The rigid body and the pendulum revisited

International Letters of Chemistry, Physics and Astronomy, 2013

In the second part of the paper, the thesis is proved to state that the existent theory describes simply a shadow of the rotating apparent mathematical pendulum. Hence, it appears, even that existent description is not sufficiently adequate. Finally, all defects of the theory, which resulted in so inadequate description of the oscillation motion of the simple mathematical pendulum, have been revealed. The necessity to re-build the existent theory has been indicated in the conclusion. Return to the source is to be the first, essential step on the new path of the cognitive action.

• General Background: A mass m hanging from a string whose length is L and a pivot point on which this mass is fixed are what a simple pendulum (which was discovered during the 10 th century by Ibn Yusuf) consists of. During the 17 th century, it is developed by some physicist, especially by Galileo. When the mass hanging from the string is released with an initial angle, it starts to move with a periodic motion. The motion can be approximated as a simple harmonic motion if the pendulum swings through a small angle (so sin (ө) can be approximated as ө). The frequency and period for the simple pendulum are the independent of the initial angle of the movement (initial position of the mass to the vertical reference line). In addition to the initial angle of the mass, the period doesn't depend on the mass of the object. However, it is affected by the length of the string which the mass is hanged on and the acceleration of gravity. The most widespread applications of the simple pendulum are for timekeeping, gravimetry (the existence of the variable g in the period equation of simple pendulum-• means that the pendulum frequency is different at different places on Earth), seismology, scholar tuning, and coupled pendula. It is also used for entertainment and religious practice. • Aim: To determine the effects or contribution of the length of the string on the period for the simple pendulum and find out a mathematical relationship between the length and the period. • Hypothesis: Since the length of the string which the mass is hanged on is shortened, the magnitude of the period for the simple pendulum gets increased. Different masses of the object hanging from the string have no effect on the period.

International Letters of Chemistry Physics and Astronomy, 2013

In the second part of the paper, the thesis is proved to state that the existent theory describes simply a shadow of the rotating apparent mathematical pendulum. Hence, it appears, even that existent description is not sufficiently adequate. Finally, all defects of the theory, which resulted in so inadequate description of the oscillation motion of the simple mathematical pendulum, have been revealed. The necessity to re-build the existent theory has been indicated in the conclusion. Return to the source is to be the first, essential step on the new path of the cognitive action.

Journal of Physics A: Mathematical and Theoretical, 2007

The motion of a classical pendulum in a gravitational field of strength g is explored. The complex trajectories as well as the real ones are determined. If g is taken to be imaginary, the Hamiltonian that describes the pendulum becomes PTsymmetric. The classical motion for this PT -symmetric Hamiltonian is examined in detail. The complex motion of this pendulum in the presence of an external periodic forcing term is also studied.

Physical Review Letters, 2009

Geometry, Integrability and Quantization

In the present paper we investigate an alternative two-axes decomposition method for rotations that has been proposed in our earlier research. It is shown to provide a convenient parametrization for many important physical systems. As an example, the kinematics of a rotating rigid body is considered and a specific class of solutions to the Euler dynamical equations are obtained in the case of symmetric inertial ellipsoid. They turn out to be related to the Rabi oscillator in spin systems well known in quantum computation. The corresponding quantum mechanical angular momentum and Laplace operator are derived as well with the aid of infinitesimal variations. Curiously, the coefficients in this new representation happen to depend only on one of the angles, which simplifies the corresponding system of ODE's emerging from separation of variables. Some applications of the hyperbolic and complex analogues of this construction in quantum mechanics and relativity are considered in a different paper cited below.

2018

Simulating the simple harmonic motion of pendulum with the Leapfrog integrator based on Hamiltonian canonical equations of first order differential equation wich gives a clear picture how the pendulum behave for different angles and amplitudes. The final proofs of the behaviour of pendulum are the graphics commented at the end.

Journal of Differential Equations, 2013

We explicitly compute the semi-global symplectic invariants near the focusfocus point of the spherical pendulum. A modified Birkhoff normal form procedure is presented to compute the expansion of the Hamiltonian near the unstable equilibrium point in Eliasson-variables. Combining this with explicit formulas for the action we find the semi-global symplectic invariants near the focus-focus point introduced by Vu Ngoc [33]. We also show that the Birkhoff normal form is the inverse of a complete elliptic integral over a vanishing cycle. To our knowledge this is the first time that semi-global symplectic invariants near a focus-focus point have been computed explicitly. We close with some remarks about the pendulum, for which the invariants can be related to theta functions in a beautiful way.

European Journal of Physics, 1999

All kinds of motion of a rigid pendulum (including swinging with arbitrarily large amplitudes and complete revolutions) are investigated both analytically and with the help of computerized simulations based on the educational software package PHYSICS OF OSCILLATIONS developed by the author (see in the web http://www.aip.org/pas). The simulation experiments of the package reveal many interesting peculiarities of this famous physical model and aid greatly an understanding of basic principles of the pendulum motion. The computerized simulations complement the analytical study of the subject in a manner that is mutually reinforcing.

Physics

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical s...

2008

Revisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructed.

International Letters of Chemistry, Physics and Astronomy, 2013

In the paper, the adequate theory of oscillator is presented, being a sort of prelude to verification of the classical theory of mechanics. The developed theory is based on a properly understood notion of energy, quantum value changes of its determined measures (potentials), as well as of such changes types of sites of full energetic states which presents the essence of the true principle of the energy conservation. In the first part of the paper the principle of energy conservation was considered. Then the energetic aspects of the oscillator motion, with an exemplary real system motion was presented. The third part was a development of kinetics of a body in the harmonic motion and verification of the adequate theory of the oscillator. At the end, this Part 4 is devoted to the determination of the gravity acceleration by means of the mathematical pendulum to confirm the previously presented findings.

One of the authors (M.S.) has been teaching the physics laboratory courses (from the Sophomore Laboratory to the Advanced (Graduate) Laboratory) for many years at the Binghamton University. Undergraduate students spend a lot of time in studying the fundamental physics in the physics courses such as the introductory physics, mechanics, electromagnetism, and so on. Even if they understand the theories in a sufficient depth to be able to apply it readily, they encounter some difficulties in understanding the essence of the experiments in the physics laboratory courses, since the conditions of the experiments are more complicated than the ideal conditions of the physics models. As far as we know, so far there are very few opportunities in discussing the gap between the real physics and ideal physics for the physics laboratory courses. The mathematics which is required for the real physics is much more complicated than the mathematics for the ideal physics. In nature there are many examp...

Physical Review Letters, 2009

European Journal of Physics, 2009

The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi), the motion of a force-free asymmetric top (Jacobi), the motion of a spherical pendulum (Weierstrass) and the motion of a heavy symmetric top with one fixed point (Weierstrass). The planar pendulum can, in fact, be used to highlight an important connection between the Jacobi and Weierstrass elliptic functions. The easy access to mathematical software by physics students suggests that they might reappear as useful mathematical tools in the undergraduate curriculum.