Governing dynamics of an invariant Lagrangian system, evolving on Lie-group, can be reduced to Lie-algebra. A similar reduction to the Lie-group is possible for the discrete cases. Starting from an invariant discrete Lagrangian, we derive equations that can be used as an integrator. Previous theory can be extended to reduce such systems when the Lagrangian is not fully invariant due to an advected term. Reducing discrete equations to Lie-group and reconstructing the path from there, makes the integrator inherently geometric. An integrator developed from these extended equations is equivalent to a variational integrator. “Reduced discrete advected equation” have been applied to the case of heavy top and precession behavior is modeled accurately.
We developed discrete version of reduction for systems with invariant Lagrangian. Later, we extended the theory to systems with broken invariance and showed that the dynamics dictated by these reduced equations is equivalent to that of a variational integrator. Owing to these excellent properties of discrete reduced equations, we can develop integrators to study behavior of important systems when explicit solution is not available.
Reduced discrete advected equations have been applied to a system of heavy top. This integrator is used to model chaotic behavior of a system under unstable equilibrium. This system respects important characteristics of the physical system like rigidity, momentum, energy conservation and agree with our prediction for variational integrators. This integrator has then been used to study complex precession behavior of a spinning heavy top. Our model was able to show good beahviour even after long intervals of time due to conservative nature of variational integrators.