We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new class of integrators, the class of aromatic Runge-Kutta methods, that extends the class of Runge-Kutta methods, and have aromatic B-series expansion but are not B-series methods. Finally, those results are partially extended to the case of more general affine group equivariance.

A geometric analysis of the SHAKE and RATTLE methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises SHAKE and RATTLE to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting.In order for SHAKE and RATTLE to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.

We develop Lie-Poisson integrators for general Hamiltonian systems on R 3 equipped with the rigid body bracket. The method uses symplectic realisation of R 3 on T * R 2 and application of symplectic Runge-Kutta schemes. As a side product, we obtain simple symplectic integrators for general Hamiltonian systems on the sphere S 2 .

Butcher series, also called B-series, are a type of expansion, fundamental in the analysis of numerical integration. Numerical methods that can be expanded in B-series are defined in all dimensions, so they correspond to sequences of maps-one map for each dimension. A long-standing problem has been to characterise those sequences of maps that arise from B-series. This problem is solved here: we prove that a sequence of smooth maps between vector fields on affine spaces has a B-series expansion if and only if it is affine equivariant, meaning it respects all affine maps between affine spaces.

We present a symplectic integrator, based on the implicit midpoint method, for classical spin systems where each spin is a unit vector in R 3 . Unlike splitting methods, it is defined for all Hamiltonians and is O(3)-equivariant, i.e., coordinate-independent. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields a new integrable discretization of the spinning top.Symplectic integrators for the computer simulation of Hamiltonian dynamics are widely used in computational physics [1,2]. For canonical Hamiltonian systems, with phase space R 2N and canonical coordinates (q i , p i ), simple and effective symplectic integrators are known. For noncanonical systems, like spin systems with phase space (S 2 ) N , some symplectic integrators are known. These are, however, either (i) based on local coordinates and not rotationally invariant, (ii) defined only for special Hamiltonians, or (iii) excessively complicated with many auxiliary variables. Here we solve the computational physics problem of providing a globally-defined, rotationally invariant, minimal-variable symplectic integrator for general spin systems. The method is surprisingly simple and depends only on the vector field of the system at hand. It is a rare example of a generating function for symplectic maps on a noncanonical phase space: a noncanonical analogue of the Poincaré generating function of classical mechanics. The method produces new discrete-time physical models, such as a new completely integrable discrete spinning top, and unveils new directions for symplectic integrators, discrete physics, and symplectic geometry.Classical spin systems are a class of noncanonical Hamiltonian systems with phase space (S 2 ) N and symplectic form the sum of the standard area elements on each sphere. If the spheres are realized as s i 2 = 1, s i ∈ R 3 , and H is the Hamiltonian on (S 2 ) N arbitrarily extended to (R 3 ) N , the equations of motion take the formṡ Our main result is a new integrator for (1) given byThis spherical midpoint method is globally defined and preserves many structural properties of the exact flow.Before explaining these properties we review symplectic integrators for canonical and noncanonical systems. Symplectic integrators for canonical Hamiltonian systems fall into two main classes: explicit methods, based on splitting the Hamiltonian into integrable terms and composing their flows, and implicit methods, typically based on generating functions. (Discrete Lagrangians can generate both types of method.) The leapfrog or Störmer-Verlet method, almost universally used in molecular dynamics, is an example of an explicit method, whereas the classical midpoint methodforż = F (z) with z ∈ R 2N , is an example of an implicit method. The classical midpoint method (3) has a number of striking features: (i) it is defined for all Hamiltonians in a uniform way (splitting methods are only defined for separable Hamiltonians); (ii) it conserves quadratic invariants; (iii) it is equivariant with respect t...