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Simulating Lagrangian Mechanics Directly



We integrate numerically a system in a Lagrangian form directly,
without deriving the underlying equations of motion explicitly. From a
C++ specification of a Lagrangian function and algebraic constraints,
our ''Lagrangian'' facility applies automatic differentiation to
prepare a differential-algebraic equation (DAE) system, which is then
solved by our high-index differential-algebraic equation (DAE) solver

Lagrangian equations of the first kind contain algebraic constraints,
resulting in an index-3 DAE; Lagrangian equations of the second kind
are constraint-free, resulting in a system of ordinary differential
equation (ODEs). The former are usually much simpler and easier to
construct (in particular when using cartesian coordinates) than the
latter, which typically involve angle coordinates and non-trivial
transformations to eliminate constraints. However, integrating an
index-3 DAE is substantially more difficult than integrating an ODE.

DAETS solves a high-index DAE as easily as an ODE. We model and
simulate rigid-body mechanisms - mechanical systems with linked rigid
parts and possible other parts such as springs - from a constrained
Lagrangian formulation and using cartesian coordinates. As a result,
we have compact models and avoid lengthy symbolic transformations that
are typically applied to derive a system of ODEs.

We illustrate by examples in 2D (such as the Andrews Squeezer
Mechanism, one of the MBS Benchmark problems) and 3D, and report
results of numerical solution by this method, with animations.

Joint work with J. Pryce, Cardiff University, UK



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McMaster University - School of Computational Science and Engineering