . 9 For a discussion on the difference between the area law and the "velocity law", see [12]. 10 W is also such that CW is parallel to SP since the angle of CW with the x-axis is If Q is taken as the origin of * V ω , the extremity of this vector follows remarkably the same trajectory as the planet (figure 3a).…”

An algebra and trigonometry -based proof of Kepler's First Law

“…Several elementary derivations have been provided by others, based on calculus, geometry, or algebra and trigonometry. [1][2][3][4][5] However, analytical derivations seen in the literature tend to reduce the dynamical equations to an equation of the ellipse that is not typically encountered by the student or enthusiast in their coursework, and the justification that this equation describes an ellipse has either been taken as a standard mathematical result, 2 or has been derived by applying calculus. 1 While it is still true in this work that we employ an unusual equation of the ellipse, we use trigonometry to derive this equation from two well-known properties.…”