Kepler’s Laws without Calculus

W. G. Unruh
Newton in the Principia used the inverse squared force law (and Galileo’s idea of compound motion) to derive Kepler’s laws. As usual for him, the proof is a purely geometric proof, using no calculus. Maxwell [2] gave a very different proof. In the 1960’s, Feynman[3] gave a geometric proof very similar to Maxwell’s. Finally, Vogt[4] in the American Journal of Physics also carried out a derivation which started out with the energy conservation equation and the angular momentum conservation to again present a geometric proof. In all of these cases, the derivation that the orbit actually is an ellipse was somewhat torturous, difficult to follow, and non obvious. In the following, following initially the Maxwell-Feynman’s approach, the derivation that the orbit is an ellipse is simplified about as much as possible.


Leonardo da Vinci’s studies of friction

Sketches from two different pages in Leonardo’s notebooks: (a, b) from Codex Atlanticus, Biblioteca Ambrosiana, Milan (CA folio 532r c. 1506-8), and (c) from Codex Arundel, British Library, London (Arundel folio 41r c. 1500-05)

Sketches from two different pages in Leonardo’s notebooks: (a, b) from Codex
Atlanticus, Biblioteca Ambrosiana, Milan (CA folio 532r c. 1506-8), and (c) from Codex Arundel, British Library, London (Arundel folio 41r c. 1500-05)

Ian M. Hutchings
Based on a detailed study of Leonardo da Vinci’s notebooks, this review examines the development of his understanding of the laws of friction and their application. His work on friction originated in studies of the rotational resistance of axles and the mechanics of screw threads.
He pursued the topic for more than 20 years, incorporating his empirical knowledge of friction into models for several mechanical systems. Diagrams which have been assumed to represent his experimental apparatus are misleading, but his work was undoubtedly based on experimental measurements and probably largely involved lubricated contacts.
Although his work had no influence on the development of the subject over the succeeding centuries, Leonardo da Vinci holds a unique position as a pioneer in tribology.

Read also: Study reveals Leonardo da Vinci’s ‘irrelevant’ scribbles mark the spot where he first recorded the laws of friction

“Walking” along a free rotating bicycle wheel (Round and round)

Marta mouse
Julio Guemez, Manuel Fiolhais
We describe the kinematics, dynamics and also some energetic issues related to the Marta mouse motion when she walks on top of a horizontal bicycle wheel, which is free to rotate like a merry-to-go round, as presented recently by Paul Hewitt in the Figuring Physics section of this magazine….

Plane Pendulum and Beyond by Phase Space Geometry

Bradley Klee
By careful analysis, the small angle approximation leads to a wildly inaccurate prediction for the period of a simple plane pendulum. We make a perturbation ansatz for the phase space trajectory of a one-dimensional, anharmonic oscillator and apply conservation of energy to set undetermined functions. Iteration of the algorithm yields, to arbitrary precision, a solution to the equations of motion and the period of oscillation. Comparison with Jacobian elliptic functions leads to multidimensional applications such as the construction of approximate Seiffert spirals. Throughout we develop a quantum/classical analogy for the purpose of comparing time-independent perturbation theories.

Duality symmetries behind solutions of the classical simple pendulum

Román Linares
The solutions that describe the motion of the classical simple pendulum have been known for very long time and are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of the solutions is time and it can be considered either as a real variable or as a purely imaginary one, which introduces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions the symmetry is codified in the functional form of its modulus, and is described mathematically by the six dimensional coset group Γ/Γ(2) where Γ is the modular group and Γ(2) is its congruence subgroup of second level. In this paper we discuss the physical consequences this symmetry has on the pendulum motions and it is argued they have similar properties to the ones termed as duality symmetries in other areas of physics, such as field theory and string theory. In particular a single solution of pure imaginary time for all allowed value of the total mechanical energy is given and obtained as the S-dual of a single solution of real time, where S stands for the S generator of the modular group.

The simple plane pendulum constitutes an important physical system whose analytical solutions are well known.
Historically the first systematic study of the pendulum is attributed to Galileo Galilei, around 1602. Thirty years later he discovered that the period of small oscillations is approximately independent of the amplitude of the swing, property termed as isochronism, and in 1673 Huygens published the mathematical formula for this period. However, as soon as 1636, Marin Mersenne and Rene Descartes had stablished that the period in fact does depend of the amplitude. The mathematical theory to evaluate this period took longer to be established.
The Newton second law for the pendulum leads to a nonlinear differential equation of second order whose solutions are given in terms of either Jacobi elliptic functions or Weierstrass elliptic functions …
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