Energy-preserving affine connections

Transcription

1 2 A. D. Lewis Enery-preservin affine connections Anrew D. Lewis 28/07/1997 Abstract A Riemannian affine connection on a Riemannian manifol has the property that is preserves the kinetic enery associate with the metric. However, there are other affine connections which have this property, an here we characterise them. A class of such enery-preservin affine connections, relate to mechanical systems with constraints, is provie. Keywors. mechanics, conservation of enery, affine connections, istributions AMS Subject Classifications. 53B20, 53C22, 70H03 1. Introuction On a Riemannian manifol (M, ), eoesics of the Levi-Civita affine connection, or more enerally, a Riemannian affine connection (i.e., not necessarily torsion-free) leave invariant the kinetic enery of the Riemannian metric. Precisely, if Z enotes the eoesic spray of a Riemannian connection, an if we efine a function K(v) = 1 2(v, v) on T M, then L Z K = 0 where L Z is the Lie erivative with respect to Z. It is interestin to consier to what extent this characterises Riemannian affine connections. Here we provie analytical conitions for a eneral affine connection to leave invariant the kinetic enery. Motivate by a construction of Syne [1928], iven a istribution D on M, we construct a natural enery-preservin affine connection on M which restricts to D. This construction has been explore in etail by the author in [Lewis 1998]. Applications to control theory are provie in [Lewis 2000]. In this paper we will follow the ifferential eometric notation an conventions of Abraham, Marsen, an Ratiu [1988], an we refer to [Kobayashi an Nomizu 1963] for backroun on affine connections. We enote by C (M) the set of C functions on a manifol M, by T (M) the C vector fiels, an by T (M) the C one-forms. If (M, ) is a Riemannian manifol, we enote by the Levi-Civita affine connection. We recall that in coorinates the Christoffel symbols of are Γjk i = 1 ( lj 2 il x k + lk x j ) jk x l. (1.1) If S is an (r, s) tensor fiel an is an arbitrary affine connection, we enote by S the (r, s + 1) tensor fiel efine by S(α 1,..., α r, X 0, X 1,..., X s ) = X0 S(α 1,..., α r, X 1,..., X s ). Preprint Assistant Professor, Department of Mathematics an Statistics, Queen s University, Kinston, ON K7L 3N6, Canaa anrew@mast.queensu.ca, URL: Research supporte by EPSRC Grant GR/K Characterisin enery-preservin affine connections Let (M, ) be a finite-imensional Riemannian manifol. If c: [a, b] M is a curve on M, efine a function E c alon c by E c (t) = 1 2(ċ(t), ċ(t)). An affine connection on M is enery-preservin if for every eoesic c of we have t E c = 0. One may reaily verify that this is equivalent to sayin, as we i in the introuction, that L Z K = 0 where Z is the eoesic spray of an K(v) = 1 2(v, v). The efinition we ive here is better suite to the computations we will here perform. To motivate our first result, let us perform a coorinate computation. The computation we perform comes from one mae in [Murray, Li, an Sastry 1994, Lemma 4.2 1] to erive stability results for certain control laws in robotics. We let M = R n an let be a Riemannian metric on R n. In stanar coorinates (x 1,..., x n ) for R n we have E c = 1 2 ijẋ i ẋ j for a curve c: t (x 1 (t),..., x n (t)). If c is a eoesic of then ẍ i = Γ jkẋj i ẋ k. Now, for a eoesic c of we compute ij t E c = ij ẋ i ẍ j x k ẋi ẋ j ẋ k = 1 2 Usin (1.1) we reaily compute ij x k 2 ilγjk l = jk x i ( ij x k 2 il Γjk l ) ẋ i ẋ j ẋ k. (2.1) ik x j. (2.2) Skew-symmetry of this expression in the inices i an j immeiately implies that t E c = 0. This raises the question which we aress in this paper; namely the question of exactly when oes an affine connection preserve enery. It is helpful to have the followin result from multilinear alebra. We enote by S k the permutation roup on k symbols. If A is a (0, k) tensor on a vector space V efine the symmetric part of A by Sym(A)(v 1,..., v k ) = 1 A(v k! σ(1),..., v σ(k) ). Of course, the same construction applies to (0, k) tensor fiels on manifols. 2.1 Lemma: Let V be a vector space over a fiel of characteristic zero an let A be a (0, k) tensor on V. The followin are equivalent: (i) A(v,..., v) = 0 for every v V ; (ii) Sym(A) = 0. Proof: (i) = (ii) Let A be a (0, k) tensor with the property that A(v,..., v) = 0 for all v V. Usin the latter property of A, for v 1,..., v k V we have ( k k ) A v i1,..., v ik = A(v σ(1),..., v σ(k) )+ i 1=1 i k=1 k 1 l=1 j 1,...,j l A(v j1 + + v jl, v j1 + + v jl ) 1

2 Enery-preservin affine connections 3 4 A. D. Lewis where the inner sum in the final term is over all istinct subsets j 1,..., j l {1,..., k} with j 1 <... j i < < j l. This then ives A(v σ(1),..., v σ(k) ) = 0 for v 1,..., v k V an so Sym(A) = 0. (ii) = (i) For v V we have k A(v,..., v) = ka(v,..., v) = 0 i=1 so A(v,..., v) = 0 since V is over a fiel of characteristic zero. Now we can easily ive the followin characterisation of enery-preservin affine connections. 2.2 Proposition: Let (M, ) be a Riemannian manifol an let be an arbitrary affine connection on M. The followin are equivalent: (i) is enery-preservin; (ii) Sym( ) = 0; (iii) for every chart (U, φ) for M with coorinates (x 1,..., x n ) we have iσ(1) i σ(2) x i 2 iσ(2) lγ l i σ(3) σ(3) i σ(1) = 0 σ S 3 for i 1, i 2, i 3 = 1,..., n. Here Γ i jk are the Christoffel symbols of in the chart (U, φ). Proof: (i) (ii) The affine connection is enery-preservin if an only if for every eoesic c: [a, b] M of we have t E c(t) = 1 2 ( ċ(t))(ċ(t), ċ(t)) + ( ċ(t) ċ(t), ċ(t)) = 1 2 ( ċ(t))(ċ(t), ċ(t)) = 0. Thus is enery-preservin if an only if (v, v, v) = 0 for every v T x M an x M. The result now follows from Lemma 2.1. (i) (iii) Proceein exactly as we i in the computation of (2.1) we see that is enery-preservin if an only if ( ) ij x k 2 jlγ l ki (x)v i v j v k = 0 for every x M, every chart (U, φ) aroun x with coorinates (x 1,..., x n ), an every v = v i x i T x M. The result now follows from Lemma 2.1. This shows, in particular, that if ij x k 2 jlγ l ki = ji x k + 2 ilγ l kj then is enery-preservin. In particular, by (2.2) this means that the Levi-Civita affine connection is enery preservin. It is also clear that any Riemannian affine connection is enery preservin. Inee, a Riemannian affine connection is characterise by = 0, so (ii) shows that is enery-preservin. 3. A class of enery-preservin affine connections In this section we provie a collection of non-riemannian enery-preservin affine connections. These arise in a natural way if one is iven a istribution D on M. First we review some of the ieas of Lewis [1998] Affine connections an istributions. Let be an affine connection on M an let D be a istribution on M. We enote by D the set of vector fiels on M takin their values in D. We say that restricts to D if X Y D for every Y D. Thus, if restricts to D then we may rear as a connection in the vector bunle D. It is important that one not require an affine connection which restricts to a istribution to be torsion-free; one reaily sees that this implies that D is interable. A notion weaker than restrictin to a istribution is that of eoesic invariance. We shall say D is eoesically invariant if for every eoesic c: [a, b] M of, ċ(t) D c(a) implies that ċ(t) D c(t) for t (a, b]. The followin result characterises eoesically invariant istributions. 3.1 Theorem: ([Lewis 1998]) D is eoesically invariant if an only if X Y + Y X D for every X D Restrictin the Levi-Civita connection to a istribution. Let (M, ) be a Riemannian manifol an let D be a istribution on M. Let D enote the orthoonal complement of D an enote by P : T M T M an P : T M T M the orthoonal projections onto D an D, respectively. If : T M T M an : T M T M are the musical isomorphisms, we efine ˆD = D an ˆD = (D ), the coistributions of annihilators of D an D, respectively. We write the projections onto ˆD an ˆD as ˆP : T M T M an ˆP : T M T M, respectively. If enotes the Levi-Civita connection, we wish to efine a new affine connection on M which has the properties 1. restricts to D, an 2. X Y X Y D for Y D. State otherwise, X Y = P ( X Y ) for Y D. Vershik [1984] efines P ( X Y ) as a vector bunle connection, an makes the point that there is no natural way to exten this to an affine connection on M. Here we escribe what such an extension shoul look like. The followin result is iven in [Lewis 1998]. 3.2 Proposition: Let D be a istribution on a Riemannian manifol (M, ). Let be the Levi-Civita connection an suppose that another affine connection has the properties (i) X Y D for every Y D, an (ii) X Y X Y D for every Y D. Then X Y = X Y + ( X P )(Y ) + S(X, Y ) for some (1, 2) tensor fiel S such that P (S(X, Y )) = 0 for Y D. Conversely, if is of this form, then it satisfies (i) an (ii).

3 Enery-preservin affine connections 5 6 A. D. Lewis Proof: We may write any affine connection on M as X Y = X Y + B(X, Y ) for some (1, 2) tensor fiel B. In particular, an affine connection satisfyin (i) an (ii) must be of this form. For Y D an any vector fiel X we have We also have Usin (i), (ii), an (3.1) we have P (Y ) = 0 = ( X P )(Y ) + P ( X Y ) = 0 = P ( X Y ) = ( X P )(Y ). (3.1) X Y = X Y + B(X, Y ). P ( X Y ) + B(X, Y ) = 0 = B(X, Y ) = ( X P )(Y ). Thus X Y = X Y + ( X P )(Y ) + S(X, Y ) for some S such that P (S(X, Y )) = 0 for Y D. Now we show that satisfies (i) an (ii) if it is of the iven form. Let X an Y be vector fiels on M. Then If Y D then P ( X Y ) = P ( X Y ) + P ( X P )(Y ). (3.2) P (Y ) = 0 = ( X P )(Y ) + P ( X Y ) = 0 (3.3) = P ( X P )(Y ) + P ( X Y ) = 0 (3.4) since P P = P. Substitutin (3.4) into (3.2) we see that P ( X Y ) = 0 for X T (M) an Y D. Therefore, X Y D. Now let Y D. From (3.3) we have ( X P )(Y ) + P ( X Y ) = 0 = P ( X P )(Y ) = 0 This may be verifie as follows. Let Y D. Then P (Y ) = Y = ( X P )(Y ) + P ( X Y ) = X Y = P ( X P )(Y ) + P ( X Y ) = P ( X Y ) = P ( X P )(Y ) = 0. Let us see how this relates to nonholonomic mechanics. We consier a mechanical system on M with Laranian L(v) = 1 2(v, v). The istribution D efines a nonholonomic constraint for the mechanical system. Thus we require the velocities of the system to lie in D. The Larane- Alembert principle for constraine systems states that solutions c: [a, b] M of the constraine problem satisfy ċ(t) ċ(t) = λ(t) P (ċ(t)) = 0 (3.6a) (3.6b) where λ is a section of D alon c (a Larane multiplier, if you like). We now show how solutions to (3.6) are relate to the affine connections escribe by Proposition 3.2. This is an intrinsic version of the construction of Syne [1928]. 3.3 Proposition: A curve c: [a, b] M is a solution to (3.6) if an only if ċ(a) D c(a) an c is a eoesic of any of the affine connections characterise by Proposition 3.2. Proof: Proposition 3.2(i) an Theorem 3.1 imply that D is eoesically invariant. Thus, if is one of the affine connections characterise by Proposition 3.2, every eoesic of whose initial velocity lies in D will satisfy (3.6b). Now suppose that c is a solution of (3.6) an let be an affine connection as in Proposition 3.2. We may ifferentiate (3.6b) to obtain ( ċ(t) P )(ċ(t)) + P ( ċ(t) ċ(t)) = 0. From (3.6a) we also have P ( ċ(t) ċ(t)) = λ(t) since λ is a section of D. We then see that so that c satisfies ċ(t) ċ(t) + ( ċ(t) P )(ċ(t)) = 0. λ(t) = ( ċ(t) P )(ċ(t)) (3.7) since P P = 0. Thus ( X P )(Y ) D for Y D. We will also fin it useful to have the followin formula: If we efine a (1, 2) tensor fiel S by S(X, Y ) = X Y X Y ( X P )(Y ) then we have ċ(t) ċ(t) + ( ċ(t) P )(ċ(t)) + S(ċ(t), ċ(t)) = 0 ( X P )(Y ) D, Y D. (3.5) by the properties of. Thus c is a eoesic of. Now suppose c: [a, b] M is a eoesic of an affine connection efine by Proposition 3.2 an that ċ(a) D c(a). Then c satisfies (3.6b) by Proposition 3.2(i). Also, c satisfies (3.6a) with λ efine by (3.7). This completes the proof.

4 Enery-preservin affine connections 7 8 A. D. Lewis 3.3. Enery-preservin affine connections which restrict to a istribution. In this section we combine all that we have one in previous sections an construct an affine connection of the type escribe by Proposition 3.2 an which preserves enery. The followin lemma will be useful. 3.4 Lemma: Let (M, ) be a Riemannian manifol with the Levi-Civita affine connection an another affine connection efine by X Y = X Y + A(X, Y ) for a (1, 2) tensor fiel A. For vector fiels X, Y, Z on M we have Proof: We compute Also ( X )(Y, Z) = (A(X, Y ), Z) (Y, A(X, Z)). L X ((Y, Z)) = ( X )(Y, Z) + ( X Y, Z) + (Y, X Z). (3.8) L X ((Y, Z)) = ( X )(Y, Z) + ( X Y, Z) + (Y, X Z) = ( X )(Y, Z) + ( X Y, Z) + (Y, X Z)+ (A(X, Y ), Z) + (Y, A(X, Z)). (3.9) Subtractin (3.8) from (3.9) an usin the fact that X = 0 we obtain the esire result. To characterise the affine connections we are intereste in, it will be convenient to introuce some notation. If A is an (r, s) tensor fiel on a Riemannian manifol (M, ) then we efine an (0, r + s) tensor fiel à by With this we state the followin result. Ã(X 1,..., X r+s ) = A(X 1,..., X r, X r+1,..., X s ). 3.5 Proposition: An affine connection : (X, Y ) X Y + ( X P )(Y ) + S(X, Y ) of the form specifie by Proposition 3.2 is enery-preservin if an only if Sym( S) + Sym( P ) = 0. Proof: Apply Proposition 2.2(ii) an Lemma 3.4. Of course this oes not uarantee the existence of enery-preservin affine connections of the form specifie by Proposition 3.2. The followin result, however, ives an example of just such an affine connection. 3.6 Proposition: Let (M, ) be a Riemannian manifol with D a istribution on M. There exists an affine connection on M with the followin properties: (i) satisfies (i) an (ii) of Proposition 3.2; (ii) is enery-preservin. Furthermore, the affine connection D efine by α( D X Y ) = α( X Y ) + P ( ˆP (α), X, P (Y )) P ( ˆP (Y ), X, P (α )). for X, Y T (M) an α T (M), satisfies these properties. Proof: We note that P (α, X, Y ) = P ( ˆP (α), X, P (Y )) + P ( ˆP (α), X, P (Y ))+ P ( ˆP (α), X, P (Y )) + P ( ˆP (α), X, P (Y )) = P ( ˆP (α), X, P (Y )) + P ( ˆP (α), X, P (Y )) by Proposition 3.2(ii) an (3.5). Thus D can be efine by α( D X Y ) = α( X Y ) + P (α, X, Y ) P ( ˆP (α), X, P (Y )) P ( ˆP (Y ), X, P (α )). By Proposition 3.2, D satisfies (i). Next we show that D is enery preservin. Let c be a eoesic of D an compute t E c(t) = 1 2 ( D ċ(t) )(ċ(t), ċ(t) + ( D ċ(t) ċ(t), ċ(t)) = P ( ˆP (ċ(t) ), ċ(t), P (ċ(t))) P ( ˆP (ċ(t) ), ċ(t), P (ċ(t))) = 0 usin Lemma 3.4 an the fact that c is a eoesic of. D This establishes the existence of an affine connection with the specifie properties. We immeiately have the followin result. 3.7 Corollary: For every solution c of (3.6), t E c = 0. That is, enery is conserve for constraine systems. The affine connection D iven in Proposition 3.6 is istinuishe in another way than its bein enery-preservin. Recall that if is an arbitrary affine connection on M, a iffeomorphism φ: M M is an affine transformation for if φ ( X Y ) = φ Xφ Y for every X, Y T (M). A iffeomorphism φ is compatible with a istribution D if T x φ(d x ) = D φ(x) for every x M. 3.8 Proposition: If a iffeomorphism φ: M M is an affine transformation for an is compatible with D, then φ is an affine transformation for D. Proof: We first claim that φ (P (X)) = P (φ X) for every X T (M). Inee, if we write X = X 1 + X 2 where X 1 D an X 2 D we have P (X) = X 1. Also, φ X 1 D since φ is compatible with D. For any Y 1 D an Y 2 D we compute 0 = φ ((Y 1, Y 2 )) = φ (φ Y 1, φ Y 2 ) = (φ Y 1, φ Y 2 )

5 Enery-preservin affine connections 9 10 A. D. Lewis which shows, in particular, that φ X 2 D. Therefore, P (φ X) = φ X 1. This emonstrates our claim that φ (P (X)) = P (φ X) for every X T (M). In like fashion we have φ (P (X)) = P (φ X) for every X T (M). From this it also follows that φ P = P. We now claim that φ P = P. For X, Y T (M) an α T (M) we have φ ( P (φ α, φ X, φ Y )) = (φ P )(α, X, Y ). Syne, J. L. [1928] Geoesics in nonholonomic eometry, Mathematische Annalen, 99, Vershik, A. M. [1984] Classical an Non-classical Dynamics with Constraints, paes , number 1108 in Lecture Notes in Mathematics, Spriner-Verla, New York- Heielber-Berlin. We also have φ (( φ XP )(φ Y )) = φ ( φ X(P (φ Y ))) φ (P ( φ Xφ Y )) = X (P (Y )) P ( X Y ) = ( X P )(Y ) usin the fact that φ is an affine transformation for an that φ P = P. This then ives φ ( P (φ α, φ X, φ Y )) = α ( φ (( φ XP )(φ Y )) ) = α ( ( X P )(Y ) ) = P (α, X, Y ). This shows that (φ P )(α, X, Y ) = P (α, X, Y ) for every X, Y T (M) an α T (M). In other wors, φ P = P. The result now follows easily from the above computations an the efinition of D. Lewis [1998] iscusses D-affine transformations by restrictin interest only to D. Here we have shown that these notions can be extene to all of T M. Acknowleements The author wishes to thank Richar Murray for a few years ao askin the question, What is the meanin of Ṁ 2C? (the quantity Ṁ 2C bein rouhly the non-intrinsic object escribe in coorinates by (2.2)). References Abraham, R., Marsen, J. E., an Ratiu, T. S. [1988] Manifols, Tensor Analysis, an Applications, secon eition, number 75 in Applie Mathematical Sciences, Spriner- Verla, ISBN Kobayashi, S. an Nomizu, K. [1963] Founations of Differential Geometry, Volume I, number 15 in Interscience Tracts in Pure an Applie Mathematics, Interscience Publishers, New York, ISBN Lewis, A. D. [1998] Affine connections an istributions with applications to nonholonomic mechanics, Reports on Mathematical Physics, 42(1/2), [2000] Simple mechanical control systems with constraints, Institute of Electrical an Electronics Enineers. Transactions on Automatic Control, 45(8), Murray, R. M., Li, Z. X., an Sastry, S. S. [1994] A Mathematical Introuction to Robotic Manipulation, CRC Press, 2000 Corporate Blv., N.W., Boca Raton, Floria 33431, ISBN