LAGRANGIAN MECHANICS WITHOUT ORDINARY DIFFERENTIAL EQUATIONS

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1 Vol. 57 (2006) REPORTS ON MATHEMATICAL PHYSICS No. 3 LAGRANGIAN MECHANICS WITHOUT ORDINARY DIFFERENTIAL EQUATIONS GEORGE W. PATRICK Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N5E6, Canada ( math.usask.ca) (Received December 5, 2005) A variational proof is provided of the existence and uniqueness of evolutions of regular Lagrangian systems. Keywords: Lagrangian systems, existence and uniqueness, implicit function theorem. Introduction Let Q be a smooth, finite-dimensional manifold and L:TQ-~ N be a smooth function. The evolutions of the Lagrangian system defined by L are the C 1 curves q : [0, h] -+ ~ which are critical points of the action Sh = L o q'(t) dt, subject to the constraint that q(0) and q(h) are constant. Standard ODE theory provides existence and uniqueness of the corresponding initial value problem because the derivatives q~(t) of the evolution curves q(t) are the integral curves of the corresponding Lagrangian vector field XE. Given two nearby ql, q2 6 Q, does there exist a unique evolution curve q(t) such that q(0) = q~ and q(h) = q2? This local boundary value problem occurs, for example, in the following two contexts: 1. If Q is a Riemannian manifold with metric g, and L(v) = lg(v,v), then the Lagrangian evolution curves are constant-speed reparameterizations of the geodesics, and the local boundary value problem becomes that of locating the unique local geodesic connecting two sufficiently nearby points. 2. Type 1 generating functions St(q2, ql) for the Hamiltonian flow are defined by St(q2, ql) = fo l Loq(s)ds, *Partially supported by the Natural Sciences and Engineering Research Council, Canada. [4371

2 438 G.w. PATRICK where q(s) is the solution to the local boundary value problem with q(0)= ql, q(t) = q2. The solutions to the local boundary value problem may be obtained from the Lagrangian flow F xe by solving the equation rqf fe (Uql) = q2 for vql ~ Tql Q as a function of ql, q2, t, where rq : T Q --+ Q is the canonical projection [6]. This is not a straightforward application of the implicit function theorem because the relevant derivative is singular at t --0, ql = q2. Using the initial value problem seems like a circuitous route to the local boundary value problem, especially considering that the boundary values ql and q2 are constraints of the original variational formulation. Regular constrained optimization problems have critical points which persist as smooth functions of the constraint values. Why not solve the boundary value problem directly by persistence from h = 0, and avoid an excursion into the initial value problem? However, the problem of finding the critical points of Sh subject to the constraint q(o) = qo, q(h) = qa, is nonregular at h = 0. For example, the objective function Sh is zero when h , and the constraint function q(t) ~ (q(0), q(h)) cannot be a submersion, because it maps into the diagonal of Q Q at h = 0. One cannot perturb from such a degenerate landscape. This article provides a direct variational proof of existence and uniqueness of the local boundary value problem, using a regularization procedure which is adapted from the one used in [3] for a similar problem in the context of discrete Lagrangian systems. The procedure results in the replacement of the variational problem with an equivalent one which is regular at h = 0, after which methods used in the proof of the infinite dimensional Morse lemma, and the (infinite dimensional) implicit function theorem, give the result. 1. Regularization Assume that L : Q --~ R is C r with r > 2. Since the aim is for a local result, also assume that Q is an open subset of Rn, and that 0 ~ Q is given. One seeks a perturbative approach from t = 0, ql = q2 = 0, so that ql and q2 will be near c). The regularization procedure is, step-by-step, as follows: 1. Transform the variational problem from one for curves in Q to one for curves in T Q, with an additional first-order constraint. The transformation is simply to seek critical points (q(t), v(t)) ~ TQ of the objective Sh = L(q(t), v(t)) dt, subject to the constraints v(t) = d-~q (t), q(o) = ql, q(h) = q2. at

3 LAGRANGIAN MECHANICS WITHOUT ORDINARY DIFFERENTIAL EQUATIONS 439 There are additional freedoms inherent in the use of curves in T Q rather than curves in Q, and these are important for the regularization. 2. Reparameterize, so that the solution curves, which are defined on [0, h], do not disappear as h The curves (q(t), v(t)), t c [0, h], are replaced by the curves (Q(t), V(t)), t c [0, 1], using Q(u) = q(hu), V(u) = v(hu), u ~ [0, 1]. For h > O, the new curve (Q(u), V(u)) satisfies an equivalent variational problem, which can be worked out as follows. First, substitute u = t/h and divide by h to obtain L(q(t),v(t))dt= lf0 1 L(Q(u), V(u))du. Since h is constant for the variational principle, one can use the right-hand side as an objective for (O(u), V(u)). The first-order constraint is 0 = --~q(t) -- v(t) = - --~u(q(u)) - V(u) u=t/h" Multiplying this by h, the reparameterized variational principle is for curves (Q(u), V(u)) with values in TQ which are critical points of the objective subject to the constraints S h ~-- f0 1 L(Q(u), V(u)) du, ~u(o(u)) d - hv(u) = O, O(O) = ql, Q(1) = q2. Notice that both the objective and the constraints are smooth through h----0 and that, at h = 0, the first-order constraint is equivalent to the constraint that the curve (Q(u), V(u)) lies in a fiber of TQ, i.e. the curve is vertical. 3. Restrict the remaining (boundary) constraints to the submanifold defined by the regularized first-order constraint, and regularize the result. The first-order constraint may be solved smoothly through h = 0 by integration, U Q(u) = Q(O) + h V(s) ds. (1) do Thus the set of curves (Q(u), V(u)) may be replaced by the set of curves {V(u)}. Also, q2 may be replaced by ql + hz, z ~ IR n. Under (1), Q(1) = Q(O) + h fo V(s) ds,

4 440 G.w. PATRICK so the objective and the boundary constraint become, respectively, Sh = f01 L ( ql + h fo V(s) ds, V(u) ) du, f01 V(u) du = z, (2) where h and q~ appear as parameters. This completes the regularization procedure, since the variational problem (2) is formally regular through h = 0. Indeed, at h = 0, (2) becomes finding the curves V(u) which are critical points of the constrained problem f0' f0' So = L(ql, g(u)) du, V(u) du = z. Using a Lagrange multiplier Z, the solutions are given by setting f0 -~v(ql, V(u))6V(u)du = fo' Z 8V(u)du for all 6V(u), i.e. OL (ql, V(u)) : Z. oat; If L is a regular Lagrangian, this implies V(u) is constant, and the constraint then implies V(u)= z. Thus the solution to (2) at constraint value z is the constant curve u ~ z, and there is exactly one critical point for each constraint value. 2. Solution of the regularized problem Consider Sh from (2) on the Banach space Ck([0, 1], R') of curves V(u), where 0_ k _< r. Since L is C r, the Omega Lemma ([2], page 102) implies that the integrand of Sh is C r as a map into C ([0, 1], IR). Since integration on C ([0, 1], ~) is bounded linear (and therefore C~), it follows that Sh is C r, irrespective of the value of k. I specialize the method for proving the infinite-dimensional Morse lemma [4, 7, 8] to the constrained variational problem (2). One first calculates the gradient of Sh from the derivative dsh using the C2([0, 1], R n) weak inner product ((v, w)) = fo v.w. That computation is as follows (for short, below Q(u) means the right-hand side of (1)): dsh ([V (u) ])(~ V (u) fo ) -- L ql + h V(s) + e ~V(s) ds, V(u) + e 6V(u) du d~ 1 ['OL, u u OL

5 from which fol ~v (Q(u), V(u))(~V(u) du IaL" 8V(u) = (av(u)- f01 8V(u) ) f01 SV(u). u fo ~v (Q ( ), V(u))3V(u)du Sh has a critical point on the level sets of the constraint if and only if the orthogonal projection l?e0 of VSh to the kernel E0 is zero, i.e. for the solutions to the constrained variational problem (2), one solves for V0 near LAGRANGIAN MECHANICS WITHOUT ORDINARY DIFFERENTIAL EQUATIONS = h~q(q(u), V(u))6V(s)duds 4- _ 1 lhol " s -- fo i -3--q (Q( ), V(s))SV(u)dsdu 4- = h~q(q(s), V(s))ds 4- ~v(q(u), V(u)) 8V(u)du, 3L I OL s VSh = ~v(q(u), V(u)) 4- h ~q(q( ), V(s))ds, Q(u) = ql 4- h V(s) ds. By the same reasoning as was used to find the differentiability of Sh, the gradient VSh (best thought of as a vector field) is a C r-1 map from Ck([0, 1],R n) to C ([0, 1], IRn), also irrespective of the value of k, 0 < k < r - 1. The constraint of (2) (i.e. the second equation) is C a because it is bounded linear, and its derivative is 3V(u) ~ /o' 8V(u) du. The kernel of this derivative, say E0, is the tangent space to the constraint set, and it splits Ck([0, 1], ]R n) orthogonally with respect to the metric ((,)) (the complement is the subspace of constant functions) by PEoV&(Vo (~ vl) = o Vo=0, V1 =0, z =0, ql =c~, h =0. To use the implicit function theorem, one requires that the appropriate partial derivative of I?~oVSh is a linear isomorphism. Remembering to set h = 0, that derivative is d E=0 3L 02L d~-- ]~E 5-~v (q' ~ avo(u)) = ]PEo S-~v2 (q, o)avo(u) 32L f1021, = ava(q,o)avo(u)- -~vi(q,o)avo(u) a2l = av 2 (q, o)~vo(u). /0 441

6 442 G.W. PATRICK If L is regular, this is a linear isomorphism of E0, with inverse _[ 02L _ )-1 3Vo(u) ~-~ ~vz(q, O) 3Vo(u). Thus, the implicit function theorem provides neighborhoods W1 c IR n x ]R n x ]R = {(ql,z, h)} containing (0, 0,0) and W2 c Ck([0, 1],IR ~) of the constant curve u ~ 0, and a C r-j map ~ : W1 --+ W2 such that for all (ql,z, h) E W1, ~(ql, z, h) E Ck([0, 1], R ") is the unique critical point in W2 of the constrained variational problem (2). By setting k = 0 and then k = r- 1, one can arrange that W2 is a C r-1 neighborhood, ~ has values in W2, and hence in the C r-i curves, 7t is C r-1 with the C r-1 topology, but that ~ provides the unique solution among the C o curves in a C o open neighborhood, say {v(u) : IV(u)l < el, of the constant curve u ~ 0. Now reverse the regularization. Pick an h > 0 such that (0, 0, h) 6 W1, set and define 17(1 -= {(ql, q2) : (ql, (q2 -- ql)/h, h) E W1} 1 ) ~(ql,q2)(t) :ql -F ~ ql, ~(q2 -ql),h (u)du. do Then (c~, q) E is defined for (ql, q2) E l'~rl and t 6 [0, h], is a first-order curve in T Q which has base integral curve a Lagrangian evolution. This evolution is unique among the continuous curves corresponding to [V(u)L < e, i.e. among C 1 curves q(t) such that [q'(t)[ < e/h, so C 1 curves q(t) in some C 1 neighborhood of the constant curve u ~ Remarks The regularization can be formulated in invariant terms on the manifold Q, using a tubular neighborhood of the antisymmetric normal bundle of the diagonal of Q x Q to accomplish the subtraction q2- ql. Replacing (Q(u), V(u)) with its TQ version V(u), the regularized variational problem, at h = 0, becomes f0 f0 So = L o V, V(u) = z, rq o V = constant. It is a pretty result that the variational principle on Q regularizes to this trivial one on the fibers of T Q. The map 7) is defined only for small z = (q2- ql)/h, and since ~(ql,q2) is a solution which goes from ql to q2 in time h, the velocity of this solution is also, approximately, (q2- ql)/h. Thus regularizing only at z = 0 provides evolutions which correspond only to velocities near zero. This is unacceptable if the objective is to solve the initial value problem by first solving the local boundary value

7 LAGRANGIAN MECHANICS WITHOUT ORDINARY DIFFERENTIAL EQUATIONS 443 problem, because it assigns evolutions only to those initial data corresponding to velocities near zero, whereas from ODE theory there is a unique integral curve of the Lagrangian vector field corresponding to any velocity. However, minor extensions of the above show that the variational principle regularizes at all z. The local solutions so obtained along the solutions u w-~ z of the regularized variational principle at h = 0, may be glued together using a technique that can be found for example in [5], page 97, thus providing solutions starting at any velocity. This is an important step in the discrete Lagrangian context [3], where the discrete initial value problem is addressed by first solving the local boundary value problem. REFERENCES [1] R. Abraham and J. E. Marsden: Foundations of Mechanics, Addison-Wesley, second edition, [2] R. Abraham, J. E. Marsden and T. S. Ratiu: Manifolds, Tensor Analysis, and Applications, Springer, second edition, [3] C. Cuell and G. W. Patrick: Discretizations of Lagrangian mechanics, in preparation. [4] M. Golubitsky and J. E. Marsden: The Morse lemma in infinite dimensions via singularity theory, SlAM J. Math. Anal. 14 (1983), [5] S. Lang: Differential Manifolds, Addison-Wesley, [6] G. W. Patrick: Two axially symmetric coupled rigid bodies: relative equilibria, stability, bifurcations, and a momentum preserving symplectic integrator, PhD thesis, University of California at Berkeley, [7] A. J. Tromba: Almost-Riemannian structures on Banach manifolds: the Morse lemma and the Darboux theorem, Canad. J. Math. 28 (1976), [8] A. J. Tromba: A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem, Math. Ann. 263 (1983),