A Tree-based Hamiltonian for Fast Symplectic Integration

A method of deriving a Hamiltonian from a tree that can be used for fast symplectic maps of N-body systems is presented. A full description of the Hamiltonian is given along with how the derivatives of that Hamiltonian can be used to implement a symplectic map based upon it. Results show that this alternate Hamiltonian can be used in place of the full Hamiltonian on some simple systems without loss of the dynamics of the system. Speed tests for how the method scales with particle count are also presented and show that the tree based Hamiltonian scales better than the O(N) of the standard Hamiltonian. In addition, even with the overhead of the tree, the new scheme can outperform the standard scheme with as few as 1000 particles in the integration. Because of the superior scaling, the tree based scheme achieves far superior performance when tens of thousands of particles, or more, are involved.