Abstract
The discrete gradient approach is generalized to yield first integral preserving methods for differential equations in Lie groups.
Highlights
Our point of departure is the system of differential equations x = F (x) = f (x) ⋅ x = x ⋅ f(x), (1)
Any differential equation (1) on a Lie group having H as first integral can be formulated via a bivector ω and the differential of H as x = F (x) = ω(dH, ⋅) = dH ⌟ ω
If the Lie group G and its Lie algebra g both are realized as m × m-matrices, the dual elements can be represented as matrices and the duality pairing could be given as ⟨p, v⟩ = trace(pT v)
Summary
Any differential equation (1) on a Lie group having H as first integral can be formulated via a bivector (dual two-form) ω and the differential of H as x = F (x) = ω(dH, ⋅) = dH ⌟ ω (2). In coordinates one can represent the symplectic two-form by a skew-symmetric matrix SΩ in a similar way as in (3) and (4) with respect to the basis {dxi ∧ dxj, i < j}. By definition, this matrix will be invertible, and its inverse is precisely Sω = SΩ−1. By considering high order methods, a larger class of manifolds and further examples
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