A new variational principle for mechanical systems subject to holonomic constraints is presented. The newly proposed GGL principle is closely related to the often used Gear-Gupta-Leimkuhler (GGL) stabilization of the differential–algebraic equations governing the motion of constrained mechanical systems. The GGL variational principle relies on an extension of the Livens principle (sometimes also referred to as Hamilton–Pontryagin principle) to mechanical systems subject to holonomic constraints. In contrast to the original GGL stabilization, the new approach facilitates the design of structure-preserving integrators. In particular, new variational integrators are presented, which result from the direct discretization of the GGL variational principle. These variational integrators are symplectic and conserve momentum maps in the case of systems with symmetry. In addition to that, a new energy–momentum scheme is developed, which results from the discretization of the Euler–Lagrange equations pertaining to the GGL variational principle. The numerical properties of the newly devised schemes are investigated in representative examples of constrained mechanical systems.
We present an extension of the Livens variational principle (sometimes also referred to as Hamilton-Pontryagin principle) to mechanical systems subject to holonomic constraints. The newly proposed principle embodies an index reduction in the spirit of the often-applied GGL stabilization and thus may be termed "GGL principle". The Euler-Lagrange equations of the GGL principle assume the form of differential-algebraic equations (DAEs) with differentiation index two. In contrast to the original GGL-DAEs, the present formulation fits into the Hamiltonian framework of mechanics. Therefore, the GGL principle facilitates the design of symplectic integrators. In particular, it offers the possibility to construct variational integrators. This is illustrated with the development of a new first-order scheme which is symplectic by design. The numerical properties of the newly devised scheme are investigated in a representative example of a constrained mechanical system.
We present an extension of the Livens variational principle (sometimes also referred to as Hamilton-Pontryagin principle) to mechanical systems subject to holonomic constraints. The newly proposed principle embodies an index reduction in the spirit of the often-applied GGL stabilization and thus may be termed “GGL principle”. The Euler-Lagrange equations of the GGL principle assume the form of differential-algebraic equations (DAEs) with differentiation index two. In contrast to the original GGL-DAEs, the present formulation fits into the Hamiltonian framework of mechanics. Therefore, the GGL principle facilitates the design of structure-preserving integrators. In particular, it offers the possibility to construct variational integrators. This is illustrated with the development of a new first-order scheme which is symplectic by design. The numerical properties of the newly devised scheme are investigated for representative examples of constrained mechanical systems.
The Python package mechkit is a toolkit for researchers in the field of continuum mechanics and material modeling. mechkit contains methods and operators for elementary tasks concerning tensor algebra and tensor notation, e.g., linear mapping, base transformations and active as well as passive transformations of tensors.
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