Geometric integrator

In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.

We can motivate the study of geometric integrators by considering the motion of a pendulum.

Assume that we have a pendulum whose bob has mass ${\displaystyle m=1}$ and whose rod is massless of length ${\displaystyle \ell =1}$. Take the acceleration due to gravity to be ${\displaystyle g=1}$. Denote by ${\displaystyle q(t)}$ the angular displacement of the rod from the vertical, and by ${\displaystyle p(t)}$ the pendulum's momentum. The Hamiltonian of the system, the sum of its kinetic and potential energies, is

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