Herpolhode

Abstract: The current paper aims at presenting a free online stereoscopic 3D simulation developed by the author. The simulation visualizes the Poinsot construction in free rigid body motion. The student is helped in understanding the famous construction and thus better comprehend the Newtonian mechanics and master its underlying mathematical formalism. Stereoscopic 3D simulations are a fruitful method for observation of phenomena hard to realize in laboratory conditions such as weightlessness. The Poinsot construction is rendered in stereo 3D graphics in the web browser and the simulation shows the construction’s inherent elements, such as invariant ellipsoids, invariant plane, polhode, herpolhode, etc. The latter are watched along with a large number of involved parameters: vectors and scalars. The presented material is directed towards university students taking the Analytical (Mathematical) Mechanics courses in the Faculty of Mathematics and Informatics and also the students from the Theoretical Physics and General Physics courses in the Faculty of Physics in Sofia University, Bulgaria, but is not restricted from use in other universities, because the simulation has free access on the Internet. The simulation, discussed in this paper can be viewed and used from http://ialms.net/sim/ web address.

Abstract: Introduction Unusually in physics, there is no pithy phrase that sums up the study of dynamics (the way in which forces produce motion), kinematics (the motion of matter), mechanics (the study of the forces and the motion they produce), and statics (the way forces combine to produce equilibrium). We will take the phrase dynamics and mechanics to encompass all the above, although it clearly does not! To some extent this is because the equations governing the motion of matter include some of our oldest insights into the physical world and are consequentially steeped in tradition. One of the more delightful, or for some annoying, facets of this is the occasional use of arcane vocabulary in the description of motion. The epitome must be what Goldstein calls “the jabberwockian sounding statement” the polhode rolls without slipping on the herpolhode lying in the invariable plane , describing “Poinsot's construction” – a method of visualising the free motion of a spinning rigid body. Despite this, dynamics and mechanics, including fluid mechanics, is arguably the most practically applicable of all the branches of physics. Moreover, and in common with electromagnetism, the study of dynamics and mechanics has spawned a good deal of mathematical apparatus that has found uses in other fields. Most notably, the ideas behind the generalised dynamics of Lagrange and Hamilton lie behind much of quantum mechanics.

Abstract: . Guided by the Jacobi’s work published one year before his death about the rotation of a rigid body, the behavior of the rotation matrix describing the dynamics of the free rigid body is studied. To illustrate this dynamics one draws on a unit sphere the trace of the three unit vectors, in the body system along the principal directions of inertia. A minimal set of properties of Jacobi’s elliptic functions are used, those which allow to compute with the necessary precision the dynamics of the rigid body without torques, the so called Euler’s top. Emphasis is on the paper published by Jacobi in 1850 on the explicit expression for the components of the rotation matrix. The tool used to compute the trajectories to be drawn are the Jacobi’s Fourier series for theta and eta functions with extremely fast convergence. The Jacobi’s sn, cn and dn functions, which are better known, are used also as ratios of theta functions which permit quick and accurate computation. Finally the main periodic part of the herpolhode curve was computed and graphically represented.

Abstract: Guided by the Jacobi's work published the year before his death about the rotation of a rigid body, the behavior of the rotation matrix describing the dynamics of the free rigid body is studied. To illustrate this dynamics one draws on a unit sphere the trace of the three unit vectors, in the body system along the principal directions of inertia. A minimal set of properties of Jacobi's elliptic functions are used, those which allow to compute with the necessary precision the dynamics of the rigid body without torques, the so called Euler's top. Emphasis is on the paper published by Jacobi in 1850 on the explicit expression for the components of the rotation matrix. The tool used to compute the trajectories to be drawn are the Jacobi's Fourier series for {\sl theta} and {\sl eta} functions with extremely fast convergence. The Jacobi's {\sl sn}, {\sl cn} and {\sl dn} functions, which are better known, are used also as ratios of {\sl theta} functions which permit quick and accurate computation. Finally the main periodic part of the herpolhode curve was computed and graphically represented.