
@article{kinon_ggl_2023,
  author  = {Kinon, Philipp L. and Betsch, P. and Schneider, S.},
  title   = {The {GGL} variational principle for constrained mechanical systems},
  journal = {Multibody System Dynamics},
  volume  = {57},
  pages   = {211--236},
  year    = {2023},
  doi     = {10.1007/s11044-023-09889-6}
}

@article{betsch2001constrained,
  title     = {Constrained integration of rigid body dynamics},
  author    = {Betsch, P and Steinmann, P},
  journal   = {Computer methods in applied mechanics and engineering},
  volume    = {191},
  number    = {3-5},
  pages     = {467--488},
  year      = {2001},
  publisher = {Elsevier}
}

@article{kinon_2024_conserving,
  title    = {Conserving Integration of Multibody Systems with Singular and Non-Constant Mass Matrix Including Quaternion-Based Rigid Body Dynamics},
  author   = {Kinon, Philipp L. and Betsch, Peter},
  year     = {2024},
  journal  = {Multibody System Dynamics},
  volume   = {63},
  number   = {1},
  pages    = {303--340},
  issn     = {1573-272X},
  doi      = {10.1007/s11044-024-10001-9},
  abstract = {Mechanical systems with singular and/or configuration-dependent mass matrix can pose difficulties to Hamiltonian formulations, which are the standard choice for the design of energy-momentum conserving time integrators. In this work, we derive a structure-preserving time integrator for constrained mechanical systems based on a mixed variational approach. Livens' principle (or sometimes called Hamilton--Pontryagin principle) features independent velocity and momentum quantities and circumvents the need to invert the mass matrix. In particular, we take up the description of rigid body rotations using unit quaternions. Using Livens' principle, a new and comparatively easy approach to the simulation of these problems is presented. The equations of motion are approximated by using (partitioned) midpoint discrete gradients, thus generating a new energy-momentum conserving integration scheme for mechanical systems with singular and/or configuration-dependent mass matrix. The derived method is second-order accurate and algorithmically preserves a generalized energy function as well as the holonomic constraints and momentum maps corresponding to symmetries of the system. We study the numerical performance of the newly devised scheme in representative examples for multibody and rigid body dynamics.}
}

@article{betsch_2009_rigid,
  title      = {Rigid Body Dynamics in Terms of Quaternions: {{Hamiltonian}} Formulation and Conserving Numerical Integration},
  shorttitle = {Rigid Body Dynamics in Terms of Quaternions},
  author     = {Betsch, Peter and Siebert, Ralf},
  year       = {2009},
  month      = jul,
  journal    = {International Journal for Numerical Methods in Engineering},
  volume     = {79},
  number     = {4},
  pages      = {444--473},
  issn       = {00295981, 10970207},
  doi        = {10.1002/nme.2586},
  urldate    = {2022-01-17},
  abstract   = {In the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unitlength constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential-algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non-singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion-based energy--momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion-based variational integrator, a director-based energy--momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations. Copyright q 2009 John Wiley \& Sons, Ltd.},
  langid     = {english},
  file       = {/Users/philipp/Zotero/storage/EBCUGGU9/Betsch and Siebert - 2009 - Rigid body dynamics in terms of quaternions Hamil.pdf;/Users/philipp/Zotero/storage/XQ99H3AE/Betsch and Siebert - 2009 - Rigid body dynamics in terms of quaternions Hamiltonian formulation and conserving numerical integr.pdf}
}


@article{kinon_structure_2023,
  author  = {Kinon, Philipp L. and Betsch, P. and Schneider, S.},
  title   = {Structure-preserving integrators based on a new variational principle for constrained mechanical systems},
  journal = {Nonlinear Dynamics},
  year    = {2023},
  doi     = {10.1007/s11071-023-08522-7},
  note    = {doi: 10.1007/s11071-023-08522-7}
}

@article{gonzalez_time_1996,
  title   = {Time integration and discrete {Hamiltonian} systems},
  volume  = {6},
  journal = {Journal of Nonlinear Science},
  author  = {Gonzalez, O},
  year    = {1996},
  doi     = {10.1007/BF02440162},
  pages   = {449--467}
}

@book{duindam_modeling_2009,
  title      = {Modeling and {control} of {complex} {physical} {systems}: {The} {port}-{Hamiltonian} {approach}},
  isbn       = {978-3-642-03195-3},
  shorttitle = {Modeling and {Control} of {Complex} {Physical} {Systems}},
  author     = {Duindam, Vincent and Macchelli, Alessandro and Stramigioli, Stefano and Bruyninckx, Herman},
  year       = {2009},
  address    = {Berlin},
  doi        = {10.1007/978-3-642-03196-0},
  publisher  = {Springer}
}

@book{hairer_geometric_2006,
  title     = {Geometric numerical integration},
  publisher = {Springer},
  author    = {Hairer, Ernst and Lubich, Christian and Wanner, Gerhard},
  year      = {2006},
  address   = {Berlin},
  doi       = {10.1007/3-540-30666-8}
}


@article{livens_hamiltons_1919,
  title   = {{IX}. -- {O}n {H}amilton's Principle and the Modified Function in Analytical Dynamics},
  volume  = {39},
  journal = {Proceedings of the Royal Society of Edinburgh},
  author  = {Livens, G. H.},
  year    = {1920},
  pages   = {113--119},
  doi     = {10.1017/S0370164600018617}
}

@article{lew2016brief,
  title     = {A brief introduction to variational integrators},
  author    = {Lew, Adri{\'a}n J and Mata A, Pablo},
  journal   = {Structure-preserving integrators in nonlinear structural dynamics and flexible multibody dynamics},
  pages     = {201--291},
  year      = {2016},
  publisher = {Springer}
}


@article{marsden_discrete_2001,
  title   = {Discrete mechanics and variational integrators},
  volume  = {10},
  journal = {Acta Numerica},
  author  = {Marsden, J. E. and West, M.},
  year    = {2001},
  doi     = {10.1017/S096249290100006X},
  pages   = {357--514}
}

@article{leyendecker_variational_2008,
  title   = {Variational integrators for constrained dynamical systems},
  volume  = {88},
  number  = {9},
  journal = {ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik},
  author  = {Leyendecker, S. and Marsden, J.E. and Ortiz, M.},
  year    = {2008},
  pages   = {677--708},
  doi     = {10.1002/zamm.200700173}
}

@book{kunkel_differential-algebraic_2006,
  series    = {{EMS} textbooks in mathematics},
  title     = {Differential-algebraic equations: analysis and numerical solution},
  publisher = {European Mathematical Society},
  author    = {Kunkel, Peter and Mehrmann, Volker},
  year      = {2006},
  doi       = {10.4171/017}
}

@book{leimkuhler_simulating_2005,
  series    = {Cambridge {Monographs} on {Applied} and {Computational} {Mathematics}},
  title     = {Simulating {Hamiltonian} {Dynamics}},
  publisher = {Cambridge University Press},
  author    = {Leimkuhler, Benedict and Reich, Sebastian},
  year      = {2005},
  doi       = {10.1017/CBO9780511614118}
}
