Hamilton's equations and symplectic numerical methods

Description:

Hamilton's equations can be used to model many problems in the real world, and these equations could present several properties. Thus, in this talk, we are going to study numerical methods that preserve such properties, such as the Symplectic Euler Method.

Hamiltonian sistems are known for beeing divergent free, i.e, the energy of the systems remains the same throughout the trajectory, wich is precicely the level set of the Hamiltonian ate the initial point. However, when we use classical numerical methods to obtain the approximated solutions, this property is not preserved. As we can see in the next pictures, they are solutions of the pendulum with explicit Euler (left) and implicit Euler (right), the energy incrise in the first solution and decrese in se second.

So, there is some numerical method that preserve such property? Yes, the symplectic methods. Insted of finding the approximated solution of the real problem, they give us the real solution of an approximated problem. As we can see in the next picture, we used the symplectic Euler (or semi-implicit Euler), we obtained a real solution of an approximated Hamiltonian system, so the energy is conserved.

Bibliography:

Geometric Numerical Integration - Structure-Preserving Algorithms for Ordinary Differential Equations, Ernst Hairer , Gerhard Wanner , Christian Lubich, https://link.springer.com/book/10.1007/3-540-30666-8 

Classical Mechanics and Symplectic Integration, Nikolaj Nordkvist, Poul G. Hjorth, https://orbit.dtu.dk/en/publications/classical-mechanics-and-symplectic-integration