## Hamilton–Pontryagin integrators on Lie groups (2007)

Citations: | 21 - 6 self |

### BibTeX

@TECHREPORT{Bou-rabee07hamilton–pontryaginintegrators,

author = {Nawaf Bou-rabee and Jerrold E. Marsden},

title = {Hamilton–Pontryagin integrators on Lie groups},

institution = {},

year = {2007}

}

### OpenURL

### Abstract

Abstract In this paper, structure-preserving time-integrators for rigid body-type mechanical

### Citations

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Citation Context ...rategyFound Comput Math yields a Lagrangian analog of a well-known class of symplectic partitioned Runge– Kutta methods including the Lobatto IIIA-IIIB pair which generalize to higher-order accuracy =-=[8, 23, 32]-=-. In the Lie group context, one can generalize this strategy using either constrained or generalized coordinates. To use constrained coordinates, one treats the system as a holonomically constrained m... |

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Citation Context ... coadjoint action on SO(3) and a numerical estimate of the gradient of the Hamiltonian. Variational integration techniques have been used to derive structure-preserving integrators on Lie groups; see =-=[2, 3, 21, 25, 33]-=-. Moser and Veselov derived a variational integrator for the free rigid body by embedding SO(3) in the linear space of 3 × 3 matrices, R 9 , and using Lagrange multipliers to constrain the matrices to... |

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Citation Context ... coadjoint action on SO(3) and a numerical estimate of the gradient of the Hamiltonian. Variational integration techniques have been used to derive structure-preserving integrators on Lie groups; see =-=[2, 3, 21, 25, 33]-=-. Moser and Veselov derived a variational integrator for the free rigid body by embedding SO(3) in the linear space of 3 × 3 matrices, R 9 , and using Lagrange multipliers to constrain the matrices to... |

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Citation Context ... coadjoint action on SO(3) and a numerical estimate of the gradient of the Hamiltonian. Variational integration techniques have been used to derive structure-preserving integrators on Lie groups; see =-=[2, 3, 21, 25, 33]-=-. Moser and Veselov derived a variational integrator for the free rigid body by embedding SO(3) in the linear space of 3 × 3 matrices, R 9 , and using Lagrange multipliers to constrain the matrices to... |

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Citation Context ...ar space using holonomic constraints. Coadjoint-orbit and energy preserving methods of the Simo and Wong type that further preserve the symplectic structure were developed for SO(3) by Lewis and Simo =-=[16, 17]-=-. This was done by defining a one-parameter family of coadjoint and energy-preserving algorithms of the Simo and Wong type in which the free parameter is a functional. The function was specified so th... |

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Citation Context ...nting that variational integrators correctly computeFound Comput Math other statistical quantities in long-time simulations. For example, in a simulation of a coupled spring-mass lattice, Lew et al. =-=[15]-=- found that variational integrators accurately compute the time-averaged instantaneous temperature (mean kinetic energy over all particles) over long-time intervals, whereas RK4 exhibits a artificial ... |

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Citation Context ...tegies available to derive structure-preserving Lie group integrators; some of these are discussed here. One strategy involves generalizations of the classical Newmark algorithms to Lie groups due to =-=[30, 31]-=-. It was not apparent that the proposed Lie–Newmark methods had the necessary structure-preserving properties. In fact, Simo and Wong proposed another set of algorithms which preserve momentum by usin... |

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Citation Context ... an equivalence is established between critical points of s and S. IftheLagrangian is left-invariant, it is shown that this principle unifies the system’s Lie– Poisson and Euler–Poincaré descriptions =-=[6, 22]-=-. Since the reconstruction equation is a differential equation on a Lie group, one cannot directly discretize it by an RK method. However, one can discretize it using an s-stage Runge–Kutta–Munthe-Kaa... |

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Citation Context ...tegies available to derive structure-preserving Lie group integrators; some of these are discussed here. One strategy involves generalizations of the classical Newmark algorithms to Lie groups due to =-=[30, 31]-=-. It was not apparent that the proposed Lie–Newmark methods had the necessary structure-preserving properties. In fact, Simo and Wong proposed another set of algorithms which preserve momentum by usin... |

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Citation Context ...p from the Lie algebra to the Lie group. These techniques were applied to bodies with attitude-dependent potentials, discrete optimal control of rigid bodies, and extended to higher-order accuracy in =-=[13, 14]-=-. Bobenko and Suris [2] considered a more general case where the symmetry group is a subgroup of the Lie group G in the context of semidirect Euler–Poincaré theory (see [9]). They did this by writing ... |

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Citation Context ...s paper is concerned with efficient, structure-preserving time integrators for mechanical systems whose configuration space is a Lie group, based on the Hamilton–Pontryagin (HP) variational principle =-=[11, 12, 20, 34, 35]-=-. This HP principle has many attractive theoretical properties; for instance, how it handles degenerate Lagrangian systems. The present paper shows that the HP viewpoint also provides a practical way ... |

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Citation Context ...s paper is concerned with efficient, structure-preserving time integrators for mechanical systems whose configuration space is a Lie group, based on the Hamilton–Pontryagin (HP) variational principle =-=[11, 12, 20, 34, 35]-=-. This HP principle has many attractive theoretical properties; for instance, how it handles degenerate Lagrangian systems. The present paper shows that the HP viewpoint also provides a practical way ... |

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Citation Context ...They also numerically demonstrated the method’s good performance crediting it to third-order accuracy in the discrete approximation to the Lie–Poisson structure. In related work, Mclachlan and Scovel =-=[24]-=- construct reduced, coadjoint-orbit preserving integrators by reducing G-equivariant integrators on T ∗ G obtained by embedding G in a linear space using holonomic constraints. Coadjoint-orbit and ene... |

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24 | Lie group variational integrators for the full body problem
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Citation Context ...p from the Lie algebra to the Lie group. These techniques were applied to bodies with attitude-dependent potentials, discrete optimal control of rigid bodies, and extended to higher-order accuracy in =-=[13, 14]-=-. Bobenko and Suris [2] considered a more general case where the symmetry group is a subgroup of the Lie group G in the context of semidirect Euler–Poincaré theory (see [9]). They did this by writing ... |

22 | Ratiu [1998] The Euler-Poincare equations and semi direct products with applications to continuum theories, Adv - Holm, Marsden, et al. |

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Citation Context ...s paper is concerned with efficient, structure-preserving time integrators for mechanical systems whose configuration space is a Lie group, based on the Hamilton–Pontryagin (HP) variational principle =-=[11, 12, 20, 34, 35]-=-. This HP principle has many attractive theoretical properties; for instance, how it handles degenerate Lagrangian systems. The present paper shows that the HP viewpoint also provides a practical way ... |

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Citation Context ...rategyFound Comput Math yields a Lagrangian analog of a well-known class of symplectic partitioned Runge– Kutta methods including the Lobatto IIIA-IIIB pair which generalize to higher-order accuracy =-=[8, 23, 32]-=-. In the Lie group context, one can generalize this strategy using either constrained or generalized coordinates. To use constrained coordinates, one treats the system as a holonomically constrained m... |

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Citation Context ... set of algorithms which preserve momentum by using the coadjoint action on SO(3) to advance the flow. Such integrators will be referred to as coadjoint-orbit preserving methods. Later, Austin et al. =-=[1]-=- showed that the midpoint rule member of the Lie–Newmark family with a Cayley reconstruction procedure was, in fact, a coadjoint-orbit preserving method for SO(3). They also numerically demonstrated t... |

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Citation Context ... an equivalence is established between critical points of s and S. IftheLagrangian is left-invariant, it is shown that this principle unifies the system’s Lie– Poisson and Euler–Poincaré descriptions =-=[6, 22]-=-. Since the reconstruction equation is a differential equation on a Lie group, one cannot directly discretize it by an RK method. However, one can discretize it using an s-stage Runge–Kutta–Munthe-Kaa... |

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Citation Context ...derivative: δ − δ∗ Dskew(g) · δ = 2 = (δg−1g) − (δg−1g) ∗ . 2 By definition of the right-trivialized tangent of τ −1 , it then follows that dskew(x)(y) = yτ(x) − (yτ(x))∗ . (4.27) 2 Cardoso and Leite =-=[5]-=- obtained the following proposition that explicitly determines τ(ξ). Moreover, they give necessary and sufficient conditions for its existence.Found Comput Math Proposition 4.10 Given ξ ∈ so(n), a sp... |

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Citation Context ...size h = 0.05, RK4 corrupts chaotic invariant sets while the lower-order accurate VE method preserves the structure of the benchmark their virtual work [23]), as well as discrete optimal control (see =-=[19]-=- and references therein). Altogether, this description of mechanics stands as a self-contained theory of mechanics akin to Hamiltonian, Lagrangian, or Newtonian mechanics. One of the distinguishing fe... |

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Citation Context ...rategyFound Comput Math yields a Lagrangian analog of a well-known class of symplectic partitioned Runge– Kutta methods including the Lobatto IIIA-IIIB pair which generalize to higher-order accuracy =-=[8, 23, 32]-=-. In the Lie group context, one can generalize this strategy using either constrained or generalized coordinates. To use constrained coordinates, one treats the system as a holonomically constrained m... |

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Citation Context ...ng map defined a transformation which preserves the canonical symplectic form. Endowing coadjoint-orbit preserving methods with energy-preserving properties was also the subject of Engø and Faltinsen =-=[7]-=-. Specifically, they introduced integrators of the Runge–Kutta–Munthe-Kaas type that preserved coadjoint orbits and energy using the coadjoint action on SO(3) and a numerical estimate of the gradient ... |

5 |
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Citation Context ...ar space using holonomic constraints. Coadjoint-orbit and energy preserving methods of the Simo and Wong type that further preserve the symplectic structure were developed for SO(3) by Lewis and Simo =-=[16, 17]-=-. This was done by defining a one-parameter family of coadjoint and energy-preserving algorithms of the Simo and Wong type in which the free parameter is a functional. The function was specified so th... |

5 | Rowley C W and Melli-Huber J 2005 Locomotion of articulated bodies in a perfect fluid - Kanso, Marsden |