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DISCRETE HAMILTON–JACOBI THEORY
"... Abstract. We develop a discrete analogue of the Hamilton–Jacobi theory in the framework of the discrete Hamiltonian mechanics. We first reinterpret the discrete Hamilton–Jacobi equation derived by Elnatanov and Schiff in the language of discrete mechanics. The resulting discrete Hamilton– Jacobi equ ..."
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Cited by 5 (5 self)
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Abstract. We develop a discrete analogue of the Hamilton–Jacobi theory in the framework of the discrete Hamiltonian mechanics. We first reinterpret the discrete Hamilton–Jacobi equation derived by Elnatanov and Schiff in the language of discrete mechanics. The resulting discrete Hamilton– Jacobi equation is discrete only in time, and is shown to recover the Hamilton–Jacobi equation in the continuoustime limit. The correspondence between discrete and continuous Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobi’s solution to the Hamilton–Jacobi equation. We also prove a discrete analogue of the geometric Hamilton–Jacobi theorem of Abraham and Marsden. These results are readily applied to discrete optimal control setting, and some wellknown results in discrete optimal control theory, such as the Bellman equation (discretetime Hamilton–Jacobi–Bellman equation) of dynamic programming, follow immediately. We also apply the theory to discrete linear Hamiltonian systems, and show that the discrete Riccati equation follows as a special case of the discrete Hamilton–Jacobi equation.
PROLONGATIONCOLLOCATION VARIATIONAL INTEGRATORS
"... Abstract. We introduce a novel technique for constructing higherorder variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our construction of the discrete Lagrangian adopts Herm ..."
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Cited by 4 (3 self)
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Abstract. We introduce a novel technique for constructing higherorder variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our construction of the discrete Lagrangian adopts Hermite interpolation polynomials and the Euler–Maclaurin quadrature formula, and involves applying collocation to the Euler–Lagrange equation and its prolongation. Considerable attention is devoted to the order analysis of the resulting variational integrators in terms of approximation properties of the Hermite polynomials and quadrature errors. A performance comparison is presented on a selection of these integrators. 1.
General techniques for constructing variational integrators
 FRONT. MATH. CHINA
, 2011
"... The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be c ..."
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Cited by 3 (3 self)
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The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton– Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finitedimensional function space or a onestep method. We prove that the properties of the quadrature formula, finitedimensional function space, and underlying onestep method determine the order of accuracy and momentumconservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.
Generating Functionals and Lagrangian PDEs
, 2011
"... Dedicated to the memory of Jerrold E. Marsden. We introduce the concept of TypeI/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi’s solution to the HamiltonJacobi equation for field theories, and w ..."
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Cited by 2 (2 self)
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Dedicated to the memory of Jerrold E. Marsden. We introduce the concept of TypeI/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi’s solution to the HamiltonJacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the socalled multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi’s solution, and we show that this functional is a TypeII generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges. 1
SPECTRAL VARIATIONAL INTEGRATORS
"... Abstract. In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prov ..."
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Cited by 1 (1 self)
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Abstract. In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are in a certain sense optimal, converging at the same rate as the best possible approximation in a certain function space. We further prove that certain geometric invariants also converge at an optimal rate, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence. 1.
A novel formulation of point vortex dynamics on the sphere
, 2012
"... geometrical and numerical aspects ..."
Advance Access publication on November 15, 2011 Prolongation–collocation variational integrators
, 2011
"... We introduce a novel technique for constructing higherorder variational integrators for Hamiltonian systems of ordinary differential equations. In the construction of the discrete Lagrangian we adopt Hermite interpolation polynomials and the Euler–Maclaurin quadrature formula and apply collocation ..."
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We introduce a novel technique for constructing higherorder variational integrators for Hamiltonian systems of ordinary differential equations. In the construction of the discrete Lagrangian we adopt Hermite interpolation polynomials and the Euler–Maclaurin quadrature formula and apply collocation to the Euler–Lagrange equation and its prolongation. Considerable attention is devoted to the order analysis of the resulting variational integrators in terms of approximation properties of the Hermite polynomials and quadrature errors. In particular, the order of the variational integrator can be computed a priori based on the quadrature error estimate. The analysis in the paper is straightforward compared to the order theory for Runge–Kutta methods. Finally, a performance comparison is presented on a selection of these integrators.
c ○ 2011 Society for Industrial and Applied Mathematics DISCRETE HAMILTON–JACOBI THEORY ∗
"... Abstract. We develop a discrete analogue of Hamilton–Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton–Jacobi equation is discrete only in time. We describe a discrete analogue of Jacobi’s solution and also prove a discrete version of the geometric Ham ..."
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Abstract. We develop a discrete analogue of Hamilton–Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton–Jacobi equation is discrete only in time. We describe a discrete analogue of Jacobi’s solution and also prove a discrete version of the geometric Hamilton–Jacobi theorem. The theory applied to discrete linear Hamiltonian systems yields the discrete Riccati equation as a special case of the discrete Hamilton–Jacobi equation. We also apply the theory to discrete optimal control problems, and recover some wellknown results, such as the Bellman equation (discretetime HJB equation) of dynamic programming and its relation to the costate variable in the Pontryagin maximum principle. This relationship between the discrete Hamilton–Jacobi equation and Bellman equation is exploited to derive a generalized form of the Bellman equation that has controls at internal stages. Key words. Hamilton–Jacobi equation, Bellman equation, dynamic programming, discrete mechanics, discretetime optimal control
VARIATIONAL INTEGRATORS FOR HAMILTONIZABLE NONHOLONOMIC SYSTEMS
"... Abstract. We report on new applications of the Poincaré and Sundman timetransformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamil ..."
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Abstract. We report on new applications of the Poincaré and Sundman timetransformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamiltonian after a suitable time reparameterization). We show how such an application permits the usage of variational integrators for these nonvariational mechanical systems. Examples are given and numerical results are compared to the standard nonholonomic integrator results. Introduction. It is well known that the dynamical equations of motion of unconstrained mechanical systems follow from a variational principle, namely Hamilton’s principle of stationary action [1, 26]. In the 1970s and 1980s several researchers discretized this continuous variational principle and developed the discrete EulerLagrange equations (see [27] and references therein for a historical account). Like
IMA Journal of Numerical Analysis Advance Access published November 15, 2011 IMA Journal of Numerical Analysis Page 1 of 23
, 2011
"... doi:10.1093/imanum/drr042 ..."