International audienceMany differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively-correct geometry and dynamics and in the minimisation of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications

Abstract. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equationWe show that standard Runge-Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order.1. Background and notation 1.1. Introduction. The interest in solving isospectral flows is motivated by their relevance in a wide range of applications, from molecular dynamics to micromagnetics to linear algebra. The general form of an isospectral flow is the differential equationwhereThe choice of the matrix function B(L) characterizes the dynamics of the underlying flow L(t). Important special cases are the Toda lattice equations, doublebracket flows and KvM flows.Toda lattice equations in the Lax formulation (1)

We consider the Fer and the Magnus expansions for the numerical solution of the nonlinear matrix Lie-group ODE y 0 = (t; y)y; y(0) = y0; where y evolves in a matrix Lie group G and (t; y) is in the Lie algebra g. Departing from a geometrical approach, that distinguishes between those operations performed in the group and those performed in the tangent space, we construct Lie-group invariant methods based on collocation. We prove that, as long as the two expansions are correctly truncated, the collocation nodes c1; c2; : : : ; c yield numerical methods whose order is the same as in the classical setting. We also relax the collocation conditions, thereby devising explicit methods of order three. To conclude, we discuss the proposed methods in two numerical experiments that arise in Hamiltonian mechanics and inverse eigenvalue problems, comparing the results with projection techniques.

The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group.In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion.Second, we discuss additional properties of the two factors. In particular, when σ is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G.