## Symplectic Integration Of Constrained Hamiltonian Systems (0)

Citations: | 50 - 10 self |

### BibTeX

@MISC{Leimkuhler_symplecticintegration,

author = {B. Leimkuhler and S. Reich},

title = {Symplectic Integration Of Constrained Hamiltonian Systems},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

. A Hamiltonian system in potential form (H(q; p) = p t M \Gamma1 p=2 + F (q)) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in R n . In this paper, methods which reduce "Hamiltonian differentialalgebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint-invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically. Key words. differential-algebraic equations, constrained Hamiltonian systems, canonical discretization schemes, symplectic methods AMS(MOS) subj...

### Citations

1105 |
Mathematical Methods of Classical Mechanics
- Arnold
- 1978
(Show Context)
Citation Context ...ration of Hamiltonian systems q 0 = +r p H(q; p) p 0 = \Gammar q H(q; p) (1) where q; p 2 R n , and H : R 2n ! R is sufficiently smooth. In applications, typically arising in the context of mechanics =-=[3]-=-, astronomy [3], and molecular dynamics [1], the Hamiltonian H is of the form H(q; p) = p t M \Gamma1 p 2 + V (q) (2) where M is the positive definite mass matrix of the system. This results in the Ha... |

1083 | Application of Lie Groups to Differential Equations - Olver - 1986 |

516 |
Solving ordinary differential equations
- HAIRER, WANNER
- 1991
(Show Context)
Citation Context ... manifoldsM. Furthermore, all known symplectic constraint-preserving integrators suffer from step-size restrictions due to stability bounds, i.e., none of the existing methods is algebraically stable =-=[14]-=-. In this paper we suggest a projection technique that allows to apply symplectic integrators, suitable for the integration of unconstrained systems (1), to the integration of constrained systems as w... |

162 |
Construction of higher order symplectic integrators, Phys
- Yoshida
- 1990
(Show Context)
Citation Context ...mally, the map \Psi h can be considered as the time-h-map of the flow corresponding to a perturbed Hamiltonian vector field x 0 = fx; ~ Hg If \Psi h is a scheme of order , then H = ~ H +O(h ) Yoshida =-=[31]-=- first noticed that for unconstrained systems (1) higher order methods can be constructed by a proper composition of second order symmetric methods such as the implicit midpoint [14] or Verlet [30] me... |

143 | Stabilization of constraints and integrals of motion in dynamical systems - Baumgarte - 1972 |

75 | Symplectic integration of Hamiltonian systems - Channell, Scovel - 1990 |

70 |
Automatic integration of Euler-Lagrange equations with constraints
- Gear, Leimkuhler, et al.
- 1985
(Show Context)
Citation Context ...tions (see, e.g., [9]), constrained systems (8) are called differential-algebraic equations (DAEs) of index three. Discretization schemes suitable for such problems have been derived (see, e.g., [5], =-=[11]-=-). However, none of these schemes can automatically be expected to preserve the symplectic structure on M. Few results have been published so far on the symplectic integration of constrained systems o... |

66 |
Symplectic integrators for Hamiltonian problems: An overview, Acta Numerica 1
- Sanz-Serna
- 1992
(Show Context)
Citation Context ...nst. (5) along solutions. Much recent research has gone into developing numerical discretization schemes that inherit the symplectic structure of the original system (see, e.g., Sanz-Serna's overview =-=[25]-=- on this subject). It has been observed [10], [21] that symplectic methods with fixed step-size possess better long-term stability properties than non-symplectic methods or symplectic methods with var... |

64 | On the numerical integration of ordinary differential equations by symmetric composition methods
- McLachlan
- 1995
(Show Context)
Citation Context ... formulation (8) is constraint-preserving and symplectic. Higher order methods can then be obtained by a proper composition of these first and second order schemes as discussed, e.g., by McLachlan in =-=[19]-=- for unconstrained systems. While these schemes are based on the direct numerical integration of the constrained formulation (8), in Section 4 we make use of the reformulations of (8) as an unconstrai... |

64 | A canonical integration technique
- Ruth
- 1983
(Show Context)
Citation Context ... now the scheme (24) for different choices of the integrator \Psi h and for Hamiltonian functions of the form H(q; p) = p t M \Gamma1 p 2 + V (q) We start with the first order symplectic Euler method =-=[24]-=-. In this case the iteration for the q-variable becomes: q k+1 = q k +hM \Gamma1 p k \Gamma h 2 M \Gamma1 rV (q k ) \Gamma h 2 2 M \Gamma1 G(q k ) tsk subject to 0 = g(q k+1 ) The p-variable is update... |

54 | Backward analysis of numerical integrators and symplectic methods
- Hairer
- 1994
(Show Context)
Citation Context ...is constraint-preserving and symplectic. We also give a backward error analysis of the schemes derived in this paper. Similar to the unconstrained case (see, e.g., Auerbach and Friedman [6] or Hairer =-=[13]-=-), we show that the numerical solutions can be formally interpreted as the exact solution of a certain perturbed Hamiltonian system evaluated at discrete time points. 2. Mathematical Background. In th... |

50 | Symplectic numerical integrators in constrained Hamiltonian systems
- Leimkuhler, Skeel
- 1994
(Show Context)
Citation Context ...icit Runge-Kutta methods. However, numerical experiments [17] indicate that those methods suffer some drawbacks concerning the stability and preservation of the constraints and/or the Hamiltonian. In =-=[18]-=-, Leimkuhler and Skeel showed that the constrained system (8) with the particular Hamiltonian (2) can be integrated directly by proper modifications [2] of the second order Verlet scheme. The resultin... |

42 | Mathematical Methods of Classical Mechanics, Springer–Verlag (New - Arnold - 1980 |

41 | The Numerical Solution of Initial Value Problems in Differential-Algebraic Equations - Brenan, Campbell, et al. - 1989 |

40 | Stability of computational methods for constrained dynamics systems - ASCHER, PETZOLD - 1993 |

37 | Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems - Jay - 1996 |

36 | Runge-Kutta schemes for Hamiltonian systems - Sanz-Serna - 1988 |

34 | Stability of Runge-Kutta methods for trajectory problems - Cooper - 1987 |

32 |
1988] Dynamical Systems III
- Arnold
(Show Context)
Citation Context ...) (6) g : R n ! R m . One way to derive the corresponding equations of motion is to replace the constraints by a potential which grows large when the system deviates from the locus of the constraints =-=[4]-=-, [23]. This can, e.g., be achieved by using the following modified Hamiltonian HC HC (q; p) = H(q; p) + 1 2ffl g(q) t g(q) 0 ! ffl !! 1, with the corresponding equations of motion q 0 = +r p H(q; p) ... |

32 |
The development of variable-step symplectic integrators, with applications to the two-body problem
- Calvo, Sanz-Serna
- 1993
(Show Context)
Citation Context ...ch has gone into developing numerical discretization schemes that inherit the symplectic structure of the original system (see, e.g., Sanz-Serna's overview [25] on this subject). It has been observed =-=[10]-=-, [21] that symplectic methods with fixed step-size possess better long-term stability properties than non-symplectic methods or symplectic methods with varying step-size. It has also been shown that ... |

27 |
Rattle: a velocity version of the shake algorithm for molecular dynamics calculations
- Anderson
- 1983
(Show Context)
Citation Context ... of the constraints and/or the Hamiltonian. In [18], Leimkuhler and Skeel showed that the constrained system (8) with the particular Hamiltonian (2) can be integrated directly by proper modifications =-=[2]-=- of the second order Verlet scheme. The resulting scheme is symplectic and preserves the constraints. More recently it has been shown by Jay [15] and Reich [22] that partitioned Runge-Kutta methods wh... |

25 |
Dangers of multiple-time-step methods
- Biesiadecki, Skeel
- 1993
(Show Context)
Citation Context ...ype of Hamiltonian systems arise, for example, in molecular dynamics simulations [1],[27]. Any other symplectic, second order method for unconstrained problems, such as the multiple-time-step methods =-=[8]-=- etc., can now be generalized to the constrained case along the same lines. This is especially useful whenever one wishes to exploit the special structure of the Hamiltonian H. Based on these second o... |

23 | On the scope of the method of modified equations - Griffiths, Sanz-Serna - 1986 |

19 | Stabilization and projection methods for multibody dynamics - Eich, Führer, et al. - 1990 |

14 |
Canonical methods for Hamiltonian systems
- Okunbor
- 1992
(Show Context)
Citation Context ... gone into developing numerical discretization schemes that inherit the symplectic structure of the original system (see, e.g., Sanz-Serna's overview [25] on this subject). It has been observed [10], =-=[21]-=- that symplectic methods with fixed step-size possess better long-term stability properties than non-symplectic methods or symplectic methods with varying step-size. It has also been shown that these ... |

13 | Lie algebras and canonical integration - Neri - 1988 |

13 | Numerical Analysis of Parameterized Nonlinear Equations - Rheinboldt - 1986 |

5 | Symplectic methods for constrained Hamiltonian systems - Leimkuhler, Reich - 1994 |

5 | Zur Theorie und numerischen Realisierung von Losungmethoden bei Differentialgleichungen mit angekoppelten algebraischen - Mrziglod - 1987 |

4 |
Symplectic integration for macromolecular dynamics
- Skeel, Biesiadecki, et al.
- 1992
(Show Context)
Citation Context ...s with an Hamiltonian H(q; p) = p t M \Gamma1 p 2 + V 1 (q) + 1 ffl V 2 (q) where ffl ? 0 is a small number. This type of Hamiltonian systems arise, for example, in molecular dynamics simulations [1],=-=[27]-=-. Any other symplectic, second order method for unconstrained problems, such as the multiple-time-step methods [8] etc., can now be generalized to the constrained case along the same lines. This is es... |

4 |
Molecular Dynamics with Internal Coordinate Constraints
- Tobias, Brooks
- 1988
(Show Context)
Citation Context ... the computation of rV (q) is rather expensive. The most popular second order scheme for the numerical integration of the corresponding constrained system is the modified Verlet scheme of Section 3.1 =-=[28]-=-. Although it requires the solution of a nonlinear system of equations, only one evaluation of rV (q) is needed per integration step. As pointed out in Section 3.2, based on the second order Verlet sc... |

4 |
Computer experiments in classical fluids, Phys
- Verlet
(Show Context)
Citation Context ...da [31] first noticed that for unconstrained systems (1) higher order methods can be constructed by a proper composition of second order symmetric methods such as the implicit midpoint [14] or Verlet =-=[30]-=- method. The idea is the following one: Let \Psi h denote the time-h-flow defined by a symmetric second order method. Then the composed mapping \Psi c 1 h \Delta \Psi c 2 h \Delta \Psi c 1 h (13) with... |

3 |
Long-time behavior of numerically computed orbits: small and intermediate timestep analysis of one-dimensional systems
- Auerbach, Friedman
(Show Context)
Citation Context ...ty properties than non-symplectic methods or symplectic methods with varying step-size. It has also been shown that these methods preserve the Hamiltonian H to high accuracy over long periods of time =-=[6]-=-. A natural question is what happens when (1) is constrained by algebraic equations on q and/or p. In this paper, we restrict ourselves to the case of holonomic constraints 0 = g(q) (6) g : R n ! R m ... |

3 | Control of constrained Hamiltonian systems and applications to control of constrained robots, Dynamical Systems Approaches to Nonlinear - McClamroch, Bloch - 1988 |

2 |
Motion Under a Strong Constraining
- Rubin, Ungar
- 1957
(Show Context)
Citation Context ... g : R n ! R m . One way to derive the corresponding equations of motion is to replace the constraints by a potential which grows large when the system deviates from the locus of the constraints [4], =-=[23]-=-. This can, e.g., be achieved by using the following modified Hamiltonian HC HC (q; p) = H(q; p) + 1 2ffl g(q) t g(q) 0 ! ffl !! 1, with the corresponding equations of motion q 0 = +r p H(q; p) p 0 = ... |

2 | Symplectic numerical integrators for constrained molecular dynamics, Working Paper - Leimkuhler, Skeel - 1992 |

2 | On the numerical solution of the Euler-Lagrange - Potra, Rheinboldt - 1990 |

1 |
Stabilization of DAEs and Invariant Manifolds, to appear, Num
- Ascher, Chin, et al.
(Show Context)
Citation Context ... equations (see, e.g., [9]), constrained systems (8) are called differential-algebraic equations (DAEs) of index three. Discretization schemes suitable for such problems have been derived (see, e.g., =-=[5]-=-, [11]). However, none of these schemes can automatically be expected to preserve the symplectic structure on M. Few results have been published so far on the symplectic integration of constrained sys... |

1 |
Theory of Matrices
- Lancester
- 1969
(Show Context)
Citation Context ...+O(h ) with HM = H + g tsas in Section 2. 2 Let us assume that the system (30) is discretized by a symplectic Runge-Kutta method with Butcher's tableau [14] c A b t Upon using tensor product notation =-=[16], th-=-e scheme (31) becomes q k+1 = q k + h(b t\Omega I)r p HM �� p k+1 = p k \Gamma h(b t\Omega I)r q HM \Gamma h 2 G(q k ) tsk Q = e\Omega q k + h(A\Omega I)r p HM P = e\Omega p k \Gamma h(A\Omega I)r... |