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I'm interested in methods for numerically integrating Hamiltonian systems

$$\begin{align} \dot q & = +\frac{\partial H}{\partial p} \\ \dot p & = -\frac{\partial H}{\partial q} \end{align}$$

One of the really nice properties about symplectic integration schemes derived from operator splitting is that they have a kind of backwards stability property -- they sample exactly from the flow of a slightly perturbed Hamiltonian

$$H' = H + \delta t\cdot\delta H + \mathscr{O}(\delta t^2).$$

You can read off what the leading-order perturbation is from iterated Poisson brackets of the kinetic and potential energies. The usual argument that I see to justify this property is based on the Baker-Campbell-Hausdorff formula; see for example section III.5 of Geometric Numerical Integration. The application of the BCH formula to symplectic integrators seems to be that we formally replace all commutators of matrices with Poisson brackets, replace the exponential mapping with "solve Hamilton's equations of motion", and wave our hands.

How do you make this argument more rigorous? What are the Lie group and the Lie algebra? Can you prove that the series in the BCH formula converges in some sense?

My best guess is that the Lie algebra is the set of smooth scalar fields defined on the cotangent bundle $T^*Q$ of the configuration space $Q$ and the Lie bracket is $\{\nabla f, \nabla g\}$ where $\{\cdot,\cdot\}$ is the Poisson bracket. The Lie group is (probably?) the volume-preserving diffeomorphisms of $T^*Q$. But my head starts to spin when I try to think about the exponential map, especially for realistic problems like the gravitational n-body problem where the potential energy function is unbounded. My Lie group knowledge extends about as far as knowing what SO(3) is, that its Lie algebra consists of antisymmetric matrices, and that the exponential mapping is the matrix exponential.

The books by Hairer, Lubich, and Wanner and Leimkuhler and Reich both seem to say that the application of BCH is purely formal and just ask you to sort of accept it so we can get to the fun stuff. For the correspondence between matrix Lie groups and their Lie algebras, I can prove that the series defining the matrix exponential converges, but the Hairer book says that the series defining the modified Hamiltonian could diverge and I find that... mildly disturbing.

Daniel Shapero
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  • The comments using the BCH formula in Ch.3 were probably just hand-wavy, as you said. They were proving that the finite truncation (of the diverging series) still leads to small errors using backward error analysis later in Ch.10. – C Tong Dec 26 '21 at 19:31
  • I know I am very late to the party, but to get from the "application of BCH" to the expansion of the modified Hamiltonian as poisson brackets, we start by applying the BCH formula for vector fields, which tells you how exponential maps (from T*Q to Q) commute. For the linear vector field case, the exponential map is just the matrix exponential. Doing this for Hamiltonian vector fields naturally gives you the poisson bracket for poisson structure J. – Ben94 May 24 '24 at 10:17

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