Discrete Hamel’s Equations

Overview

My research is focused on numerical integration of the equations of motion of mechanical systems. It is rooted in fundamental ideas related to the study of mechanics. The first of these ideas is the use of moving frames to simplify the equations of motion. Euler used body frames to measure the angular velocity components of a rotating rigid body [7]; such frames tremendously simplify the study of rotational dynamics of a rigid body. The general form of the differential equations of motion written with respect to an arbitrary non-coordinate frame has been studied in the work of Hamel [6] (and more recently in [2]) and is referred to as Hamel’s equations; the discretization of these equations has been the goal of my research.

As an application of Hamel’s equations, consider a system of contacting rigid bodies, possibly subject to control. Under such circumstances we may prefer to use multiple frames for capturing dynamics of these bodies. The use of frames in this situation is nearly inevitable as the contact points of the bodies are moving, making modeling the body interaction difficult or even impossible in generalized coordinates. Hamel’s formalism provide a natural framework for incorporating such non-coordinate frames in the study of dynamics of complicated mechanical system.

The differential equations that describe a mechanical system are often difficult or impossible to solve analytically. In such cases numerical integrators that are discrete-time models approximating the underlying differential equations of motion allow for simulation of initial value problems.

There are many methods of deriving such discretizations, each with certain advantages. The approach I take to numerical integration follows the reasoning that since the continuous equations are naturally derived via variation of an action functional (Hamilton’s principle), discretizations of these equations might arise from a similar variation of a discretized action. Such numerical integrators are called variational integrators, see e.g. [12, 14] for details.

Drawing from these ideas, I have worked to answer the following question: given a mechan- ical system in which velocity is measured against some non-coordinate frame (possibly—but not always—a body frame), how can we find variational integrators that approximate Hamel’s equations for the choice of frame?

Variational integrators naturally mirror the features of mechanical systems they approximate. Specifically, they conserve a discrete analogue of symplectic form and, when symmetry is present, preserve momentum. There is an interest in variational integrators in the simulation of relativistic systems [4], computational control theory [5], and computer animation [9]. My work on variational integration of Hamel’s equations provides means of incorporating the method of moving frames into structure-preserving numerical simulation of such systems. A clever choice of a frame can entail significant simplifications in the equations of motion; it is anticipated that these simplifications will carry over to variational integrators constructed with the aid of moving frames.

In the early stages of my research I discovered a variational derivation of Hamel’s equations via an extension of Hamilton’s principle to velocity-momentum space established earlier by Yoshimura and Marsden in [15]; the details of my derivation are published in [1]. Variational integrators on Lie groups were derived using a discretized version of this extended action principle by Bou-Rabee and Marsden in [3]; my method of deriving Hamel’s equations thus indicated a possible route to the variational integrators we have sought. A paper is currently being prepared describing the various versions of discrete Hamel’s equations that were found in the course of my work and investigating their structure-preserving properties.

Hamel’s Equations and Variational Integrators

The configuration of a mechanical system with n degrees of freedom can be represented (locally) by $n$ generalized coordinates $(q)_{i=1}^n$ that are local coordinates on an n-dimensional differentiable manifold $Q$, the configuration space. The dynamics are dictated by a Lagrangian—a function on the tangent bundle (simply put, the position and velocity space) of $Q$; the Lagrangian has physical significance as the difference between kinetic and potential energy.

A coordinate-induced frame is a set of n vector fields that are tangent to the n coordinate lines at each point in our local neighborhood in $Q$ and as such they form a basis of the tangent space at each point in this neighborhood. We often use the notation $(q^i,\dot{q}^i)$ to denote the lift of the local coordinate chart to the tangent bundle; in doing so we assume that the components $\dot{q}^i$ of the velocity vector $\dot{q} \in T_qQ$ are measured against the coordinate-induced frame. When we use this frame as the basis to measure the velocity components the resulting equations of motion of our mechanical system are the well-known Euler–Lagrange equations.

There are many circumstances in which an alternate choice of a frame can make the equations of motion simpler to analyze. Consider the case when our configuration space is a Lie group $G$. We may choose to measure vectors using the associated Lie algebra (the tangent fiber at the group identity, $T_eG$). At each point the coordinate induced basis is transformed by the push-forward of the left or right group action. When the Lagrangian is invariant with respect to the lifted action of $G$ on $TQ$ the resulting generalization of the Euler–Lagrange equations are the Euler–Poincare equations. For systems on Lie groups, the Euler–Poincare equations are a significant improvement upon the Euler–Lagrange equations.

Variational integrators approximating the Euler–Poincare equations have been studied (see e.g. [10, 3]) and have important applications in the study of mechanical systems with symmetries and constraints. Similar structure-preserving integrators that extend the Euler–Poincare equations to the use of a general non-coordinate frame have not yet been developed. The Euler–Lagrange equations generalized to this arbitrary choice of frame are called Hamel’s equations, written locally as

$\frac{d}{dt} \frac{\partial \ell}{\partial \xi^j} = u_j[\ell] + \xi^i \frac{\partial \ell}{\partial \xi^m}c^m_{ij}(q)$

where $(\xi^i)^n_{i=1}$ are the components of velocity with respect to the frame, $\ell$ is the Lagrangian defined in terms of $(q^i,\xi^j)$, $(u^j)^n_{j=1}$ are the vector fields that define the frame, $u_j[\ell]$ are the directional derivatives of $l$, and $c^m_{ij} (q)$ are the components of the Jacobi–Lie bracket of the vector fields $u_i$ and $u_j$ at $q$. Note that the well known Euler–Lagrange equations (where velocity components are $\dot{q}^i$ instead of $\xi^i$) are a special case of Hamel’s equations. Hamel’s equations are known to be an effective tool for studying mechanical systems with velocity constraints and/or symmetries [2]; numerically integrating such systems while taking advantage of simplifications gained by introducing a non- coordinate frame is thus relevant to current research in mechanics and is a topic of interest.

Variational integrators are derived via a discrete principle of critical action; the solutions they describe are critical with respect to a discrete Lagrangian that approximates the action over a time step h; thus, our first step in deriving such integrators is defining a discretized action. Such discretizations take the form

$\int_{t_0}^{t_F} \ell(q^i,\xi^j)dt \quad \rightarrow \quad \sum_{k=0}^{N-1} \ell_d(q_k,q_{k+1}), \qquad \mbox{where } \frac{t_F-t_0}{N} = h$

with the function $\ell_d$ itself being determined by some choice of mapping $TQ \mapsto Q\times Q$ which can be thought of as a discretization of the tangent bundle. We have certain freedom in how we go about defining a discrete Lagrangian; different choices will result in different integrators. As a result, we currently have two versions for discrete Hamel’s equations that reflect the structure of the continuous Hamel’s equations. Both versions are derived from the discrete variational principle; one in particular is provably a symplectic integrator. In another discretization symplecticity has not yet been directly verified, however we observed that this discretization has good long-time behavior in simulations. A paper currently in preparation (along with my thesis) will describe the derivation of these integrators and examine their structure- preserving properties.

Future Directions

The results of my research have applications in the study of control systems and mechanical systems with nonintegrable constraints; the numerical simulation of such systems is a motivating goal of my study of variational integrators and Hamel’s equations. I hope to continue to study such systems and I expect that my results will give rise to some interesting applications and perhaps some new questions in the classical setting. Of course, variational integrators are by no means a panacea for the simulation of mechanical systems; there are a variety of approaches to such approximations each with properties that make a method more or less desirable given the circumstances. The variational approach is useful when studying constrained, forced, and dissipative systems as it results in a natural treatment for such features [12].

I also have an interest in relativity and the incorporation of numerical methods into the study of relativistic systems, and I see opportunities to expand my results in classical mechanics to such systems. First, I expect that discrete Hamel’s equations may have applications in the simulation of systems evolving in a relativistic (gravitational) field. As is the case in Lagrangian mechanical systems, a choice of non-coordinate frame may result in Hamel’s equations that are simplifications of the differential equations written with respect to the coordinate frame. The methods I have studied and developed may be applicable on certain differentiable manifolds, thus it may be an interesting future project to extend my results to a pseudo-Riemannian manifold equipped with a Lorentz metric.

Second, I intend to apply the ideas behind variational integrators of Lagrangian systems to a numerical approximation of Einstein’s field equations. There is a need for robust approximation methods of Einstein’s equations in applied astrophysics. As an example, interesting and possi- bly observable gravitational waves might be generated by relativistic binary mass systems such as closely orbiting black-hole and/or neutron star systems. The study of such complicated systems requires numerical simulation; physicists use a variety of finite-element method techniques to nu- merically simulate the associated boundary value PDE’s and predict the structure of the resulting observable waves, and are in many ways still fine-tuning such techniques.

It is known that Einstein’s equations may be derived in a variational manner from the Einstein– Hilbert action (a functional of the metric) such that solutions to Einstein’s equations are critical with respect to the action [13]. The variational viewpoint is applied most often in the study of quantum gravity because of its appealing use in a path-integral formulation. However, I conjecture the Einstein–Hilbert action may also be used to derive numerical approximations of the Einstein field equations that might have structure-preserving properties analogous to the discrete Euler– Lagrange equations and the discrete Hamel’s equations.

References

[1] Ball, K. R., D. V. Zenkov, and A. M. Bloch [2012], Variational Structures for Hamel’s Equations and Stabilization, Proceedings of the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control , 178–183.

[2] Bloch, A. M., J. E. Marsden, and D. V. Zenkov [2009], Quasivelocities and Symmetries in Nonholonomic Systems, Dynamical Systems: An International Journal 24, 187–222.

[3] Bou-Rabee, N. and J. E. Marsden [2009], Hamilton-Pontryagin Integrators on Lie Groups Part I: Introduction and Structure-Preserving Properties, Foundations of Computational Mathemat- ics 9, 2, 197–219.

[4] Brown, J. D. [2006], Midpoint rule as a variational-symplectic integrator: Hamiltonian systems, Physical Review D, 73, 2.

[5] de Le ́on, M., D. Martin de Diego, and A. Santamar ́ıa-Merino [2007], Discrete variational integrators and optimal control theory, Advances in Computational Mathematics 26, 1, 251– 268.

[6] Hamel, G. [1904], Die Lagrange-Eulersche Gleichungen der Mechanik, Z. Math. Phys. 50, 1–57. [7] Euler, L. [1752], Decouverte d’un nouveau principe de Mecanique, M ́emoires de l’acad ́emie des

sciences de Berlin 6, 185–217. [8] Hairer, E., C. Lubich, and G. Wanner [2006], Geometric Numerical Integration: Structure

Preserving-Algorithms for Ordinary Differential Equations, Springer-Verlag, Berlin, Germany.

[9] Kharevych, L., et al. [2006], Geometric, variational integrators for computer animation, Pro- ceedings of the 2006 ACM SIGGRAPH/Eurographics symposium on Computer animation, Eurographics Association, 43–51.

[10] Kobilarov, M., J. E. Marsden, and G. S. Sukhatme [2010], Geometric Discretization of Non- holonomic Systems with Symmetry, Discrete and Continuous Dynamical Systems Series S 3, 1, 61–84.

[11] Marsden, J. E., S. Pekarsky, and S. Shkoller [1999] , Discrete Euler–Poincar ́e and Lie–Poisson Equations, Nonlinearity 12, 1647-1662

[12] Marsden, J. E. and M. West [2001], Discrete Mechanics and Variational Integrators, Acta Numerica, 357–514.

[13] Wald, Robert M. [1984], Appendix E: Lagrangian and Hamiltonian Formulations of General Relativity, General Relativity.

[14] Wendlandt, J. M. and J. E. Marsden [1997], Mechanical Integrators Derived from a Discrete Variational Principle, Physica D 106, 223-246.

[15] Yoshimura, H. and J. E. Marsden [2006], Dirac Structures in Lagrangian Mechanics. Part II: Variational Structures, Journal of Geometry and Physics 57, 209–250.