Nonlinear Dyn DOI 10.1007/s11071-016-3290-3 ORIGINAL PAPER A nilpotent algebra approach to Lagrangian mechanics and constrained motion Aaron D. Schutte Received: 11 December 2015 / Accepted: 12 December 2016 © Springer Science+Business Media Dordrecht 2016 Abstract Lagrangian mechanics is extended to the Within the Lagrangian framework, one is compelled so-called nilpotent Taylor algebra T. It is shown that to construct the requisite equations of motion of a this extension yields a practical computational tech- mechanical system by first considering the kinemat- nique for the evaluation and analysis of the equations of ics of the system and then by evaluating the total sys- motion of general constrained dynamical systems. The tem kinetic energy T and potential energy U in order underlying T-algebra utilized herein permits the analy- to arrive at the Lagrangian L of the system. Invari- sis of constrained dynamical systems without the need ably, the process requires one to find the partial deriva- for analytical or symbolic differentiations. Instead, the tives of L with respect to (1) the generalized coordi- algebra produces the necessary exact derivatives inher- nates that describe the system state and (2) time. Fur- ently through binary operations, thus permitting the thermore, expressions for U may themselves require numerical analysis of constrained dynamical systems many differentiation evaluations with respect to the using only the defining scalar functions (the Lagrangian system configuration. This process can become non- L and the imposed constraints). The extension of the tractable if a large number of coordinates are involved. Lagrangian framework to the T-algebra is demon- In the presence of constraints, the situation is further strated analytically for a problem of constrained motion complicated since derivative information of the con- in a central field and numerically for the calculation of straints may be needed for both the development of Lyapunov exponents of N-pendulum systems. the constrained equations of motion and their numeri- cal integration. Implementation of analysis techniques Keywords Constrained motion · Automatic involving system linearization and sensitivity calcula- differentiation · Nilpotent algebra · Lagrangian tion of Lagrangian systems also relies on differenti- mechanics · Lyapunov exponents ation and can become quite challenging for complex systems. By extensions of the Lagrangian to the nilpo- tent Taylor algebra developed herein, it is the goal of 1 Introduction this paper to show that the derivation of equations of motion for general constrained dynamical systems and The traditional approach to Lagrangian mechanics their analyses can be transformed from an analytical involves the evaluation of real-valued scalar functions. to a purely numerical process given knowledge of only B the Lagrangian and the constraints on the motion. A. D. Schutte ( ) Schemes for numerical simulation of dynamical sys- The Aerospace Corporation, 2310 E. El Segundo Blvd, El Segundo, CA 90245-4691, USA tems typically rely on formulations wherein the sys- e-mail: [email protected] tem topology is determined before actual implementa- 123 A. D. Schutte tion. This is partly due to the fact that derivatives play structed. It is shown that the Taylor algebra T can be a central role in relating the kinematics of a system used to produce derivatives to arbitrary order of contin- to its dynamics. Practical numerical implementation uous real-valued multivariate functions. The Jacobian of the equations of motion of a constrained dynami- and Hessian numbers are constructed from truncated cal system essentially relies on three different forms T-algebras and used in the description of Lagrangian of differentiation, which include (1) analytical differ- mechanics. More importantly, it is demonstrated how entiation by hand, (2) symbolic differentiation, or (3) the algebras are used to numerically evaluate the ele- automatic differentiation. The first option is feasible if ments needed to formulate general constrained equa- all the necessary derivatives can be obtained by hand tions of motion for dynamical systems and to perform prior to implementation. However, the process must be linearization and sensitivity analyses. Finally, the exact repeated with changing system topology and is error nature of the derivatives produced by the algebras is prone for complex systems. Symbolic differentiation demonstrated analytically for a problem of constrained utilizes a computer algebra system to accomplish the motion in a central field and numerically for the calcu- same feat in a more automated fashion, and as a result lation of Lyapunov exponents for Lagrangian systems. is less error prone. Both of these forms of differenti- ation lead to algorithms that rely on concrete analyti- cal expressions. In contrast, automatic differentiation 2 Nilpotent J, H, and T algebras obtains derivatives of functions numerically. Common automatic differentiation methods apply a judicious Consider a commutative ring R. By definition, the use of the chain rule to evaluate the derivative of a binary operations of addition (+) and multiplication (·) given function by source code transformation or oper- on R satisfy the usual commutative, associative, and ator overloading [1,2]. This assumes that the function distributive axioms. The element 0 ∈ R is the additive of interest is suitably differentiable and can be imple- identity and 1 ∈ R is the multiplicative identity. While mented in a particular programming language. These all elements in R have an additive inverse, we do not methods fall into the classification of forward or reverse assume multiplicative inverses. Thus, R is not a field. mode techniques, where forward and reverse are used Elements in R that have a multiplicative inverse are to indicate the direction traversed by the chain rule. called units. It is also essential to note that an element k Software implementation of these techniques is widely ∈ R is nilpotent to order k if = 0 for minimal + available [3]. Another popular form of numerical dif- k ∈ N . ferentiation includes the complex-step derivative [4– Definition 1 Let R[x1,...,xn] be a polynomial ring 6], which has been used to numerically compute first- over the real numbers in the n algebraically indepen- and second-order derivatives. However, this approach dent indeterminates x1,...,xn. R[x1,...,xn] is called is based on the finite difference approximation and the set of polynomials over the ring R, and it is com- is subject to truncation errors, making it a generally mutative since R is commutative. unfeasible option for the numerical development and analysis of equations of motion for constrained sys- Definition 2 An ideal I is a nonempty subset of the tems. commutative ring R such that the following closure Yet another approach to automatic differentiation properties are satisfied involves the use of the so-called dual number alge- 1. ∀a, b ∈ I: a ± b ∈ I. bra. The dual numbers, which can be traced back to 2. ∀a ∈ I and ∀b ∈ R: ab ∈ I. the work of Clifford [7], have been shown to produce exact first derivatives of real-valued functions by sim- An ideal allows the construction of a quotient ring R/I ply extending the function to the dual number alge- (also written as R mod I). This is essentially a new bra. In mechanics, dual numbers have been primarily ring with the elements of the ideal I removed from used in kinematics analysis; e.g., see [8]. A dual num- the ring R. The quotient of a polynomial ring in mul- ber extension to second-order derivatives has recently tiple indeterminates is given by R[x1,...,xn]/I.This been developed [9], which was successfully applied to is an important type of ring since many rings are con- a Navier–Stokes solver. In what follows, a superset of veniently expressed as a quotient of polynomial rings. the dual numbers called the Taylor numbers is con- Here, it is assumed the ideal I = (α1,...,αm) is a 123 A nilpotent algebra approach to Lagrangian mechanics and constrained motion set of generators such that αi ∈ R[x1,...,xn].This Clearly, Eq. (7) represents a first degree Taylor polyno- allows the construction of the Taylor algebra mial. Higher degree terms vanish since higher powers of are zero. Thus, f˜(z) returns the evaluated function T(1,...,n) R[x1,...,xn] f (a0) and information about the first derivative f (a0). /{x x ···x } , , ∈N+ , (1) i j m i j m The exact derivative f (a0) is obtained by evaluating f˜(a + ). This is one of the most useful properties of where the ideal I ={xi x j ···xm}i, j,m∈N+ is both max- 0 imal and nilpotent to order m for 1 ≤ i, j,...,m ≤ n. the Jacobian algebra. An ideal is nilpotent when Ik ={0} for any k ∈ N+. Extending the Jacobian algebra to higher derivatives T Note that the T-algebra actually depends on the trun- yields another subset of a truncated –algebra, which 3 = cation order m. Obscurely, the most germane and inter- has a third-order nilpotent element such that 0. H esting property of this algebra lies in its ability to pro- In a single indeterminate , the Hessian algebra, ,is duce exact derivatives to arbitrary order of continuous given in the single indeterminate form as real-valued multivariate functions. This can be accom- H() R[x]/{x3}. (8) plished by simply extending the function of interest to The Hessian number is given by the T–algebra of appropriate truncation order. = + + 2, The dual numbers are a well-known subset of a z a0 a1 a2 (9) truncated T–algebra. Abstractly, the dual numbers are where a0, a1, a2 ∈ R. Addition and multiplication of derived from the Jacobian algebra, J, in a single inde- the Hessian number follow the definitions terminate so that (a0, a1, a2) + (b0, b1, b2) J() [ ]/{ 2}; R x x (2) = (a0 + b0, a1 + b1, a2 + b2) (10) i.e., J() is isomorphic to the quotient ring R[x]/{x2}.