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# Simulations in Analytical Mechanics

Franz Michel, Bernhard Schnizer

Lecture Notes for the Course can be found at http://www.itp.tu-graz.ac.at/LV/schnizer/Analytische_Mechanik/

The Lecture notes comprise 12 Chapters:

1. Introduction
2. Kinematics of a Point Particle
3. Dynamics of a Mass Point
4. Linear and Nonlinear Oscillations
5. Central Forces
6. Constrained Motion, Reaction Forces, Integrable Nonlinear Motion
7. Systems of Mass Points
8. Moving Frames of Reference
9. Mechanics of a Rigid Body
10. Special Relativity Theory and Relativistic Mechanics
11. Fundamental Principles of Mechanics
12. Canonical Equations of Motion. Hamilton-Jacobi Integration Theory

During this course and also in the corresponding exercises simulations are shown. Most of them have been prepared in Mathematica by F. Michel and B. Schnizer. Most of these simulation are diplayed in this site. The numerical solution of the Henon-Heiles problem including the graphical representation of the trajectory and the Poincar'e map all performed in MATLAB can be found at: http://www.itp.tu-graz.ac.at/MML/Poincare/

Also most of the drawings of the lecture notes have been prepared in Mathematica. But these are not displayed.

If Mathematica is not installed at your computer, you need the "Mathreader" which may be downloaded at http://www.wolfram.com/products/mathreader/ . This enables you to open and read the notebooks, which are large and therefore compessed.

## Style sheet

Note that all Mathematica notebooks (except K11GekoppPendel) need a special style sheet called MMLStyleSheet. This must be stored in the directory containing the other notebooks and in the Mathematica directory. The cells containing input data have a brown background color. A user may change these to see the working for different values of the data.

First edition as of 2003-12-31: MMLStyleSheet

## Chap. 5: Central Forces.

The scattering of particles by a hard sphere is shown in K5StreuungAnKugel, ReaderVersion:

The scattering of a point charge by a fixed charge, the trajectories, the scattering angle and the distribution of the final points on a spherical $4\pi$ counter are shown in K5RutherfScattering, ReaderVersion:

The motion of a point charge in the field of a fixed magnetic monopole is treated in K5E5Monopol.pdf. The motion is shown in K5E5MagnMonopol., ReaderVersion:

The motion of a point charge in the field of a fixed magnetic dipole (point poles having a finite distance) is shown in K5E5MagnDipol, ReaderVersion:

## Chap. 6: Constrained Motion. Nonlinear Oscillations.

The period of an oscillating mathematical pendulum depends on the amplitude. This is displayed in K6MathPendel, ReaderVersion:

The various types of motion of a mathematical pendulum are dicussed in K6MathPend1, ReaderVersion:

The solutions for all types of plane motion of the mathematical pendulum using elliptical functions and integrals are shown in K6MathPend2, ReaderVersion:

The solutions for non-planar motion of the spherical pendulum using elliptical functions and integrals are shown in: K6SpherPend1, ReaderVersion:

The periodic and non-periodic motion of a spherical pendulum is simulated in K6SpherPend2, ReaderVersion:

The period of a cycloidal pendulum is independent of the amplitude. The motion of a mass point on a cycloid is achieved by constraining the cord supporting the mass by a templet, which is the evolute of the original cycloid and is again a cycloid. This all is diplayed in simulations contained in K6ZykloidenPend, ReaderVersion:

## Chap.7: Systems of masses.

The inertial and the relative motion of a binary star system is displayed in http://www.itp.tu-graz.ac.at/MML/SimVoUeAM/K7Doppelst.nb, ReaderVersion:

## Chap.8: Moving Frames of Reference.

The motion of the plane of a pendulum depending on its geographical latitude is simulated for the north pole, the equator and a median latitude in http://www.itp.tu-graz.ac.at/MML/SimVoUeAM/K8FoucaultP.nb, ReaderVersion:

## Chap.9: Mechanics of a Rigid Body.

Poinsot's geometrical construction describing the motion of a rigid body is shown in a movie: http://www.itp.tu-graz.ac.at/MML/SimVoUeAM/K9RollPoinsot.mov

## Chap.11: Fundamental Principles of Mechanics

The motion of several coupled pendula is treated by Lagrangian mechanics. The equations of motion are derived and solved numerically. The solutions are plotted. The stability of motion of a pendulum whose support is oscillating is discussed; the equations of motion are solved numerically and the solutions displayed. An external author permitted us to post his notebook here: http://www.itp.tu-graz.ac.at/MML/SimVoUeAM/K11GekoppPendel.nb, ReaderVersion:

## Ch.12 Canonical Equations of Motion. Hamilton-Jacobi Integration Theory.

The action and reduced action function of a mass point moving in a homogeneous field are obtained, the motion of the mass and of the action function is simulated: http://www.itp.tu-graz.ac.at/MML/SimVoUeAM/K12Wirkungsfunk.nb, ReaderVersion:

Further simulations are planned and will be added as soon as they have been completed.

## vollständige Dokumentation

http://www.itp.tu-graz.ac.at/ --- Andreas Hirczy <ahi@itp.tu-graz.ac.at>, 2003-05-07 11:52:15