arXiv subjects

Melvin Leok

Publications and source records attributed to Melvin Leok.

At least 55 records · Page 3Linked to original sources

arXiv2016

Embedding-Based Interpolation on the Special Orthogonal Group

We study schemes for interpolating functions that take values in the special orthogonal group $SO(n)$. Our focus is on interpolation schemes obtained by embedding $SO(n)$ in a linear space, interpolating in the linear space, and mapping the result onto $SO(n)$ via the closest point projection. The resulting interpolants inherit both the order of accuracy and the regularity of the underlying interpolants on the linear space. The values and derivatives of the interpolants admit efficient evaluation via either explicit formulas or iterative algorithms, which we detail for two choices of embeddings: the embedding of $SO(n)$ in the space of $n \times n$ matrices and, when $n=3$, the identification of $SO(3)$ with the set of unit quaternions. Along the way, we point out a connection between these interpolation schemes and geodesic finite elements. We illustrate the utility of these interpolation schemes by numerically computing minimum acceleration curves on $SO(n)$, a task which is handled naturally with $SO(n)$-valued finite elements having $C^1$-continuity.

math.NA↗

arXiv2016

Computing the Fréchet Derivative of the Polar Decomposition

We derive iterative methods for computing the Fréchet derivative of the map which sends a full-rank matrix $A$ to the factor $U$ in its polar decomposition $A=UH$, where $U$ has orthonormal columns and $H$ is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fréchet derivative of the polar decomposition to the matrix sign function $\mathrm{sign}(X) = X (X^2)^{-1/2}$ applied to a certain block matrix $X$.

math.NA↗

arXiv2016

Variational integrators for interconnected Lagrange-Dirac systems

Interconnected systems are an important class of mathematical models, as they allow for the construction of complex, hierarchical, multiphysics, and multiscale models by the interconnection of simpler subsystems. Lagrange--Dirac mechanical systems provide a broad category of mathematical models that are closed under interconnection, and in this paper, we develop a framework for the interconnection of discrete Lagrange--Dirac mechanical systems, with a view towards constructing geometric structure-preserving discretizations of interconnected systems. This work builds on previous work on the interconnection of continuous Lagrange--Dirac systems (Jacobs and Yoshimura 2014) and discrete Dirac variational integrators (Leok and Ohsawa 2011). We test our results by simulating some of the continuous examples given in Jacobs and Yoshimura 2014.

math.NA↗

arXiv2016

Interpolation on Symmetric Spaces via the Generalized Polar Decomposition

We construct interpolation operators for functions taking values in a symmetric space -- a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition -- a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically.

math.NA↗

arXiv2015

Global Formulations of Lagrangian and Hamiltonian Mechanics on Two-Spheres

This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolving on the product of an arbitrary number of two-spheres. Four types of Euler-Lagrange equations and Hamilton's equations are developed in a coordinate-free fashion on two-spheres, without relying on local parameterizations that may lead to singularities and cumbersome equations of motion. The proposed intrinsic formulations of Lagrangian and Hamiltonian dynamics are novel in that they incorporate the geometry of two-spheres, resulting in equations of motion that are expressed compactly, and they are useful in analysis and computation of the global dynamics.

math.DS↗

arXiv2014

Geometric Representations of Whitney Forms and their Generalization to Minkowski Spacetime

In this work, we present two alternative yet equivalent representation formulae for Whitney forms that are valid for any choice of coordinates, and generalizes the original characterization of Whitney forms in Whitney (1957) that requires the use of barycentric coordinates. In addition, we demonstrate that these formulae appropriately generalize the notion of Whitney forms and barycentric coordinates to Minkowski spacetime, and naturally to any other flat pseudo-Riemannian manifold. These alternate forms are related to each other through a duality between the exterior algebras on vectors and covectors. In addition, these two formulae have a geometrically intuitive interpretation which provide interesting insights into their structure. Furthermore, we obtain an explicit characterization of the Hodge dual of the space of Whitney forms on Minkowski spacetime, and this opens the door to treating the theory of classical electromagnetism in a fully covariant fashion through the combination of multisymplectic variational integration and spacetime finite-element exterior calculus (FEEC) techniques. We conclude the paper with the results from an $\mathbb{R}^{1+1}$ wave equation simulation, as a proof-of-concept of the spacetime formulation described herein.

math.NA↗

arXiv2014

Lie Group Spectral Variational Integrators

We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic, momentum preserving, and can be constructed to be of arbitrarily high-order, or can be made to converge geometrically. Furthermore, these methods are stable and accurate for very large time steps. We demonstrate the construction of one such variational integrator for the rigid body, and discuss how this construction could be generalized to other related Lie group problems. We close with several numerical examples which demonstrate our claims, and discuss further extensions of our work.

math.NA↗

arXiv2013

Symplectic Semiclassical Wave Packet Dynamics

The paper gives a symplectic-geometric account of semiclassical Gaussian wave packet dynamics. We employ geometric techniques to "strip away" the symplectic structure behind the time-dependent Schrödinger equation and incorporate it into semiclassical wave packet dynamics. We show that the Gaussian wave packet dynamics is a Hamiltonian system with respect to the symplectic structure, apply the theory of symplectic reduction and reconstruction to the dynamics, and discuss dynamic and geometric phases in semiclassical mechanics. A simple harmonic oscillator example is worked out to illustrate the results: We show that the reduced semiclassical harmonic oscillator dynamics is completely integrable by finding the action--angle coordinates for the system, and calculate the associated dynamic and geometric phases explicitly. We also propose an asymptotic approximation of the potential term that provides a practical semiclassical correction term to the approximation by Heller. Numerical results for a simple one-dimensional example show that the semiclassical correction term realizes a semiclassical tunneling.

math-ph↗

arXiv2013

A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge--Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.

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arXiv2013

Generating Functionals and Lagrangian PDEs

We introduce the concept of Type-I/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 Hamilton-Jacobi 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 so-called multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II 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.

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arXiv2013

The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories

We introduce a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit Euler-Lagrange equations for fields obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Lastly, we show a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields, Maxwell's equations, and elastostatics.

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arXiv2012

Spectral Variational Integrators

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.

math.NA↗

arXiv2012

Dirac Structures and Hamilton-Jacobi Theory for Lagrangian Mechanics on Lie Algebroids

This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\T^EE^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.

math-ph↗

arXiv2012

Dynamics and Control of a Chain Pendulum on a Cart

A geometric form of Euler-Lagrange equations is developed for a chain pendulum, a serial connection of $n$ rigid links connected by spherical joints, that is attached to a rigid cart. The cart can translate in a horizontal plane acted on by a horizontal control force while the chain pendulum can undergo complex motion in 3D due to gravity. The configuration of the system is in $(\Sph^2)^n \times \Re^2$. We examine the rich structure of the uncontrolled system dynamics: the equilibria of the system correspond to any one of $2^n$ different chain pendulum configurations and any cart location. A linearization about each equilibrium, and the corresponding controllability criterion is provided. We also show that any equilibrium can be asymptotically stabilized by using a proportional-derivative type controller, and we provide a few numerical examples.

math.OC↗

arXiv2012

Hamel's Formalism and Variational Integrators on a Sphere

This paper discusses Hamel's formalism and its applications to structure-preserving integration of mechanical systems. It utilizes redundant coordinates in order to eliminate multiple charts on the configuration space as well as nonphysical artificial singularities induced by local coordinates, while keeping the minimal possible degree of redundancy and avoiding integration of differential-algebraic equations.

math.NA↗

arXiv2012

Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints

We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton-Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic Hamilton-Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac-Hamilton-Jacobi equation can be used to exactly integrate the equations of motion.

math-ph↗

arXiv2011

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.

math.SG↗

arXiv2011

Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3)

This paper provides nonlinear tracking control systems for a quadrotor unmanned aerial vehicle (UAV) that are robust to bounded uncertainties. A mathematical model of a quadrotor UAV is defined on the special Euclidean group, and nonlinear output-tracking controllers are developed to follow (1) an attitude command, and (2) a position command for the vehicle center of mass. The controlled system has the desirable properties that the tracking errors are uniformly ultimately bounded, and the size of the ultimate bound can be arbitrarily reduced by control system parameters. Numerical examples illustrating complex maneuvers are provided.

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