## DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS

Citations: | 13 - 10 self |

### BibTeX

@MISC{Leok_discretehamiltonian,

author = {Melvin Leok and Jingjing Zhang},

title = {DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS},

year = {}

}

### OpenURL

### Abstract

Abstract. We derive a variational characterization of the exact discrete Hamiltonian, which is a Type II generating function for the exact flow of a Hamiltonian system, by considering a Legendre transformation of Jacobi’s solution of the Hamilton–Jacobi equation. This provides an exact correspondence between continuous and discrete Hamiltonian mechanics, which arise from the continuous and discrete-time Hamilton’s variational principle on phase space, respectively. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators, which include the symplectic partitioned Runge–Kutta methods. This extends the framework of variational integrators to Hamiltonian systems with degenerate Hamiltonians, for which the standard theory of Lagrangian variational integrators cannot be applied. We also characterize the group invariance properties of discrete Hamiltonians which lead to a discrete Noether’s theorem. 1.

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Citation Context ...g functions. The discrete Hamiltonian perspective allows one to avoid some of the technical difficulties associated with the singularity associated with Type I generating functions at time t = 0 (see =-=[19]-=-, p. 177). Example 1. To illustrate the difficulties associated with degenerate Hamiltonians, consider H(q, p) = qp, with Legendre transformation given by FH : T ∗ Q → T Q, (q, p) ↦→ (q, ∂H/∂p) = (q, ... |

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Citation Context ...analogues of Lagrangian and Hamiltonian mechanics, which are derived from discrete variational principles, yield a class of geometric numerical integrators [13] referred to as variational integrators =-=[15, 21]-=-. The discrete variational approach to constructing numerical integrators is of interest as they automatically yield methods that are symplectic, and by a backward error analysis, exhibit bounded ener... |

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Citation Context ...2 −4π π−2 2π π−4 π 2 −2π 0 1 2 1 4 1 π π−4 4π 1 2 π−2 2π 1 π 0 1 π−2 2π 2 π π−2 2π π 2 −2π+4 2π 2 −4π π−2 2π 1 π−2 0 π−2 2π 2 π π−2 2π π−2 2π 2 π π−2 2π Example 5. Chebyshev quadrature (see p. 415 of =-=[14]-=-) is designed to approximate weighted integrals of the form ∫ 1 −1 f(x)w(x)dx = b s∑ f(xi) + E[f(x)], with an equally weighted sum of the function values at the quadrature points xi, and an error term... |

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Citation Context ...tion 1.1. Discrete Mechanics. Discrete-time analogues of Lagrangian and Hamiltonian mechanics, which are derived from discrete variational principles, yield a class of geometric numerical integrators =-=[13]-=- referred to as variational integrators [15, 21]. The discrete variational approach to constructing numerical integrators is of interest as they automatically yield methods that are symplectic, and by... |

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Citation Context ...the finite-dimensional function space Cs d , then by interpolation, we have that 1 = ∑s i=1 f(ci)φi(τ) = ∑s i=1 φi(τ). Thus, ∑s i=1 bi = ∫ 1 ∑s 0 i=1 φi(τ)dτ = 1. Partitioned Runge–Kutta order theory =-=[5]-=- states that the condition ∑s i=1 bi = 1 implies that the variational integrator (42) is at least first-order. Therefore, to obtain a consistent method, it is sufficient that the constant function is ... |

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Citation Context ...and truncating. Then, a term-by-term comparison allows one to determine the coefficients in the series expansion of S2, from which one constructs a symplectic map that approximates the exact flow map =-=[7, 11, 24, 9]-=-. However, approximating Jacobi’s solution on the Lagrangian side, or the exact discrete right Hamiltonian S(q0, pT ) in (4), in terms of their variational characterization provides an elegant method ... |

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Citation Context ...and truncating. Then, a term-by-term comparison allows one to determine the coefficients in the series expansion of S2, from which one constructs a symplectic map that approximates the exact flow map =-=[7, 11, 24, 9]-=-. However, approximating Jacobi’s solution on the Lagrangian side, or the exact discrete right Hamiltonian S(q0, pT ) in (4), in terms of their variational characterization provides an elegant method ... |

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Citation Context ...llows one to apply the Galerkin construction of variational integrators to Hamiltonian systems directly, and may potentially generalize to variational integrators for multisymplectic Hamiltonian PDEs =-=[4, 17, 18]-=-. Discrete Lagrangian mechanics is expressed in terms of a discrete Lagrangian, which can be viewed as a Type I generating function of a symplectic map, and discrete Hamiltonian mechanics is naturally... |

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Citation Context ...cal integrators is of interest as they automatically yield methods that are symplectic, and by a backward error analysis, exhibit bounded energy errors for exponentially long times (see, for example, =-=[12]-=-). When the discrete Lagrangian or Hamiltonian is group-invariant, they will yield numerical methods that are momentum preserving. Discrete Hamiltonian mechanics can be derived from discrete Lagrangia... |

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Citation Context ...llows one to apply the Galerkin construction of variational integrators to Hamiltonian systems directly, and may potentially generalize to variational integrators for multisymplectic Hamiltonian PDEs =-=[4, 17, 18]-=-. Discrete Lagrangian mechanics is expressed in terms of a discrete Lagrangian, which can be viewed as a Type I generating function of a symplectic map, and discrete Hamiltonian mechanics is naturally... |

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Citation Context ...al convergence study, and provides a numerical verification of the forth-order accuracy of Cheby4 and GauLe4 methods. In Figure 2, we chose initial conditions (q0, p0) = (2, 2), and the time interval =-=[0, 3]-=-. The global error for the position (q) and momentum (p) components at six step sizes h are plotted on a log-log scale. The global error is given by the difference between the numerical solution and e... |

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Citation Context ...llows one to apply the Galerkin construction of variational integrators to Hamiltonian systems directly, and may potentially generalize to variational integrators for multisymplectic Hamiltonian PDEs =-=[4, 17, 18]-=-. Discrete Lagrangian mechanics is expressed in terms of a discrete Lagrangian, which can be viewed as a Type I generating function of a symplectic map, and discrete Hamiltonian mechanics is naturally... |

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Citation Context ...and truncating. Then, a term-by-term comparison allows one to determine the coefficients in the series expansion of S2, from which one constructs a symplectic map that approximates the exact flow map =-=[7, 11, 24, 9]-=-. However, approximating Jacobi’s solution on the Lagrangian side, or the exact discrete right Hamiltonian S(q0, pT ) in (4), in terms of their variational characterization provides an elegant method ... |

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Citation Context ... an infinite-dimensional function space, with a finite-dimensional function space, and uses numerical quadrature to approximate the integral.14 MELVIN LEOK AND JINGJING ZHANG Let {ψi(τ)} s i=1 , τ ∈ =-=[0, 1]-=-, be a set of basis functions for a s-dimensional function space Cs d . We also choose a numerical quadrature formula with quadrature weights bi, and quadrature points ci, i.e., ∫ 1 f(x)dx ≈ 0 ∑s j=1 ... |

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Citation Context ...alogues of Lie-Poisson reduction. In particular, the constrained variational formulation of continuous Lie-Poisson reduction [6] appears to be related to the Hamilton–Pontryagin variational principle =-=[27]-=-. It would be interesting to develop discrete Lie–Poisson reduction [18] from the Hamiltonian perspective, in the context of the discrete Hamilton– Pontryagin principle [16, 25]. • Extensions to Multi... |

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Citation Context ...diagram above commutes when the Hamiltonian is hyperregular. An added benefit is that such an approach would remain valid even if the Hamiltonian is degenerate, as is the case for point vortices (see =-=[22]-=-, p. 22), and no corresponding Lagrangian formulation exists. The Galerkin construction for Lagrangian variational integrators is attractive, since it provides a general framework for constructing a l... |

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Citation Context ...analogues of Lagrangian and Hamiltonian mechanics, which are derived from discrete variational principles, yield a class of geometric numerical integrators [13] referred to as variational integrators =-=[15, 21]-=-. The discrete variational approach to constructing numerical integrators is of interest as they automatically yield methods that are symplectic, and by a backward error analysis, exhibit bounded ener... |

18 |
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Citation Context ...hich it is impossible to obtain a Lagrangian and apply the method in [21] to derive Hamiltonian variational integrators. The derivation on p. 209 of the book [13], which is analogous to the result in =-=[26]-=-, generalizes the approach in [21] by considering discrete Lagrangian SPRK methods without the restriction that the Runge–Kutta coefficients are obtained from integrals of Lagrange polynomials. It is ... |

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Citation Context ...l method that is momentum preserving, it is natural to consider discrete analogues of Lie-Poisson reduction. In particular, the constrained variational formulation of continuous Lie-Poisson reduction =-=[6]-=- appears to be related to the Hamilton–Pontryagin variational principle [27]. It would be interesting to develop discrete Lie–Poisson reduction [18] from the Hamiltonian perspective, in the context of... |

12 |
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Citation Context |

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(Show Context)
Citation Context ...amiltonian for a given time interval in terms of discrete Hamiltonians for the subintervals. This can be viewed as the Type II analogue of the discrete Hamilton–Jacobi equation that was introduced in =-=[10]-=-. 3. Discrete Variational Hamiltonian Mechanics 3.1. Discrete Type II Hamilton’s Variational Principle in Phase Space. The Lagrangian formulation of discrete variational mechanics is based on a discre... |

5 | Discrete Hamilton–Jacobi theory
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(Show Context)
Citation Context ...Equation. A discrete analogue of the Hamilton–Jacobi equation was first introduced in [10], and the connections to discrete Hamiltonian mechanics, and discrete optimal control theory were explored in =-=[23]-=-. In essence, the discrete Hamilton–Jacobi equation therein can be viewed as a composition theorem that relates the discrete Hamiltonians that generate the maps over subintervals, with the discrete La... |

4 | Discrete Hamilton–Pontryagin mechanics and generating functions on Lie groupoids
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(Show Context)
Citation Context ...agin variational principle [27]. It would be interesting to develop discrete Lie–Poisson reduction [18] from the Hamiltonian perspective, in the context of the discrete Hamilton– Pontryagin principle =-=[16, 25]-=-. • Extensions to Multisymplectic Hamiltonian PDEs. Multisymplectic integrators have been developed in the setting of Lagrangian variational integrators [17], and Hamiltonian multisymplectic integrato... |

3 | Discrete Dirac structures and variational discrete Dirac mechanics
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(Show Context)
Citation Context ...elds discrete Hamiltonian mechanics [15]. Alternatively, the second-order curve condition can be imposed using Lagrange multipliers, and this corresponds to the discrete Hamilton–Pontryagin principle =-=[16]-=-. In contrast to the prior literature on discrete Hamiltonian mechanics, which typically start from the Lagrangian setting, we will focus on constructing Hamiltonian variational integrators from the H... |

2 |
Discrete variational integrators and optimal control theory
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(Show Context)
Citation Context ...nd (9), and applying (5) yields (10) ∂S2 ∂t = ˙p(t)q(t) + H(q(t), p(t)) − ˙p(t)∂S2 ∂p ( ) ∂S2 = H(q(t), p(t)) = H , p . ∂p □ The Type II Hamilton–Jacobi equation also appears on p. 201 of [13] and in =-=[8]-=-. However, this equation has generally been used in the construction of symplectic integrators based on Type II generating functions by considering a series expansion of S2 in powers of t, substitutin... |

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