Variational Integrators

Transcription

1 Variational Integrators Thesis by Matthew West In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasaena, California 2004 (Submitte May 28, 2004)

2 ii c 2004 Matthew West All Rights Reserve

3 iii Acknowlegements Many people have contribute to this work. In particular, I thank my avisor Jerry Marsen, with whom it has been a true pleasure an honour to work, an my co-authors Harish Bhat, Fehmi Cirak, Razvan Fetecau, Sameer Jalnapurkar, Couro Kane, Sanjay Lall, Melvin Leok, Arian Lew, Marcel Oliver, Michael Ortiz, Sergey Pekarsky, Steve Shkoller, an Clauia Wulff, all of whom have contribute markely to my eucation an enjoyment of research. In aition, I woul like to thank Darryl Holm, Arieh Iserles, Ben Leimkuhler, Christian Lubich, Robert McLachlan, Richar Murray, Sebastian Reich, Mark Roberts, Peter Schröer, Bob Skeel, an Yuri Suris for their valuable suggestions an help along the way. Finally, this work woul not have been possible without the love an support of my parents Owen an Juy, my sisters Kate an Anna, an of course Nicole.

4 iv Abstract Variational integrators are a class of iscretizations for mechanical systems which are erive by iscretizing Hamilton s principle of stationary action. They are applicable to both orinary an partial ifferential equations, an to both conservative an force problems. In the absence of forcing they conserve (multi-)symplectic structures, momenta arising from symmetries, an energy up to a boune error. In this thesis the basic theory of iscrete variational mechanics for orinary ifferential equations is evelope in epth, an is use as the basis for constructing variational integrators an analyzing their numerical properties. This is then taken as the starting point for the evelopment of a new class of asynchronous time stepping methos for soli mechanics, known as Asynchronous Variational Integrators (AVIs). These explicit methos time step ifferent elements in a finite element mesh with fully inepenent an ecouple time steps, allowing the simulation to procee locally at the fastest rate allowe by local stability restrictions. Numerical examples of AVIs are provie, emonstrating the excellent properties they posess by virtue of their variational erivation.

5 v Contents Acknowlegements Abstract iii iv 1 Introuction Variational integrators History an literature Work associate with this thesis Outline of thesis Discrete Dynamics an Variational Integrators Continuous time Lagrangian ynamics Discrete time Lagrangian ynamics Variational integrators Examples of iscrete Lagrangians Constraine systems Forcing an issipation Conservation Properties of Variational Integrators Noether s theorem an momentum conservation Discrete time Noether s theorem an iscrete momenta Continuous time symplecticity Discrete time symplecticity Backwar error analysis Multisymplectic systems an variational integrators Variational multisymplectic mechanics Multisymplectic iscretizations Discrete variational mechanics Backgroun: Lagrangian mechanics Basic efinitions

6 vi Lagrangian vector fiels an flows Lagrangian flows are symplectic Lagrangian flows preserve momentum maps Discrete variational mechanics: Lagrangian viewpoint Discrete Lagrangian evolution operator an mappings Discrete Lagrangian maps are symplectic Discrete Noether s theorem Backgroun: Hamiltonian mechanics Hamiltonian mechanics Hamiltonian form of Noether s theorem Legenre transforms Generating functions Coorinate expression Discrete variational mechanics: Hamiltonian viewpoint Discrete Legenre transforms Momentum matching Discrete Hamiltonian maps Discrete Lagrangians are generating functions Corresponence between iscrete an continuous mechanics Backgroun: Hamilton-Jacobi theory Generating function for the flow Hamilton-Jacobi equation Jacobi s solution Discrete variational mechanics: Hamilton-Jacobi viewpoint Variational integrators Introuction Implementation of variational integrators Equivalence of integrators Backgroun: Error analysis Local error an metho orer Global error an convergence Orer calculation Variational error analysis Local variational orer Discrete Legenre transform orer

7 vii Variational orer calculation The ajoint of a metho an symmetric methos Exact iscrete Lagrangian is self-ajoint Orer of ajoint methos Composition methos Multiple steps Single step, multiple substeps Single step Examples of variational integrators Mipoint rule Störmer-Verlet Newmark methos Explicit symplectic partitione Runge-Kutta methos Symplectic partitione Runge-Kutta methos Galerkin methos Forcing an constraints Backgroun: Force systems Force Lagrangian systems Force Hamiltonian systems Legenre transform with forces Noether s theorem with forcing Discrete variational mechanics with forces Discrete Lagrange- Alembert principle Discrete Legenre transforms with forces Discrete Noether s theorem with forcing Exact iscrete forcing Integration of force systems Backgroun: Constraine systems Constraine Lagrangian systems Constraine Hamiltonian systems: Augmente approach Constraine Hamiltonian systems: Dirac theory Legenre transforms Conservation properties Discrete variational mechanics with constraints Constraine iscrete variational principle

8 viii Augmente Hamiltonian viewpoint Direct Hamiltonian viewpoint Conservation properties Constraine exact iscrete Lagrangians Constraine variational integrators Constraine geometric integration Variational integrators for constraine systems Low-orer methos SHAKE an RATTLE Composition methos Constraine symplectic partitione Runge-Kutta methos Constraine Galerkin methos Backgroun: Force an constraine systems Lagrangian systems Hamiltonian systems Legenre transforms Conservation properties Discrete variational mechanics with forces an constraints Lagrangian viewpoint Discrete Hamiltonian maps Exact force constraine iscrete Lagrangian Noether s theorem Variational integrators with forces an constraints Multisymplectic continuum mechanics Multisymplectic continuum mechanics Configuration geometry Variations an ynamics Horizontal variations Conservation laws Multisymplectic forms Multisymplectic form formula Spatial multisymplectic form formula an reciprocity Temporal multisymplectic form formula an symplecticity Noether s theorem Symmetries an momentum maps

9 ix 6 Multisymplectic asynchronous variational integrators Asynchronous variational integrators (AVIs) Systems of particles Asynchronous time iscretizations Implementation of AVIs Momentum conservation properties Numerical examples Complexity an convergence Multisymplectic iscretizations Discrete Configuration Geometry Discrete Variations an Dynamics Horizontal Variations Discrete conservation laws Discrete Multisymplectic Forms Discrete Multisymplectic Form Formula Discrete Reciprocity an Time Symplecticity Discrete Noether s Theorem Proof of AVI convergence Asychronous splitting methos (ASMs) AVIs as ASMs Convergence proof

10 x List of Figures 1.1 Average kinetic energy for a nonlinear spring-mass lattice system Error in numerically approximate heat for the lattice system Energy evolution for a issipative system using a variational integrator Energy for a conservative system using a variational integrator Phase space plots using a variational integrator A section of a fiber bunle Example mesh for a multisymplectic iscretization A configuration as a section of a fiber bunle Spacetime iagram of an asynchronous mesh Algorithm implementing AVI time stepping Mesh of the helicopter blae Time evolution of the helicopter blae, rigi case Time evolution of the helicopter blae, intermeiate case Time evolution of the helicopter blae, soft case Number of elemental upates for the helicopter blae Time evolution of the total energy for the helicopter blae Geometry an mesh of the slab use for the cost/accuracy example L 2 errors for the trajectory of the slab Decomposition of the error use in proving ASM convergence

11 xi List of Tables 6.1 Perio of rotation of the helicopter blae Number of elemental upates for the helicopter blae

12 1 Chapter 1 Introuction 1.1 Variational integrators This thesis futher evelops the subject of variational integrators as it applies to mechanical systems of engineering interest. The iea behin this class of algorithms is to iscretize the variational formulation of a given problem, rather than the ifferential equations. These problems may be either conservative or issipative an force, an may be cast as either orinary or partial ifferential equations. For conservative problems, we focus on iscretizing Hamilton s principle of stationary action in Lagrangian mechanics, while for issipative or force problems we iscretize the Lagrange Alembert principle. While the iea of iscretizing variational formulations of mechanics is stanar for elliptic problems, in the form of Galerkin an finite element methos (e.g., Johnson [1987], Hughes [1987]), it has only been applie relatively recently to erive variational time stepping algorithms for mechanical systems. Avantages of variational integrators. The variational metho for eriving integrators means that the resulting algorithms automatically have a number of properties. In particular, they are symplectic methos 1, they exactly preserve momenta associate to symmetries of the system, an they have excellent longtime energy stability. These properties make them ieal for simulating physical systems which are either conservative or near-conservative. In such cases, one frequently wishes to compute certain average or statistical properties of the system, such as the spee of a shock-front, an maintaining goo conservation properties where appropriate can be essential. In aition, the variational methoology allows one to easily an cleanly erive goo integrators even in extremely complex geometries, such as the asynchronous space-time meshes use in 6.1. This applies whether the mechanical system is conservative or not. Finally, the variational methoology serves as a unifying framework from which to consier the 1 Symplecticity is explaine in more etail in 1.6.3, 1.6.4, 2.1.3, an

13 2 Lagrangian sie of symplectic integration theory. This fact erives from generating function theory, which shows that all symplectic integrators for Lagrangian systems are in fact variational. Geometric integration. Variational integrators are examples of methos which preserve geometric structures of the continuous system, an as such they fall within the general class of geometric integration techniques. For mechanical systems the primary quantities on which most work has focusse are momenta, energy, an symplectic structures. An important result ue to Ge an Marsen [1988] shows that an integrator can either preserve energy an momenta, or symplectic structure an momenta, but not all three. For this reason, the geometric integration community has largely split into those working on symplectic-momentum methos, incluing variational integrators, an those working on energy-momentum methos. PDEs an multisymplectic iscretizations. Variational integrators for ODEs can be extene to PDEs by iscretizing both space an time variationally. An elegant framework within which this can be performe is given by multisymplectic mechanics, which is a local space-time viewpoint of mechanics. As with ODEs, variational methos for multisymplectic PDEs conserve (multi-) symplecticity, momenta, an have excellent energy behavior. Asynchronous Variational Integrators (AVIs). In this thesis, the primary application of variational iscretizations of multisymplectic PDEs is the evelopment of Asynchronous Variational Integrators (AVIs) for soli mechanics. These methos use fully ecouple asynchronous space-time meshes, which means that in a finite element setting all elements can be time steppe inepenently of each other, subject only to local CFL conitions. This provies a large efficiency gain, as it is no longer necessary for the time step of the whole mesh to be ictate by the smallest element, as woul orinarily be the case for explicit time steppers. Furthermore, the variational metho of erivation ensures that they asynchronicity is introuce without losing any of the goo properties of stanar methos such as the Newmark algorithm. 1.2 History an literature Of course, the variational view of mechanics goes back to Euler, Lagrange an Hamilton. The form of the variational principle most important for continuous mechanics that we use in this article is ue, of course, to Hamilton [1834]. We refer to Marsen an Ratiu [1999] for aitional history, references an backgroun on geometric mechanics. There have been many attempts at the evelopment of a iscrete mechanics an corresponing integrators that we will not attempt to survey in any systematic fashion. The theory of iscrete variational mechanics in the form we shall use it (that uses two copies Q Q of the configuration space

14 3 for the iscrete analogue of the velocity phase space) has its roots in the optimal control literature of the 1960s: see, for example, Joran an Polak [1964], Hwang an Fan [1967] an Cazow [1970]. In the context of mechanics early work was one, often inepenently, by Cazow [1973], Logan [1973], Maea [1980, 1981a,b], an Lee [1983, 1987], by which point the iscrete action sum, the iscrete Euler-Lagrange equations an the iscrete Noether s theorem were clearly unerstoo. This theory was then pursue further in the context of integrable systems in Veselov [1988, 1991] an Moser an Veselov [1991], an in the context of quantum mechanics in Jaroszkiewicz an Norton [1997a,b] an Norton an Jaroszkiewicz [1998]. The variational view of iscrete mechanics an its numerical implementation is further evelope in Wenlant an Marsen [1997a,b] an then extene in Kane, Marsen, an Ortiz [1999a], Marsen, Pekarsky, an Shkoller [1999b,a], Bobenko an Suris [1999a,b] an Kane, Marsen, Ortiz, an West [2000]. The beginnings of an extension of these ieas to a nonsmooth framework is given in Kane, Repetto, Ortiz, an Marsen [1999b], an is carrie further in Fetecau, Marsen, Ortiz, an West [2003a]. Other applications inclue Rowley an Marsen [2002], an an investigation of convergence is given in Müller an Ortiz [2003]. Other iscretizations of Hamilton s principle are given in Mutze [1998], Cano an Lewis [1998] an Shibberu [1994]. Other versions of iscrete mechanics (not necessarily iscrete Hamilton s principles) are given in, for instance, Itoh an Abe [1988], Labue an Greenspan [1974, 1976a,b], an MacKay [1992]. Of course, there have been many works on symplectic integration, largely one from other points of view than that evelope here. We will not attempt to survey this in any systematic fashion, as the literature is simply too large with too many points of view an too many intricate subtleties. We give a few highlights an give further references in the boy of the thesis. For instance, we shall connect the variational view with the generating function point of view that was begun in De Vogelaére [1956]. Generating function methos were evelope an use in, for example, Ruth [1983], Forest an Ruth [1990] an in Channell an Scovel [1990]. See also Berg, Warnock, Ruth, an Forest [1994], an Warnock an Ruth [1992, 1991]. For an overview of symplectic integration, see Hairer, Lubich, an Wanner [2002], as well as Sanz-Serna [1992b] an Sanz-Serna an Calvo [1994]. Qualitative properties of symplectic integration of Hamiltonian systems are given in Gonzalez, Higham, an Stuart [1999] an Cano an Sanz-Serna [1997]. Longtime energy behaviour for oscillatory systems is stuie in Hairer an Lubich [2000]. Longtime behaviour of symplectic methos for systems with issipation is given in Hairer an Lubich [1999]. A numerical stuy of preservation of aiabatic invariants is given in Reich [1999b] an Shimaa an Yoshia [1996]. Backwar error analysis is stuie in Benettin an Giorgilli [1994], Hairer [1994], Hairer an Lubich [1997] an Reich [1999a]. Other ieas connecte to the above literature inclue those of Baez an Gilliam [1994], Gilliam [1996], Gillilan an Wilson [1992]. For other references see the large literature on symplectic methos in

15 4 molecular ynamics, such as Schlick, Skeel, Brunger, Kale, Boar, Hermans, an Schulten [1999], an for various applications, see Hary, Okunbor, an Skeel [1999], Leimkuhler an Skeel [1994], Barth an Leimkuhler [1996] an references therein. A single-step variational iea that is relevant to our approach is given in Ortiz an Stainier [1999], an evelope further in Raovitzky an Ortiz [1999], an Kane et al. [1999b, 2000]. Direct iscretizations on the Hamiltonian sie, where one iscretizes the Hamiltonian an the symplectic structure, are evelope in Gonzalez [1996b,a] an further in Gonzalez [1999] an Gonzalez et al. [1999]. This is evelope an generalize much further in McLachlan, Quispel, an Robioux [1998, 1999]. Multisymplectic mechanics has its origins in the mi-twentieth century (see Kijowski an Tulczyjew [1979] an references therein for a representative view). There has been a recent explosion in work, however, riven by numerical applications, beginning with inepenent work by Marsen, Patrick, an Shkoller [1998] an Briges an Reich [2001a]. Non-numerical applications have also avance, with work by Marsen an Shkoller [1999], Kouranbaeva an Shkoller [2000], Briges an Laine-Pearson [2001], Hyon [2001], Briges an Derks [2002], an Binz, e Leon, e Diego, an Socolescu [2002]. The work on numerical applications has avance on several main fronts, incluing that in Briges an Reich [2001b], Reich [2000a], an Reich [2000b], as well as Islas an Schober [2002] an Islas, Karpeev, an Schober [2001], an the series of papers by Sun an Qin [2000], Wang an Qin [2001], Guo, Li, an Wu [2001b], Chen [2001], Guo, Ji, Li, an Wu [2001a], Guo, Li, Wu, an Wang [2002a], Guo, Li, Wu, an Wang [2002b], Guo, Li, Wu, an Wang [2002c], Chen an Qin [2002], Liu an Qin [2002], Chen [2002], Hong an Qin [2002], Wang an Qin [2002], an Chen [2003]. Finally, there is the work associate with this thesis, which is iscusse in the next section. The asynchronous methos evelope in this thesis have much in common with multi-time step integration algorithms, sometimes terme subcycling methos. These algorithms have been evelope in Neal an Belytschko [1989] an Belytschko an Mullen [1976], mainly to allow high-frequency elements to avance at smaller time steps than the low-frequency ones. In its original version, the metho groupe the noes of the mesh an assigne to each group a ifferent time step. Ajacent groups of noes were constraine to have integer time step ratios (see Belytschko an Mullen [1976]), a conition that was relaxe in Neal an Belytschko [1989] an Belytschko [1981]. Recently an implicit multi-time step integration metho was evelope an analyze in Smolinski an Wu [1998]. We also mention the relate work of Hughes an Liu [1978] an Hughes, Pister, an Taylor [1979]. There are also many connections between the multi-time step impulse metho (also known as Verlet- I an r-respa), which is popular in molecular ynamics applications, an the AVI algorithm (see Grubmüller, Heller, Winemuth, an Schulten [1991] an Tuckerman, Berne, an Martyna [1992]).

16 1.3 Work associate with this thesis 5 Work incorporate in an arising from this thesis is: Kane, Marsen, Ortiz, an West [2000] stuie the Newmark algorithm from the perspective of variational integrators an prove that the Newmark metho with γ = 1/2 is itself variational an hence symplectic. This paper also inclue the first introuction of the iea of a iscrete Lagrange- Alembert principle for force an issipative systems. Marsen, Pekarsky, Shkoller, an West [2001] lai the grounwork for variational multisymplectic integrators of continuum theories by reformulating classical continuum mechanics in an intrinsic multisymplectic framework. The two examples treate in etail were ieal fluis an hyper-elasticity, an much attention was pai to the role of incompressibility. This necessitate the evelopment of the first theory of constraine multisymplectic systems. Marsen an West [2001] provie a survey of the existing theory for iscrete variational mechanics an variational integrators for ODEs. It also presente a great eal of new theory for the first time, incluing much work on the numerical properties of variational integrators such as high-orer methos an approximation accuracy results. Fetecau, Marsen, Ortiz, an West [2003a] consiere Lagrangian systems with trajectories which are continuous but nonsmooth, with collision problems being the main focus. This lea to the evelopment of both a continuous variational theory for such systems as well as variational integrators for such problems. These were the first geometric integrators evelope which were applicable to nonsmooth systems. Fetecau, Marsen, an West [2003b] extene the nonsmooth ODE theory from Fetecau et al. [2003a] to the multisymplectic PDE setting. This provies a formulation which encompasses such problems as flui-soli interactions, internal shock waves in continua an collisions of elastic boies. Lew, Marsen, Ortiz, an West [2003a] introuce the concept of Asynchronous Variational Integrators (AVI). This paper also evelope further the concept of full variations for multisymplectic systems an the conservation properties associate with horizontal variations. Lew, Marsen, Ortiz, an West [2003b] further investigate the numerics of AVIs. This involve stuying a large scale simulation of a helicopter rotor blae as well as proviing a proof of convergence an an analysis of the computational complexity of AVIs. This was the first proof of convergence for a fully asynchronous integration time integration metho. Jalnapurkar, Leok, Marsen, an West [2003] evelope a iscrete version of Routh reuction theory for systems with abelian symmetry groups.

17 6 Oliver, West, an Wulff [2003] stuie the approximate momentum conservation properties of variational iscretizations of nonlinear wave equations, giving both numerical examples an theory explaining the observe behavior. This represente the first rigorous results concerning the backwar error analysis of spatial PDE iscretizations. Cirak an West [2003] use the ieas in Fetecau et al. [2003a] as the basis for constructing a very efficient technique for explicit time stepping of finite element moels with collisions, an emonstrate the algorithms on large parallel simulations of colliing shells an solis. Pekarsky an West [2003] evelope groupoi iscretizations of iffeomorphism groups an use these to construct the first exactly circulation preserving integrators for ieal fluis. Lall an West [2003] unifie the iscretizations of the calculus of variations use in iscrete mechanics an in iscrete optimal control. This lea to the evelopment of the Hamiltonian sie of iscrete mechanics an the concept of a iscrete Hamilton-Jacobi equation for iscrete mechanics. 1.4 Outline of thesis We begin in 1.5 to 1.7 with a simple overview of variational integrators for ODEs an PDEs. This material is a summary of 2 to 6. In 2 we evelop iscrete variational mechanics for ODEs, incluing extensive comparisons with continuous-time Lagrangian an Hamiltonian mechanics. This is then use in 3 as the basis for variational integrators, whose numerical properties are investigate in etail. Next, 4 consiers both iscrete mechanics an variational integrators for systems with forcing an constraints. The final two chapters eal with variational mechanics an integrators for PDEs. In 5 we formulate continuum mechanics within the context of variational multisymplectic mechanics, while in 6 we evelop Asynchronous Variational Integrators (AVIs) as a special case of variational multisymplectic iscretizations. 1.5 Discrete Dynamics an Variational Integrators In this section we give a brief overview of how iscrete variational mechanics can be use to erive variational integrators. We begin by reviewing the erivation of the Euler-Lagrange equations, an then show how to mimic this process on a iscrete level.

18 1.5.1 Continuous time Lagrangian ynamics 7 For concreteness, consier the Lagrangian system L(q, q) = 1 2 qt M q V (q), where M is a symmetric positive-efinite mass matrix an V is a potential function. We work in R n or in generalize coorinates an will use vector notation for simplicity, so q = (q 1,q 2,...,q n ). In the stanar approach of Lagrangian mechanics, we form the action function by integrating L along a curve q(t) an then compute variations of the action while holing the enpoints of the curve q(t) fixe. This gives T δs(q) = δ L ( q(t), q(t) ) T t = 0 = 0 T 0 [ L L δq + q [ L q ( L t q ] q δ q t )] δq t + [ ] T L q δq, (1.1) 0 where we have use integration by parts. The final term is zero because we assume that δq(t) = δq(0) = 0. Requiring that the variations of the action be zero for all δq implies that the integran must be zero for each time t, giving the well-known Euler-Lagrange equations L q (q, q) ( ) L (q, q) = 0. (1.2) t q For the particular form of the Lagrangian chosen above, this is just M q = V (q), which is Newton s equation: mass times acceleration equals force. It is well known that the system escribe by the Euler-Lagrange equations has many special properties. In particular, the flow on state space is symplectic, meaning that it conserves a particular two-form, an if there are symmetry actions on phase space, then there are corresponing conserve quantities of the flow, known as momentum maps. We will return to these ieas later in this work in Discrete time Lagrangian ynamics We will now see how iscrete variational mechanics performs an analogue of the above erivation. Rather than taking a position q an velocity q, consier now two positions q 0 an q 1 an a time step t R. These positions shoul be thought of as being two points on a curve at time t apart, so that q 0 q(0) an q 1 q( t). We now consier a iscrete Lagrangian L (q 0,q 1, t), which we think of as approximating the action integral along the curve segment between q 0 an q 1. For concreteness, consier the very simple approximation to the integral T 0 Lt given by using the rectangle rule2 (the length of the 2 As we shall see later, more sophisticate quarature rules lea to higher-orer accurate integrators.

19 interval times the value of the integran with the velocity vector replace by (q 1 q 0 )/ t): [ 1 L (q 0,q 1, t) = t 2 8 ( ) T q1 q 0 M t ( ) ] q1 q 0 V (q 0 ). (1.3) t Next consier a iscrete curve of points {q k } N k=0 an calculate the iscrete action along this sequence by summing the iscrete Lagrangian on each ajacent pair. Following the continuous erivation above, we compute variations of this action sum with the bounary points q 0 an q N hel fixe. This gives N 1 δs ({q k }) = δ L (q k,q k+1, t) = k=0 N 1 [ ] D1 L (q k,q k+1, t) δq k + D 2 L (q k,q k+1, t) δq k+1 k=0 N 1 [ = D2 L (q k 1,q k, t) + D 1 L (q k,q k+1, t) ] δq k k=1 + D 1 L (q 0,q 1, t) δq 0 + D 2 L (q N 1,q N, t) δq N, (1.4) where we have use a iscrete integration by parts (rearranging the summation). Henceforth, D i L inicates the slot erivative with respect ot the i-th argument of L. If we now require that the variations of the action be zero for any choice of δq k with δq 0 = δq N = 0, then we obtain the iscrete Euler-Lagrange equations D 2 L (q k 1,q k, t) + D 1 L (q k,q k+1, t) = 0, (1.5) which must hol for each k. For the particular L chosen above, we compute ( ) qk q k 1 D 2 L (q k 1,q k, t) = M t [ ( ) ] qk+1 q k D 1 L (q k,q k+1, t) = M + ( t) V (q k ), t an so the iscrete Euler-Lagrange equations are ( ) qk+1 2q k + q k 1 M ( t) 2 = V (q k ). This is clearly a iscretization of Newton s equations, using a simple finite ifference rule for the erivative. If we take initial conitions (q 0,q 1 ), then the iscrete Euler-Lagrange equations efine a recursive rule for calculating the sequence {q k } N k=0. Regare in this way, they efine a map (q k,q k+1 ) (q k+1,q k+2 ) which we can think of as a one-step integrator for the system efine by the continuous Euler-Lagrange equations. This viewpoint is consiere in epth in 3.

20 9 Average kinetic energy t = 0.5 t = 0.2 t = 0.1 t = 0.05 RK4 VI1 lskjf slkfj t = 0.05 t = 0.1 t = 0.2 t = Time Figure 1.1: Average kinetic energy (1.6) as a function of T for a nonlinear spring-mass lattice system, using a first-orer variational integrator (VI1) an a fourth-orer Runge-Kutta metho (RK4) an a range of time steps t. Observe that the Runge-Kutta metho suffers substantial numerical issipation, unlike the variational metho. Heat calculation example. As we will consier in etail in 1.6, variational integrators are interesting because they inherit many of the conservative properties of the original Lagrangian system. As an example of this, we consier the numerical approximation of the heat of a couple spring-mass lattice moel. The numerical heat for time T is efine to be the numerical approximation of K(T) = 1 T T q 2 t, (1.6) while the true heat of the system is the limit of the quantity, K = lim K(T). (1.7) T The temperature of the system, which is an intensive as oppose to extensive quantity, is the heat K ivie by the heat capacity n, where n is the number of masses an is the imension of space. We assume that the system is ergoic an that this limit exists. In Figure 1.1 we plot the numerical approximations to (1.6) at T = 10 5 compute using a first-orer variational integrator (VI1) an a fourth-orer Runge-Kutta metho (RK4). As the time step is ecrease the numerical solution tens towars the true solution. Note, however, that the lack of issipation in the variational integrator means that for quite large time steps it computes the average kinetic energy much better. To make this precise, we consier the harmonic approximation to the lattice system (that is, the linearization), for which we can compute the limit (1.7) analytically. The error in the numerically compute heat is plotte in Figure 1.2 for a range of ifferent time steps t an final times T, using the same first-orer variational metho (VI1) an fourth-orer Runge-Kutta metho (RK4), as well as a fourth-orer

21 10 Temperature error (relative) 10 0 VI VI4 RK VI1 VI4 RK RK4 VI VI Cost for T = Cost for T = Cost for T = 1000 Figure 1.2: Error in numerically approximate heat for the harmonic (linear) approximation to the lattice system from Figure 1.1, using a first-orer variational integrator (VI1), a fourth-orer Runge- Kutta metho (RK4) an a fourth-orer variational integrator (VI4). The three plots have ifferent final times T, while the cost is increase within each plot by ecreasing the time step t. For each T the ashe horizontal line is the exact value of K(T) K, which is the minimum error that the numerical approximation can achieve without increasing T. Observe that the low-orer variational metho VI1 beats the traitional RK4 metho for larger errors, while the high-orer variational metho VI4 combines the avantages of both high-orer an variational structure to always win. variational integrator (VI4). To compute the heat (1.7) numerically we must clearly let T an t 0. Both of these limits increase the cost of the simulation, an so there is a traeoff between them. As we see in Figure 1.2, for a fixe T there is some t which aequately resolves the integral (1.6), an so the error cannot ecrease any further without increasing T. To see this, take a numerical approximation K(T, t) to (1.6) an ecompose the error as K(T, t) }{{ K = } K(T, t) K(T) + }{{} K(T) K. (1.8) }{{} total error iscretization error limit error Decreasing t will reuce the iscretization error, but at some point this will become negligible compare to the limit error, which will only ten to zero as T is increase. The striking feature of Figure 1.2 is that the variational integrators perform far better than a traitional Runge-Kutta metho. For large error tolerances, such as 1% or 5% error (10 2 or in Figure 1.2), the first-orer variational metho is very cheap an simple. For higher precision, the fourth-orer Runge-Kutta metho eventually becomes cheaper than the first-orer variational integrator, but the fourth-orer variational metho combines the avantages of both an is always the metho of choice. Of course, such sweeping statements as above have to be interprete an use with great care, as in the precise statements in the text that follows. For example, if the integration step size is too large, then sometimes energy can behave very baly, even for a variational integrator (see, for

22 11 example, Gonzalez an Simo [1996]). It is likewise well known that energy conservation oes not guarantee trajectory accuracy. These points will be iscusse further below Variational integrators We are primarily intereste in iscrete Lagrangian mechanics for eriving integrators for mechanical systems. Any integrator which is the iscrete Euler-Lagrange equation for some iscrete Lagrangian is calle a variational integrator. As we have seen above, variational integrators can be implemente by taking two configurations q 0 an q 1 of the system, which shoul approximate q 0 q(t 0 ) an q 1 q(t 0 + t), an then solving the iscrete Euler-Lagrange equations (1.5) for q 2. This process can then be repeate to calculate an entire iscrete trajectory. The map (q k 1,q k ) (q k,q k+1 ) efine by the iscrete Euler-Lagrange equations is known as the iscrete evolution map. Position-momentum form. For mechanical systems it is more common to specifiy the initial conitions as a position an a velocity (or momentum), rather than two positions. To rewrite a variational integrator in a position-momentum form we first observe that we can efine the momentum at time step k to be p k = D 2 L (q k,q k+1, t) = D 1 L (q k 1,q k, t). (1.9) The two expressions for p k are equal because this equality is precisely the iscrete Euler-Lagrange equations (1.5). Using this efinition we can write the position-momentum form of a variational integrator as p k = D 1 L (q k,q k+1, t) p k+1 = D 2 L (q k,q k+1, t). (1.10a) (1.10b) Given an initial conition (q 0,p 0 ) we can solve the implicit equation (1.10a) to fin q 1, an then evaluate (1.10b) to give p 1. We then have (q 1,p 1 ) an we can repeat the proceure. The sequence {q k } N k=0 so obtaine will clearly satisfy the regular iscrete Euler-Lagrange equations (1.5) for all k, ue to the efinition (1.9) of p k. This equality is further elaborate in Orer of accuracy. We remarke above that a iscrete Lagrangian shoul be thought of as approximating the continuous action integral. We will now make this statement precise. We say that a iscrete Lagrangian is of orer r if L (q 0,q 1, t) = t 0 L(q(t), q(t))t + O( t) r+1, (1.11)

23 12 where q(t) is the unique solution of the Euler-Lagrange equations for L with q(0) = q 0 an q( t) = q 1. It can then be proven (see Theorem 3.3) that if L is of orer r, then the corresponing variational integrator is also of orer r, so that q k = q(k t) + O( t) r+1. To esign high-orer variational integrators, we must therefore construct iscrete Lagrangians which accurately approximate the action integral. Symmetric methos. One useful observation when calculating the orer of integrators is that symmetric methos always have even orer. We say that a iscrete Lagrangian is symmetric if L (q 0,q 1, t) = L (q 1,q 0, t). (1.12) This implies [Marsen an West, 2001, Theorem 2.4.1] that the resulting variational integrator will also be symmetric, an will thus automatically be of even orer. We will use this fact below. Geometric asie. The efinition (1.9) of p k efines a map Q Q T Q. In fact we can efine two such maps, known as the iscrete Legenre transforms, by FL + (q 0,q 1 ) = (q 1,D 2 L (q 0,q 1, t)) an FL (q 0,q 1, t) = (q 1, D 1 L (q 0,q 1 )). This is iscusse further in The position-momentum form (1.10) of the iscrete Euler-Lagrange equations is thus given by an is a map F t L : T Q T Q, where F t L F t L = FL ± F t L (FL ± ) 1 : Q Q Q Q is the iscrete evolution map. This shows that variational integrators are really one-step methos, although they may initially appear to be two-step. This form of the integrator is calle the iscrete Hamilton map an is investigate in Examples of iscrete Lagrangians We now consier some examples of iscrete Lagrangians. Generalize mipoint rule. The classical mipoint rule for the system ẋ = f(x) is given by x k+1 x k = ( t)f((x k+1 +x k )/2). If we a a parameter α [0,1] where the force evaluation occurs (so α = 1/2 is the stanar mipoint), then we can write the corresponing iscrete Lagrangian ( L mp,α (q 0,q 1, t) = ( t)l (1 α)q 0 + αq 1, q 1 q 0 t = t 2 ( ) T q1 q 0 M t ( q1 q 0 t ) ) ( t)v ((1 α)q 0 + αq 1 ). (1.13)

24 13 The iscrete Euler-Lagrange equations (1.5) are thus ( ) qk+1 2q k + q k 1 M ( t) 2 ) ) = (1 α) V ((1 α)q k + αq k+1 α V ((1 α)q k 1 + αq k (1.14) an the position-momentum form (1.10) of the variational integrator is ( ) qk+1 q k p k = M t ( qk+1 q k p k+1 = M t ) + (1 α)( t) V ((1 α)q k + αq k+1 (1.15a) ) ) α( t) V ((1 α)q k + αq k+1. (1.15b) This is always an implicit metho, an for general α [0,1] it is first-orer accurate. When α = 1/2 it is easy to see that L mp,α is symmetric, an thus the integrator is secon-orer. Generalize trapezoial rule. Rather than evaluating the force at an average location, we coul instea average the evaluate forces. Doing so at a parameter α [0,1] gives a generalization of the trapezoial rule ( L tr,α (q 0,q 1, t) = ( t)(1 α)l q 0, q 1 q 0 t = t 2 ( ) T q1 q 0 M t ( q1 q 0 t Computing the iscrete Euler-Lagrange equations (1.5) gives with corresponing position-momentum (1.10) form ) ( + ( t)αl q 1, q 1 q 0 t ) ) ( ( t) (1 α)v (q 0 ) + αv (q 1 ) ). (1.16) ( ) qk+1 2q k + q k 1 M ( t) 2 = V (q k ) (1.17) ( qk+1 q k p k = M t ( qk+1 q k p k+1 = t ) + ( t)(1 α) V (q k ) (1.18a) ) ( t)α V (q k ). (1.18b) This metho is explicit for all α, an is generally first-orer accurate. For α = 1/2 it is symmetric, an thus becomes secon-orer accurate. Observe that there is no α in the iscrete Euler-Lagrange equations (1.17), although it oes appear in the position-momentum form (1.18). This means that the only effect of α is on the starting proceure of this integrator, as thereafter the trajectory will be entirely etermine by (1.17). If we are given an initial position an momentum (q 0,p 0 ), then we can use (1.18a) to calculate q 1 an then continue with (1.17) for future time steps. For this proceure to be secon-orer accurate it is necessary to take α = 1/2 in the use of (1.18a) for the first time step.

25 14 Newmark metho. The Newmark family of integrators, originally given in Newmark [1959], are wiely use in structural ynamics coes. They are usually written (see, for example, Hughes [1987]) for the system L = 1 2 qt M q V (q) as maps (q k, q k ) (q k+1, q k+1 ) satisfying the implicit relations q k+1 = q k + ( t) q k ( t)2 [(1 2β)a(q k ) + 2βa(q k+1 )] q k+1 = q k + ( t)[(1 γ)a(q k ) + γa(q k+1 )] a(q) = M 1 ( V (q)), (1.19a) (1.19b) (1.19c) where the parameters γ [0,1] an β [0, 1 2 ] specify the metho. It is simple to check that the metho is secon-orer if γ = 1/2 an first-orer otherwise, an that it is generally explicit only for β = 0. The β = 0, γ = 1/2 case is well known to be symplectic (see, for example, Simo, Tarnow, an Wong [1992]) with respect to the canonical symplectic form Ω L. This can be easily seen from the fact that this metho is a rearrangement of the position-momentum form of the generalize trapezoial rule with α = 1/2. Note that this metho is the same as the velocity Verlet metho, which is popular in molecular ynamics coes. As we remarke above, if the metho (1.18) is implemente by taking one initial step with (1.18a) as a starting proceure, an then continue with (1.17), then this will give a metho essentially equivalent to explicit Newmark. To be exactly equivalent, however, an to be secon-orer accurate, one must take α = 1/2 in the use of (1.18a). This will be of importance in It is also well known (for example, Simo et al. [1992]) that the Newmark algorithm with β 0 oes not preserve the canonical symplectic form. Nonetheless it can be shown Kane et al. [2000] that the Newmark metho with γ = 1/2 an any β can be generate from a iscrete Lagrangian, an it thus preserves a non-canonical symplectic structure. An alternative an inepenent metho of analyzing the symplectic members of Newmark has been given by Skeel, Zhang, an Schlick [1997], incluing an interesting nonlinear analysis in Skeel an Srinivas [2000]. The Newmark metho is iscusse in greater etail in Galerkin methos an symplectic Runge-Kutta schemes. Both the generalize mipoint an generalize trapezoial iscrete Lagrangians iscusse above can be viewe as particular cases of linear finite element iscrete Lagrangians. If we take shape functions φ 0 (α) = 1 α φ 1 (α) = α, (1.20)

26 15 then a general linear Galerkin iscrete Lagrangian is given by L G,0 (q 0,q 1, t) = ( m w i L i=1 φ 0 (α i )q 0 + φ 1 (α i )q 1, φ 0 (α i )q 0 + φ 1 (α i )q 1 t ), (1.21) where (α i,w i ), i = 1,...,m, is a set of quarature points an weights. Taking m = 1 an (α 1,w 1 ) = (α,1) gives the generalize mipoint rule, while taking m = 2, (α 1,w 1 ) = (0,1 α) an (α 2,w 2 ) = (1,α) gives the generalize trapezoial rule. Taking high-orer finite element basis functions an quarature rules is one metho to construct high-orer variational integrators. In general, we have a set of basis functions φ j, j = 0,...,s, an a set of quarature points (α i,w i ), i = 1,...,m. The resulting Galerkin iscrete Lagrangian is then L G,s,full (q 0,...,q s, t) = m s w i L φ j (α i )q j, i=1 j=0 1 t s j=0 φ j (α i )q j. (1.22) This (s+1)-point iscrete Lagrangian can be use to erive a stanar two-point iscrete Lagrangian by taking where ext L G,s,full L G,s (q 0,q 1, t) = ext Q 1,...,Q s 1 L G,s,full (q 0,Q 1,...,Q s 1,q 1, t), (1.23) means that L G,s,full shoul be evaluate at extreme or critical values of Q 1,...,Q s. When s = 1 we immeiately recover (1.21). Of course, using the iscrete Lagrangian (1.22) is equivalent to a finite element iscretization in time of (1.1), as in Bottasso [1997] for example. An interesting feature of Galerkin iscrete Lagrangians is that the resulting variational integrator can always be implemente as a partitione Runge-Kutta metho (see for etails). Using this technique high-orer implicit methos can be constructe, incluing the collocation Gauss-Legenre family an the Lobatto IIIA-IIIB family of integrators Constraine systems Many physical systems can be most easily expresse by taking a larger system an imposing constraints, which we take here to mean requiring that a given constraint function g is zero for all configurations. To iscretize such problems, we can either work in local coorinates on the constraint set {q g(q) = 0}, or we can work with the full configurations q an use Lagrange multipliers to enforce g(q) = 0. Here we consier the secon option, as the first option requires no moification to the variational integrator theory 3. 3 In the event that the constraint set is not a vector space, local coorinates woul require the more general theory of iscrete mechanics on smooth manifols, as in 4.

27 16 Taking variations of the action with Lagrange multipliers ae requires that N 1 [ ] δ L (q k,q k+1,h) + λ k+1 g(q k+1 ) = 0 (1.24) k=0 an so using (1.4) gives the constraine iscrete Euler-Lagrange equations D 2 L (q k 1,q k,h) + D 1 L (q k,q k+1,h) = λ k g(q k ) g(q k+1 ) = 0 (1.25a) (1.25b) which can be solve for λ k an q 4 k+1. These equations have all of the conservation properties, such as symplecticity an momentum conservation, as the unconstraine iscrete equations. An interesting example of a constraine variational integrator is the SHAKE metho [Ryckaert, Ciccotti, an Berensen, 1977], which can be neatly obtaine by taking the generalize trapezoial rule of with α = 1/2 an forming the constraine equations as in (1.25). This is carrie out in Forcing an issipation Now we consier nonconservative systems; those with forcing an those with issipation. For problems in which the nonconservative forcing ominates there is likely to be little benefit from variational integration techniques. There are many problems, however, for which the system is primarily conservative, but where there are very weak nonconservative effects which must be accurately accounte for. Examples inclue weakly ampe systems, such as photonic rag on satellites, an small control forces, such as arise in continuous thrust technologies for spacecraft. In applications such as these the conservative behavior of variational integrators can be very important, as they o not introuce numerical issipation in the conservative part of the system, an thus accurately resolve the small nonconservative forces. Recall that the (continuous) integral Lagrange- Alembert principle is δ L(q(t), q(t))t + F(q(t), q(t)) δq t = 0, (1.26) where F(q,v) is an arbitrary force function. We efine the iscrete Lagrange- Alembert principle to be δ L (q k,q k+1 ) + [ F (q k,q k+1 ) δq k + F + (q k,q k+1 ) δq k+1 ] = 0, (1.27) 4 Observe that the linearization of the above system is not symmetric, unlike for constraine elliptic problems. This is because we are solving forwar in time, rather than for all times at once as in a bounary value problem.