The formulation of the dynamics of a mechanical system can be done by the method of the Port Controlled Hamiltonia System (PCHS), but this method still leaves a Lagrange multiplier. Furthermore, the dynamics can be formulated using another method which is more systematic, namely the Routhian Reduction method. The method illustrates a system that is subject to non-holonomic constraints and external force, so that the Lagrange multiplier can be removed from the equation. Before formulating the dynamics of a non-holonomic mechanical system, the researcher will analyze the potential energy that occurs in a system that moves in the cylinder configuration space. Potential energy is the main part that must be completed to formulate the motion system of an object, because Routhian reduction only reviews the kinetic energy and potential energy in a dynamic system. The dynamical system reviewed is an object that moves both translation and rotation with a non-holonomic constraint, namely the Tippe Top (TT). The author analyzes the potential energy of a mechanical system that moves in a cylinder configuration space with non-holonomic constraints. Method in this research is a mathematical theoretical study. This method can reduce the equation TT’s motion with and without friction that moves on the surface of the cylinder clearly in the form of a set of differential equations. According the result of this riset, the potential energy for the TT with non-holonomic constraints that move on the surface in the tube can be determined by U = mg(r(1 − cos ρ ) + (R − acos θ ) cos ρ + asin θ cos φ sin ρ), transforming the TT’s Lagrangian that moves on a flat plane (Cartesian coordinates) to the tube coordinates, with reference to the height of the plane solved by coordinate transformation.

Tippe top is an example of simple moving system of rigid body with non-holonomic constraint, but the analysis of this system is not simple. A tippe top equation has been derived with Routhian reduction method and Poincaré equation, and physics computation in finding numeric solution of the dynamics of the tippe top has also been utilized by using Maple program. However, the Poincaré equation required that quasi-coordinate of the quasi-velocity is found, while in the case of the dynamics of tippe top, there is not any exact solution of the quasi-coordinate of the quasi-velocity was found. Therefore, the tippe top equation should be reduced to solve the problem. In this research, Routhian reduction was employed so that the Routhian reduction-based Poincaré equation was used to derive the tippe top equation. The method was able to derive a tippe top equation on a flat plane and tube inner surface clearly represented differential equations.

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Physics computing can be used to help to solve complex dynamic equations, both translation and rotation. The purpose of this study was to obtain differences in the dynamics of the tippe top with and without friction moving on inner surface of a cylindrical with varying initial state based of Routhian Reduction. The equation of tippe top in flat fields with and without friction has been reduced by the Routhian reduction method with the Poincare equation with computational in the previous research, and computation has also been carried out in the search for numerical solutions to the dynamics of tippe top with friction in the Maple program. In this study the reduction used is a Routhian reduction, so that the equation used in determining the equations of tippe top motion with and without friction that moves in a curved plane in the form of a cylindrical surface with varying initial state based on maple is Poincaré’s equation based on Routhian reduction with and without friction. The effect of friction can be seen clearly through the dynamics and graph equations in the return top. This method can reduce the equation of backward motion with and without friction that moves on the surface of the cylinder clearly in the form of a set of differential equations. This research can be continued by solving the dynamic equations of the tippe top in other curved fields such as the torus and ball. The findings of this study are dynamic equations and graphs of friction with and without friction equations that move in curved fields in the inner of surfaces in cylinders with varying initial state based on maple.