On Periodic Solutions to Lagrangian System With Singularities
aa r X i v : . [ m a t h . D S ] F e b ON PERIODIC SOLUTIONS TO LAGRANGIANSYSTEM WITH SINGULARITIES
OLEG ZUBELEVICH
DEPT. OF THEORETICAL MECHANICS,MECHANICS AND MATHEMATICS FACULTY,M. V. LOMONOSOV MOSCOW STATE UNIVERSITYRUSSIA, 119899, MOSCOW, MGU
Abstract.
A Lagrangian system with singularities is considered.The configuration space is a non-compact manifold that dependson time. A set of periodic solutions has been found. Introduction
Let us start from a model example.Consider the plane R with the standard Cartesian frame Oxy, r = x e x + y e y and the standard Euclidean norm | · | .A particle of mass m moves in the plane being influenced by a forcewith the potential V ( r ) = − γ (cid:16) | r − r | n + 1 | r + r | n (cid:17) , here γ > r = 0 is a given vector. The Lagrangian ofthis system is as follows L ( r , ˙ r ) = m | ˙ r | − V ( r ) . (1.1)If n = 1 then this system is classical the two fixed centers problemand it is integrable [2]. The case of potentials of type V ( r ) ∼ − / | r | is mainly used in quantum mechanics but the classical statement is alsostudied, see [9], [5] and references therein. Mathematics Subject Classification.
Key words and phrases.
Lagrangian systems, periodic solutions, inverse squarepotential, two fixed center problem.Partially supported by grant RFBR 18-01-00887.
One of the consequences from the main Theorem 2.1 (see the nextsection) is as follows.
Proposition 1.
For any ω > , n ≥ and for any m ∈ N system(1.1) has an ω -periodic solution r ( t ) that first m times coils clockwiseabout the point r and then m times coils counterclockwise about thepoint − r . If m = 1 then the solution r ( t ) forms an -like curve inthe plane. This fact does not hold for n = 1 [4].Consider another pure classical mechanics example. Introduce inour space an inertial Cartesian frame Oxyz such that the gravity is g = − g e z .Let a particle of mass m be influenced by gravity and slides withoutfriction on a surface z = f ( r ) = − γ (cid:16) | r − r | n + 1 | r + r | n (cid:17) , r = x e x + y e y . The corresponding Lagrangian is L ( r , ˙ r ) = m (cid:16) | ˙ r | + (cid:0) ∇ f ( r ) , ˙ r (cid:1) (cid:17) − mgf ( r ) . (1.2)Proposition 1 holds for system (1.2) also.Existence problems for periodic solutions to Lagrangian systems haveintensively been studied since the beginning of the 20th century andeven earlier. There is an immense number of different results andmethods developed in this field. We mention only a few of them whichare mach closely related to this article.In [3] periodic solutions have been obtained for the Lagrangian sys-tem with Lagrangian L ( t, x, ˙ x ) = 12 g ij ( x ) ˙ x i ˙ x j − W ( t, x ) , x = ( x , . . . , x m ) ∈ R m here and in the sequel we use the Einstein summation convention. Theform g ij is symmetric and positive definite: g ij ξ i ξ j ≥ const · | ξ | . The potential is as follows W ( t, x ) = V ( x ) + g ( t ) P mi =1 x i , where V isa bounded function | V ( x ) | ≤ const and g is an ω − periodic function.The functions V, g ij are even.Under these assumptions the authors prove that there exists a non-trivial ω − periodic solution.Our main tool to obtain periodic solutions is variation technique.Variational problems and Hamiltonian systems have been studied ex-tensively. Classic references of these subjects are [10], [11], [7]. N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 3 The Main Theorem
Introduce some notations. Let x = ( x , . . . , x m ) and ϕ = ( ϕ , . . . , ϕ n )be points of the standard R m and R n respectively. Then let z stand forthe point ( x, ϕ ) ∈ R m + n . By | · | denote the standard Euclidean normof R k , k = m, m + n, that is | x | = P ki =1 ( x i ) . The variable z can consist only of x without ϕ or conversely.Assume that we are provided with a set σ ∈ R m + n such that1) If z = ( x, ϕ ) ∈ σ then − z ∈ σ and ( x, ϕ + 2 πp ) ∈ σ for any p = ( p , . . . , p n ) ∈ Z n ;2) the set σ does not have accumulation points.Introduce a manifold Σ = R m + n \ σ. Remark 1.
Actually, the space R m + n wraps a cylinder C = R m × T n ,where T n is the torus with angle variables ϕ . By the same reason, Σ can be regarded as the cylinder with several dropped away points.Nevertheless for analysis purposes we prefer to deal with R m + n . The main object of our study is the following Lagrangian systemwith Lagrangian L ( t, z, ˙ z ) = 12 g ij ˙ z i ˙ z j + a i ˙ z i − V, z = ( z , . . . , z m + n ) . (2.1) Remark 2.
The term a i ˙ z i in the Lagrangian corresponds to the socalled gyroscopic forces. For example, the Coriolis force and the Lorentzforce are gyroscopic. The functions g ij , a i depend on ( t, z ) and belong to C ( R × Σ);moreover all these functions are 2 π − periodic in each variable ϕ j and ω − periodic in the variable t, ω >
0. For all ( t, z ) ∈ R × Σ it followsthat g ij = g ji .The function V ( t, z ) ∈ C ( R × Σ) is also 2 π − periodic in each variable ϕ j and ω − periodic in the variable t .We also assume that there are positive constants C, M, A, K, P suchthat for all ( t, z ) ∈ R × Σ and ξ ∈ R m + n we have | a i ( t, z ) | ≤ C + M | z | , g ij ( t, z ) ξ i ξ j ≥ K | ξ | ; (2.2)for all ( t, z ) ∈ R × Σ , σ ′ ∈ σ it follows that V ( t, z ) ≤ A | z | − P | z − σ ′ | + C . (2.3)The Lagrangian L is defined up to an additional constant (see Defi-nition 1 below), so the constant C is not essential. OLEG ZUBELEVICH
System (2.1) obeys the following ideal constraints: f j ( t, z ) = 0 , j = 1 , . . . , l < m + n, f j ∈ C ( R m + n +1 ) . (2.4)The functions f j are also 2 π − periodic in each variable ϕ j and ω − periodicin the variable t .Introduce a set F ( t ) = { z ∈ R m + n | f j ( t, z ) = 0 , j = 1 , . . . , l } . Assume that rank (cid:16) ∂f j ∂z k ( t, z ) (cid:17) = l (2.5)for all z ∈ F ( t ). So that F ( t ) is a smooth manifold.Assume also that all the functions f j are either odd: f j ( − t, − z ) = − f j ( t, z ) (2.6)or even f j ( − t, − z ) = f j ( t, z ) . (2.7)The set σ belongs to F ( t ): σ ⊂ F ( t ) . (2.8) Remark 3.
Actually, it is sufficient to say that all the functions aredefined and have formulated above properties in some open symmet-ric vicinity of the manifold F ( t ) . We believe that this generalizationis unimportant and keep referring to the whole space R m + n just forsimplicity of exposition. Remark 4.
The set σ can be void. In this case the condition (2.8) isdropped and formula (2.3) takes the form V ( t, z ) ≤ A | z | + C . Definition 1.
We shall say that a function z ( t ) ∈ C ( R , Σ) is a solu-tion to system (2.1), (2.4) if there exists a set of functions { α , . . . , α l } ⊂ C ( R ) such that ddt ∂L∂ ˙ z i ( t, z ( t ) , ˙ z ( t )) − ∂L∂z i ( t, z ( t ) , ˙ z ( t )) = α j ( t ) ∂f j ∂z i ( t, z ( t )) (2.9) and z ( t ) ∈ F ( t ) (2.10) for all real t .In the absence of constraints (2.4) right side of (2.9) is equal to zeroand the condition (2.10) is dropped. The functions α i are defined from equations (2.9), (2.4) uniquely. N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 5
Definition 2.
Let V p ⊂ C ( R , Σ) , p = ( p , . . . , p n ) ∈ Z n stand for a set of functions z ( t ) = ( x ( t ) , ϕ ( t )) that satisfy the followingproperties for all t ∈ R :1) x ( t + ω ) = x ( t ); ϕ ( t + ω ) = ϕ ( t ) + 2 πp, z ( t ) ∈ F ( t ) .In the absence of constraints (2.4) condition 2) is omitted. Definition 3.
We shall say that two functions z , z ∈ V p are homo-topic to each other iff there exists a mapping z ( s, t ) ∈ C ([0 , × R , Σ) such that1) z ( s, · ) ∈ V p , z (0 , t ) = z ( t ) , z (1 , t ) = z ( t ) . By [ z ] we denote the homotopy class of the function z . Theorem 2.1.
Assume that1) all the functions are even: g ij ( − t, − z ) = g ij ( t, z ) , a i ( − t, − z ) = a i ( t, z ) , V ( − t, − z ) = V ( t, z );
2) the following inequality holds K − M ω √ − Aω >
0; (2.11)
3) for some ν ∈ Z n the set V ν is non-void and there is a function ˜ z ( t ) ∈ V ν ∩ C ( R , Σ); ˜ z ( − t ) = − ˜ z ( t );
4) and inf {k z − σ ′ k C [0 ,ω ] | z ∈ [˜ z ] , σ ′ ∈ σ } > . (2.12) Then system (2.1), (2.4) has an odd solution z ∗ ( t ) ∈ [˜ z ] , z ∗ ( − t ) = − z ∗ ( t ) . This assertion remains valid in the absence of constraints (2.4).When σ = ∅ then the assertion remains valid with condition 4) omitted. Actually the solution stated in this theorem is as smooth as it isallowed by smoothness of the Lagrangian L and the functions f j up to C ∞ .Loosely speaking, condition 4) of this theorem (formula (2.12)) im-plies that we look for a solution among the curves those can not beshrunk into a point. See for example problem from the Introduction. OLEG ZUBELEVICH
Remark 5.
If all the functions do not depend on t then we can choose ω to be arbitrary small and inequality (2.11) is satisfied. Taking avanishing sequence of ω , we obtain infinitely many periodic solutionsof the same homotopic type. Remark 6.
Condition 2) of the Theorem is essential. Indeed, system L ( t, x, ˙ x ) = 12 ˙ x − (cid:16) x − sin t (cid:17) obeys all the conditions except inequality (2.11). It is easy to see thatthe corresponding equation ¨ x + x = sin t does not have periodic solu-tions. Examples.
Our first example is as follows.
Figure 1. the tube and the ball
A thin tube can rotate freely in the vertical plane about a fixed hor-izontal axis passing through its centre O . A moment of inertia of thetube about this axis is equal to J . The mass of the tube is distributedsymmetrically such that tube’s centre of mass is placed at the point O .Inside the tube there is a small ball which can slide without friction.The mass of the ball is m . The ball can pass by the point O and fallout from the ends of the tube.The system undergoes the standard gravity field g . It seems to be evident that for typical motion the ball reaches anend of the tube and falls down out the tube. It is surprisingly, at leastfor the first glance, that this system has very many periodic solutionssuch that the tube turns around several times during the period.The sense of generalized coordinates φ, x is clear from Figure 1.The Lagrangian of this system is as follows L ( x, φ, ˙ x, ˙ φ ) = 12 (cid:16) mx + J (cid:17) ˙ φ + 12 m ˙ x − mgx sin φ. (2.13)Form theorem 2.1 it follows that for any constant ω > φ ( t ) , x ( t ) , t ∈ R such that1) x ( t ) = − x ( − t ) , φ ( t ) = − φ ( − t ); N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 7 x ( t + ω ) = x ( t ) , φ ( t + ω ) = φ ( t ) + 2 π. This result shows that for any ω > ω − periodicmotion such that the tube turns around once during the period. Thelength of the tube should be chosen properly.Our second example is a counterexample. Let us show that the firstcondition of the theorem 2.1 can not be omitted. Consider a mass point m that slides on a right circular cylinder ofradius r . The surface of the cylinder is perfectly smooth. The axis x ofthe cylinder is parallel to the gravity g and directed upwards. The Lagrangian of this system is L ( x, ϕ, ˙ x, ˙ ϕ ) = m (cid:16) r ˙ ϕ + ˙ x (cid:17) − mgx. (2.14)All the conditions except the evenness are satisfied but it is clear thissystem does not have periodic solutions.3. Proof of Theorem 2.1
In this section we use several standard facts from functional analysisand the Sobolev spaces theory [6], [1].By c , c , . . . we denote inessential positive constants.3.1. Recall that the Sobolev space H ( R ) consists of functions u ( t ) , t ∈ R such that u, ˙ u ∈ L ( R ). The following embedding holds H ( R ) ⊂ C ( R ).Recall another standard fact. Lemma 3.1.
Let u ∈ H ( R ) and u (0) = 0 . Then for any a > wehave k u k L (0 ,a ) ≤ a k ˙ u k L (0 ,a ) , k u k C [0 ,a ] ≤ a k ˙ u k L (0 ,a ) . Here and below the notation k ˙ u k L (0 ,a ) implies that (cid:13)(cid:13)(cid:13) ˙ u (cid:12)(cid:12)(cid:12) (0 ,a ) (cid:13)(cid:13)(cid:13) L (0 ,a ) the same is concerned to k u k C [0 ,a ] etc.3.2. Here we collect several spaces which are needed in the sequel. Definition 4. By X denote a space of functions u ∈ H ( R ) such thatfor all t ∈ R the following conditions hold u ( − t ) = − u ( t ) , u ( t + ω ) = u ( t ) . OLEG ZUBELEVICH
By virtue of lemma 3.1, the mapping u
7→ k ˙ u k L (0 ,ω ) determines anorm in X . This norm is denoted by k u k . The norm k · k is equivalentto the standard norm of H [0 , ω ]. The space ( X, k · k ) is a Banachspace. Since the norm k · k is generated by an inner product( u, v ) X = Z ω ˙ u ( t ) ˙ v ( t ) dt the space X is also a real Hilbert space, particularly this implies that X is a reflexive Banach space. Definition 5.
Let Φ stand for the space { ct + u ( t ) | c ∈ R , u ∈ X } . By the same argument, (Φ , k · k ) is a reflexive Banach space. Observealso that Φ = R ⊕ X and by direct calculation we get k ψ k = ωc + k u k , ψ ( t ) = ct + u ( t ) ∈ Φ . Observe that X ⊂ Φ. Definition 6.
Let E stand for the space X m × Φ n = { z ( t ) = ( x , . . . , x m , ϕ , . . . , ϕ n )( t ) | x i ∈ X, ϕ j ∈ Φ } . The space E is also a real Hilbert space with an inner product definedas follows ( z, y ) E = Z ω m + n X i =1 ˙ z i ( t ) ˙ y i ( t ) dt, where z = ( z k ) , y = ( y k ) ∈ E, k = 1 , . . . , m + n. We denote the corresponding norm in E by the same symbol andwrite k z k = (cid:13)(cid:13) | z | (cid:13)(cid:13) = m + n X k =1 k z k k . The space E is also a reflexive Banach space.Introduce the following set E = n ( x, ϕ ) ∈ E (cid:12)(cid:12)(cid:12) ϕ j = 2 πν j ω t + u j , ∀ x ∈ X, ∀ u j ∈ X, j = 1 , . . . , n o . This set is a closed plane of codimension n in E .If ( x, ϕ ) ∈ E then ϕ ( t + ω ) = ϕ ( t ) + 2 πν. Definition 7.
Let Y stand for the space (cid:8) u ∈ L ( R ) | u ( t ) = u ( − t ) , u ( t + ω ) = u ( t ) almost everywhere in R (cid:9) . N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 9
The space Y m + n is a Hilbert space with respect to the inner product( z, y ) Y = Z ω m + n X i =1 z i ( t ) y i ( t ) dt, where z = ( z k ) , y = ( y k ) ∈ Y m + n , k = 1 , . . . , m + n. K and consider a function u : [0 , ω ] → Σ , u ( t ) = ( u , . . . , u m + n )( t ) , u i ∈ H [0 , ω ] (3.1)such that the following two conditions hold Z ω dt | u ( t ) − ˆ u | ≤ K , k u k H [0 ,ω ] ≤ K (3.2)with some constant vector ˆ u ∈ R m + n . Lemma 3.2.
There exists a positive constant K such that for anyfunction u that satisfies (3.1) and (3.2) one has | u ( t ) − ˆ u | ≥ K vuut n + m X k =1 k u k − ˆ u k k C , / [0 ,ω ] , t ∈ [0 , ω ] , j = 1 , . . . , N. The constant K depends only on K . Here C , / [0 , ω ] is the Holder space. Proof of Lemma 3.2. By c , c , . . . we denote inessential positiveconstants.Observe that H [0 , ω ] ⊂ C , / [0 , ω ] and k · k C , / [0 ,ω ] ≤ c k · k H [0 ,ω ] . (3.3)Take arbitrary t ′ ∈ [0 , ω ]. Fix j and introduce the following notations v = ( v , . . . , v m + n ) ,v k = u k − ˆ u k , a = | v ( t ′ ) | , b = vuut n + m X k =1 k v k k C , / [0 ,ω ] . Observe that from conditions of the Lemma and formula (3.3) we get b ≤ c . (3.4)It follows that | v k ( t ) − v k ( t ′ ) | ≤ k v k k C , / [0 ,ω ] | t − t ′ | / , t ∈ [0 , ω ]and | v k ( t ) | ≤ | v k ( t ′ ) | + k v k k C , / [0 ,ω ] | t − t ′ | / . Consequently, we obtain | v ( t ) | ≤ a + b | t − t ′ | / . We then have K ≥ Z ω dt | v ( t ) | ≥ Z t ′ dt (cid:0) a + b ( t ′ − t ) / (cid:1) + Z ωt ′ dt (cid:0) a + b ( t − t ′ ) / (cid:1) . The last two integrals are computed easily with the help of the change ξ = a + b ( t ′ − t ) / for the first integral, and ξ = a + b ( t − t ′ ) / for thesecond integral respectively.Assume for definiteness that t ′ ∈ [0 , ω/ Z ωt ′ dt (cid:0) a + b ( t − t ′ ) / (cid:1) ≤ K and formula (3.4) imply a ≥ c b. The Lemma is proved.
Definition 8.
Let W stand for a set E ∩ [˜ z ] . S : W → R ,S ( z ) = Z ω L ( t, z, ˙ z ) dt. Our next goal is to prove that this functional attains its minimum.By using estimates (2.2), (2.3) we get S ( z ) ≥ Z ω (cid:0) K | ˙ z | − | ˙ z | ( C + M | z | ) − A | z | (cid:1) dt. From the Cauchy inequality and Lemma 3.1 it follows that Z ω | ˙ z || z | dt ≤ ω √ k z k , Z ω | z | dt ≤ ω k z k , Z ω | ˙ z | dt ≤ √ ω k z k . We finally obtain S ( z ) ≥ (cid:16) K − M ω √ − Aω (cid:17) k z k − C √ ω k z k . (3.5)By formula (2.11) the functional S is coercive: S ( z ) → ∞ (3.6) N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 11 as k z k → ∞ . Note that the Action Functional which corresponds to system (2.14)is also coercive but, as we see above, property (3.6) by itself does notimply existence results.3.5. Let { z k } ⊂ W be a minimizing sequence: S ( z k ) → inf z ∈ W S ( z )as k → ∞ . By formula (3.6) the sequence { z k } is bounded: sup k k z k k < ∞ . Lemma 3.3.
The following estimates hold inf {| z k ( t ) − σ ′ | | t ∈ R , σ ′ ∈ σ, k ∈ N } = inf {| z k ( t ) − σ ′ | | t ∈ [0 , ω ] , σ ′ ∈ σ, k ∈ N } > . (3.7) Proof of lemma 3.3.
The middle equality in formula (3.7) followsfrom the fact that z k ∈ V ν , we will prove inequality.It is sufficient to check inequality (3.7) for σ ′ ∈ σ ∩ Λ , Λ = [ k ∈ N { z ∈ R m + n | | z − z k ( t ) | ≤ , t ∈ [0 , ω ] } . The set σ ∩ Λ is finite.By formula (2.3) we obtain P Z ω dt | z k ( t ) − σ ′ | + Ψ k ≤ c − Z ω V ( t, z k ( t )) dt + Ψ k ≤ S ( z k ) + c . (3.8)Here Ψ k = Z ω a i ( t, z k ( t )) ˙ z k ( t ) dt, c = A k z k k L [0 ,ω ] + C ω. By formula (2.2) and lemma 3.1 this sequence is bounded: | Ψ k | ≤ c k z k k + c k z k k . So that from formulas (3.8), (2.12) and Lemma 3.2, Lemma 3.3 isproved.Since the space E is reflexive, this sequence contains a weakly conver-gent subsequence. Denote this subsequence in the same way: z k → z ∗ weakly in E . Moreover, the space H [0 , ω ] is compactly embedded in C [0 , ω ]. Thusextracting a subsequence from the subsequence and keeping the samenotation we also have max t ∈ [0 ,ω ] | z k ( t ) − z ∗ ( t ) | → , (3.9)as k → ∞ ; and inf {| z ∗ ( t ) − σ ′ | | t ∈ R , σ ′ ∈ σ } > . The set E is convex and strongly closed therefore it is weakly closed: z ∗ ∈ E . By continuity (3.9) one also gets z ∗ ∈ W (see Lemma 4.1 inthe Appendix).3.6. Let us show that inf z ∈ W S ( z ) = S ( z ∗ ) . Lemma 3.4.
Let a sequence { u k } ⊂ Φ weakly converge to u ∈ Φ (or u k , u ∈ X and u k → u weakly in X ); and also max t ∈ [0 ,ω ] | u k ( t ) − u ( t ) | → as k → ∞ .Then for any f ∈ C ( R ) and for any v ∈ L (0 , ω ) it follows that Z ω f ( u k ) ˙ u k vdt → Z ω f ( u ) ˙ uvdt, as k → ∞ . Indeed, Z ω f ( u k ) ˙ u k vdt = Z ω (cid:0) f ( u k ) − f ( u ) (cid:1) ˙ u k vdt + Z ω f ( u ) ˙ u k vdt. The function f is uniformly continuous in a compact set h min t ∈ [0 ,ω ] { u ( t ) } − c, max t ∈ [0 ,ω ] { u ( t ) } + c i with some constant c >
0. Consequently we obtainmax t ∈ [0 ,ω ] | f ( u k ( t )) − f ( u ( t )) | → . Since the sequence { u k } is weakly convergent it is bounded:sup k k u k k < ∞ particularly, we get k ˙ u k k L (0 ,ω ) < ∞ . N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 13
So that as k → ∞ (cid:12)(cid:12)(cid:12) Z ω (cid:0) f ( u k ) − f ( u ) (cid:1) ˙ u k vdt (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) v (cid:0) f ( u k ) − f ( u ) (cid:1)(cid:13)(cid:13) L (0 ,ω ) · k ˙ u k k L (0 ,ω ) → . To finish the proof it remains to observe that a function w Z ω f ( u ) ˙ wvdt belongs to Φ ′ (or to X ′ ). Indeed, (cid:12)(cid:12)(cid:12) Z ω f ( u ) ˙ wvdt (cid:12)(cid:12)(cid:12) ≤ max t ∈ [0 ,ω ] | f ( u ( t )) | · k v k L (0 ,ω ) k w k . Lemma 3.5.
Let a sequence { u k } ⊂ Φ (or { u k } ⊂ X ) be such that max t ∈ [0 ,ω ] | u k ( t ) − u ( t ) | → as k → ∞ . Then for any f ∈ C ( R ) and for any v ∈ L (0 , ω ) it follows that Z ω f ( u k ) vdt → Z ω f ( u ) vdt as k → ∞ . p k ( t, ξ ) = L ( t, z k , ˙ z ∗ + ξ ). The function p k is a quadratic polynomial of ξ ∈ R m + n , so that p k ( t, ξ ) = L ( t, z k , ˙ z ∗ ) + ∂L∂ ˙ z i ( t, z k , ˙ z ∗ ) ξ i + 12 ∂ L∂ ˙ z j ∂ ˙ z i ( t, z k , ˙ z ∗ ) ξ i ξ j . The last term in this formula is non-negative: ∂ L∂ ˙ z j ∂ ˙ z i ( t, z k , ˙ z ∗ ) ξ i ξ j = g ij ( t, z k ) ξ i ξ j ≥ . We consequently obtain p k ( t, ξ ) ≥ L ( t, z k , ˙ z ∗ ) + ∂L∂ ˙ z i ( t, z k , ˙ z ∗ ) ξ i . It follows that S ( z k ) = Z ω p k ( t, ˙ z k − ˙ z ∗ ) dt ≥ Z ω L ( t, z k , ˙ z ∗ ) dt + Z ω ∂L∂ ˙ z i ( t, z k , ˙ z ∗ )( ˙ z ik − ˙ z i ∗ ) dt. (3.10) From Lemma 3.4 and Lemma 3.5 it follows that Z ω L ( t, z k , ˙ z ∗ ) dt → Z ω L ( t, z ∗ , ˙ z ∗ ) dt as k → ∞ , and Z ω ∂L∂ ˙ z i ( t, z k , ˙ z ∗ )( ˙ z ik − ˙ z i ∗ ) dt → k → ∞ . Passing to the limit as k → ∞ in (3.10) we finally yieldinf z ∈ W S ( z ) ≥ S ( z ∗ ) = ⇒ inf z ∈ W S ( z ) = S ( z ∗ ) . Remark 7.
Basing upon these formulas one can estimate the norm k z ∗ k . Indeed, take a function ˆ z ∈ W then due to formula (3.5) oneobtains S (ˆ z ) ≥ S ( z ∗ ) ≥ (cid:16) K − M ω √ − Aω (cid:17) k z ∗ k − C √ ω k z ∗ k , here S (ˆ z ) is an explicitly calculable number. v ∈ X m + n such that ∂f j ∂z k ( t, z ∗ ) v k ( t ) = 0it follows that ddε (cid:12)(cid:12)(cid:12) ε =0 S ( z ∗ + εv ) = 0 . Introduce a linear functional b : X m + n → R , b ( v ) = ddε (cid:12)(cid:12)(cid:12) ε =0 S ( z ∗ + εv ) , and a linear operator A : X m + n → X l , ( Av ) j = ∂f j ∂z k ( t, z ∗ ) v k . It is clear, both these mappings are bounded andker A ⊂ ker b. Lemma 3.6.
The operator A maps X m + n onto X l that is A ( X m + n ) = X l . Proof.
By ˜ A ( t ) denote the matrix ∂f j ∂z k (cid:0) t, z ∗ ( t ) (cid:1) . N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 15
It is convenient to consider our functions to be defined on the circle t ∈ S = R / ( ω Z ) . Fix an element w ∈ X l . Let us cover the circle S with open intervals U i , i = 1 , . . . , N such that there exists a set of functions v i ∈ H ( U i ) , ˜ A ( t ) v i ( t ) = w ( t ) , t ∈ U i , i = 1 , . . . , N. And let ψ i be a smooth partition of unity subordinated to the covering { U i } . A function ˜ v ( t ) = P Ni =1 ψ i ( t ) v i ( t ) belongs to H ( S ) and for each t it follows that ˜ A ( t )˜ v ( t ) = w ( t ). But the function ˜ v is not obliged tobe odd.Since ˜ A ( − t ) = ˜ A ( t ) we have˜ A ( t ) v ( t ) = w ( t ) , v ( t ) = ˜ v ( t ) − ˜ v ( − t )2 ∈ X m + n . The Lemma is proved.3.10. Recall a lemma from functional analysis [8].
Lemma 3.7.
Let
E, H, G be Banach spaces and A : E → H, B : E → G be bounded linear operators; ker A ⊆ ker B. If the operator A is onto then there exists a bounded operator Γ : H → G such that B = Γ A. Thus there is a linear function Γ ∈ ( X l ) ′ such that b ( v ) = Γ A ( v ) , v ∈ X m + n . Or by virtue of the Riesz representation theorem, there exists a setof functions { γ , . . . , γ l } ⊂ X such that ddε (cid:12)(cid:12)(cid:12) ε =0 S ( z ∗ + εv ) = Z ω ˙ γ j ( t ) ddt (cid:16) ∂f j ∂z k ( t, z ∗ ) v k ( t ) (cid:17) dt for all v ∈ X m + n .3.11. Every element v ∈ X m + n is presented as follows v ( t ) = Z t y ( s ) ds, where y ∈ Y m + n is such that Z ω y ( s ) ds = 0 . Introduce a linear operator h : Y m + n → R m + n by the formula h ( y ) = Z ω y ( s ) ds. Define a linear functional q : Y m + n → R by the formula q ( y ) = ( b − Γ A ) v, v ( t ) = Z t y ( s ) ds. Now all our observations lead toker h ⊆ ker q. Therefore, there exists a linear functional λ : R m + n → R such that q = λh Let us rewrite the last formula explicitly. There are real constants λ k such that for any y k ∈ Y one has Z ω (cid:16) ∂L∂ ˙ z k ( t, z ∗ , ˙ z ∗ ) y k ( t ) + ∂L∂z k ( t, z ∗ , ˙ z ∗ ) Z t y k ( s ) ds (cid:17) dt = Z ω ˙ γ j ( t ) ∂f j ∂z k ( t, z ∗ ) y k ( t ) dt + Z ω ˙ γ j ( t ) ddt (cid:16) ∂f j ∂z k ( t, z ∗ ) (cid:17) Z t y k ( s ) dsdt + λ k Z ω y k ( s ) ds. Z ω ∂L∂ ˙ z k ( t, z ∗ , ˙ z ∗ ) y k ( t ) dt + Z ω y k ( s ) Z ωs ∂L∂z k ( t, z ∗ , ˙ z ∗ ) dtds = Z ω ˙ γ j ( t ) ∂f j ∂z k ( t, z ∗ ) y k ( t ) dt + Z ω y k ( s ) Z ωs ˙ γ j ( t ) ddt (cid:16) ∂f j ∂z k ( t, z ∗ ) (cid:17) dtds + λ k Z ω y k ( s ) ds. (3.11)In this formula the functions ∂L∂ ˙ z k ( t, z ∗ , ˙ z ∗ ) , ˙ γ j ( t ) ∂f j ∂z k ( t, z ∗ )are even and ω -periodic functions of t .The functions ˙ γ j ( t ) ddt (cid:16) ∂f j ∂z k ( t, z ∗ ) (cid:17) , ∂L∂z k ( t, z ∗ , ˙ z ∗ )are odd and ω -periodic.Employ the following trivial observation. Proposition 2. If w ∈ L ( R ) is an ω − periodic and odd function thenfor any constant a ∈ R a function t Z ta w ( s ) ds N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 17 is also ω − periodic. So that the functions Z ωs ˙ γ j ( t ) ddt (cid:16) ∂f j ∂z k ( t, z ∗ ) (cid:17) ds, Z ωs ∂L∂z k ( t, z ∗ , ˙ z ∗ ) ds are even and ω -periodic in s .Therefore, equation (3.11) is rewritten as ( y, η ) Y = 0 for any y =( y , . . . y m + n ) ∈ Y m + n and η = ( η , . . . , η m + n ) stands for η k = ∂L∂ ˙ z k ( t, z ∗ , ˙ z ∗ ) + Z ωt ∂L∂z k ( s, z ∗ , ˙ z ∗ ) ds − ˙ γ j ( t ) ∂f j ∂z k ( t, z ∗ ) − Z ωt ˙ γ j ( s ) dds (cid:16) ∂f j ∂z k ( s, z ∗ ) (cid:17) ds − λ k ∈ Y. Consequently we obtain the following system ∂L∂ ˙ z k ( t, z ∗ , ˙ z ∗ ) + Z ωt ∂L∂z k ( s, z ∗ , ˙ z ∗ ) ds = ˙ γ j ( t ) ∂f j ∂z k ( t, z ∗ ) + Z ωt ˙ γ j ( s ) dds (cid:16) ∂f j ∂z k ( s, z ∗ ) (cid:17) ds + λ k . (3.12)here k = 1 , . . . , m + n. If we formally differentiate both sides of equations (3.12) in t thenwe obtain the Lagrange equations (2.9) with α j = ¨ γ j .Equations (3.12) hold for almost all t ∈ (0 , ω ) but all the functionscontained in (3.12) are defined for all t ∈ R .The functions Z ωt ˙ γ j ( s ) dds (cid:16) ∂f j ∂z k ( s, z ∗ ) (cid:17) ds, Z ωt ∂L∂z k ( s, z ∗ , ˙ z ∗ ) ds are ω − periodic by Proposition 2. Equation (3.12) holds for almost all t ∈ R .3.13. Let g ij stand for the components of the matrix inverse to ( g ij ) : g ij g ik = δ jk . Present equation (3.12) in the form˙ z j ∗ ( t ) = g kj ( t, z ∗ ( t )) · (cid:16) λ k + ˙ γ i ( t ) ∂f i ∂z k ( t, z ∗ ( t )) + Z ωt ˙ γ i ( s ) dds (cid:16) ∂f i ∂z k ( s, z ∗ ( s )) (cid:17) ds − Z ωt ∂L∂z k ( s, z ∗ ( s ) , ˙ z ∗ ( s )) ds − a k ( t, z ∗ ( t )) (cid:17) . (3.13)Together with equation (3.13) consider equations ∂f j ∂t ( t, z ∗ ( t )) + ∂f j ∂z k ( t, z ∗ ( t )) ˙ z k ∗ ( t ) = 0 . (3.14) These equations follow from (2.4).Recall that by the Sobolev embedding theorem, z ∗ ∈ X ⊂ C ( R ) . Due to (2.5) we havedet B ( t, z ∗ ) = 0 , B ( t, z ∗ ) = (cid:16) g kj ( t, z ∗ ) ∂f i ∂z k ( t, z ∗ ) ∂f l ∂z j ( t, z ∗ ) (cid:17) for all t . Substituting ˙ z ∗ from (3.13) to (3.14) we can express ˙ γ j andsee ˙ γ j ∈ C ( R ). Thus from (3.13) it follows that ˙ z ∗ ∈ C ( R ) . Applying this argument again we obtain ¨ γ j , ¨ z ∗ ∈ C ( R ) . This proves the theorem for the case of odd constraints.3.14. Let us discuss the proof of the theorem under the assumptionthat the constraints are even (2.7).
Definition 9. By Z denote a space of functions u ∈ H ( R ) such thatfor all t ∈ R the following conditions hold u ( − t ) = u ( t ) , u ( t + ω ) = u ( t ) . The space Z l is a real Hilbert space with respect to an inner product( u, v ) Z = l X i =1 Z ω (cid:0) u i ( t ) v i ( t ) + ˙ u i ( t ) ˙ v i ( t ) (cid:1) dt. This is the standard inner product in H [0 , ω ].So what is changed now? The operator A takes the space X m + n ontothe space Z l . The proof of this fact is the same as in Lemma 3.6.By the Riesz representation theorem, there exists a set of functions { γ , . . . , γ l } ⊂ Z such that b ( v ) = Z ω ˙ γ j ( t ) ddt (cid:16) ∂f j ∂z k ( t, z ∗ ) v k ( t ) (cid:17) dt + Z ω γ j ( t ) ∂f j ∂z k ( t, z ∗ ) v k ( t ) dt for all v = Z t y ( s ) ds, y ∈ Y m + n , Z ω y ( s ) ds = 0 . So that equation (3.12) is replaced with the following one ∂L∂ ˙ z k ( t, z ∗ , ˙ z ∗ ) + Z ωt ∂L∂z k ( s, z ∗ , ˙ z ∗ ) ds = ˙ γ j ( t ) ∂f j ∂z k ( t, z ∗ ) + Z ωt ˙ γ j ( s ) dds (cid:16) ∂f j ∂z k ( s, z ∗ ) (cid:17) ds + Z ωt γ j ( s ) ∂f j ∂z k ( s, z ∗ ) ds + λ k . N PERIODIC SOLUTIONS TO LAGRANGIAN SYSTEM 19
Here k = 1 , . . . , m + n. By the same argument the functions Z ωt γ j ( s ) ∂f j ∂z k ( s, z ∗ ) ds, Z ωt ˙ γ j ( s ) dds (cid:16) ∂f j ∂z k ( s, z ∗ ) (cid:17) ds are ω − periodic and one can put α j = ¨ γ j − γ j . Other argument is the same as above. The theorem is proved.4.
Appendix
Lemma 4.1.
Fix a positive constant δ .Let z , z be functions from V p such that inf {| z i ( t ) − σ ′ | | t ∈ [0 , ω ] , σ ′ ∈ σ, i = 1 , } ≥ δ. There exists a positive number ε > such that if max t ∈ [0 ,ω ] | z ( t ) − z ( t ) | < ε then these functions are homotopic.Proof of lemma 4.1. Our argument is quite standard. So we presenta sketch of the proof.It is convenient to consider z , z as functions with values in C (seeRemark 1). In the same sense F ( t ) is a submanifold in C and thefunctions z , z define a pair of closed curves in C .Choose a Riemann metric in C , for example as follows dτ = m + n X k =1 ( dz k ) . This metric inducts a metric in F ( t ).Under the conditions of the Lemma any two points z ( t ) , z ( t ) ∈ F ( t ) are connected in F ( t ) with a unique shortest piece of geodesic χ ( ξ, t ) , ξ ∈ [0 , ˜ ξ ( t )] such that χ (0 , t ) = z ( t ) , χ ( ˜ ξ ( t ) , t ) = z ( t ) . Here ξ is the arc-length parameter.Define the homotopy as follows z ( s, t ) = χ ( s ˜ ξ ( t ) , t ) , s ∈ [0 , Acknowledgments.
The author wishes to thank Professor E. I. Ku-gushev for useful discussions.