Analytisk mekanik med datoralgebra
In this project we have used the theory of analytical mechanics to derive equations of motion for complex mechanical systems in order to study their behaviour over time. Even if the theory provides powerful tools to tackle tough problems which would be very hard to solve by methods of Newtonian mechanics, you are still relatively limited without running any computer simulations. One realises that in a faster rate it will be an incredibly difficult task to find an analytical solution when the degrees of freedom increases. Even if one would succeed to derive the equations, one still has to solve them to get the information of the systems behavior. Of experience one knows that the equations in general are nonlinear and hence must be integrated numericaly. These nonlinearities preposses the sensible dependence on initial conditions which means that small perturbations from equilibrium will lead to divergent outcomes in the continuing evolution of the motion. The motion is then what you call chaotic. Since we have been dealing with nonlinear problems, numerical and symbolical analysis has been performed with the computer aid. Hence, a big part of this project has been to get familiar with the Maple based simulation program ?Sophia?, developed by the department of mechanics at KTH [7]. With this program we could simulate different kinds of pendulum systems and by plotting their solutions, investigate the influence on the overall motion by the parameter values and initial conditions.