Open Access

Periodic solutions for N + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq1_HTML.gif-body problems with N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq2_HTML.gif fixed centers

Boundary Value Problems20132013:129

DOI: 10.1186/1687-2770-2013-129

Received: 26 January 2013

Accepted: 2 May 2013

Published: 17 May 2013

Abstract

In this paper, we prove the existence of a new periodic solution for N + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq1_HTML.gif-body problems with N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq2_HTML.gif fixed centers and strong-force potentials. In this model, N particles with equal masses are fixed at the vertices of a regular N-gon and the ( N + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq3_HTML.gifth particle is fixed at the center of the N-gon, the ( N + 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq4_HTML.gifth particle winding around N particles.

MSC:34C15, 34C25, 70F10.

Keywords

N + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq1_HTML.gif-body problems with N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq2_HTML.gif-fixed centers minimizing variational methods strong-force potentials

1 Introduction and main results

In the eighteenth century, the 2-fixed center problem was studied by Euler [13]. Here, let us consider the N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq2_HTML.gif-fixed center problem: We assume N particles q 1 , q 2 , , q N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq5_HTML.gif with equal masses 1 are fixed at the vertices e 1 2 π N j = ( cos 2 π j N , sin 2 π j N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq6_HTML.gif ( j = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq7_HTML.gif) of a regular polygon and the ( N + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq3_HTML.gifth particle q N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq8_HTML.gif is fixed at the origin ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq9_HTML.gif, the ( N + 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq4_HTML.gifth particle with mass m N + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq10_HTML.gif is attracted by the other particles, and moves according to Newton’s second law and a more general power law than the Newton’s universal gravitational square law. In this system, the position q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq11_HTML.gif for the ( N + 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq4_HTML.gifth particle satisfies the following equation:
m N + 2 q ¨ ( t ) = i = 1 N + 1 m i m N + 2 ( q ( t ) q i ) | q ( t ) q i | α + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ1_HTML.gif
(1.1)
Equivalently,
q ¨ ( t ) = i = 1 N ( q ( t ) q i ) | q ( t ) q i | α + 2 + m N + 1 ( q ( t ) q N + 1 ) | q ( t ) q N + 1 | α + 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ2_HTML.gif
(1.2)
q ¨ ( t ) = U ( q ) q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ3_HTML.gif
(1.3)
where
α > 0 and U ( q ) = i = 1 N 1 | q ( t ) q i | α + m N + 1 | q ( t ) q N + 1 | α . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equa_HTML.gif

The type of system (1.2) is called a singular Hamiltonian system which attracts many researchers (see [110] and [1116]).

Specially, Gordon [10] proved the Keplerian elliptical orbits are the minimizers of Lagrangian action defined on the space for non-zero winding numbers.

In this paper, we use a variational minimizing method to look for a periodic solution for the ( N + 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq4_HTML.gifth particle which winds around the q i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq12_HTML.gif ( i = 1 , , N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq13_HTML.gif).

Definition 1.1 [10]

Let C : x ( t ) : [ a , b ] R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq14_HTML.gif be a given oriented closed curve, and p C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq15_HTML.gif. Define φ : C S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq16_HTML.gif:
φ ( t ) = x ( t ) p | x ( t ) p | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equb_HTML.gif

When some point on C goes around the curve once, its image point φ ( x ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq17_HTML.gif will go around S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq18_HTML.gif a number of times. This number is defined as the winding number of the curve C relative to the point p and is denoted by deg ( x ( t ) p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq19_HTML.gif.

Let
f ( q ) = 0 1 [ 1 2 | q ˙ ( t ) | 2 + U ( q ) ] d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ4_HTML.gif
(1.4)
q Λ 1 = { q W 1 , 2 ( R / Z , R 2 ) , q ( t ) q i , for  i = 1 , , N + 1 , q ( t + k N ) = ( cos ( 2 k π N ) sin ( 2 k π N ) sin ( 2 k π N ) cos ( 2 k π N ) ) q ( t ) , deg ( q ( t ) q i ) = 1 , for  i = 1 , , N , deg ( q ( t ) q N + 1 ) = 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ5_HTML.gif
(1.5)
q Λ 2 = { q W 1 , 2 ( R / Z , R 2 ) , q ( t ) q i , for  i = 1 , , N + 1 , q ( t + k N ) = ( cos ( 2 k π N ) sin ( 2 k π N ) sin ( 2 k π N ) cos ( 2 k π N ) ) q ( t ) , deg ( q ( t ) q i ) = 0 , for  i = 1 , , N , deg ( q ( t ) q N + 1 ) = 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ6_HTML.gif
(1.6)
q Λ 3 = { q W 1 , 2 ( R / Z , R 2 ) , q ( t ) q i , for  i = 1 , , N + 1 , q ( t + k N ) = ( cos ( 2 k π N ) sin ( 2 k π N ) sin ( 2 k π N ) cos ( 2 k π N ) ) q ( t ) , deg ( q ( t ) q i ) = 1 , for  i = 1 , , N , deg ( q ( t ) q N + 1 ) = 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ7_HTML.gif
(1.7)
q Λ 4 = { q W 1 , 2 ( R / Z , R 2 ) , q ( t ) q i , for  i = 1 , , N + 1 , q ( t + k N ) = ( cos ( 2 k π N ) sin ( 2 k π N ) sin ( 2 k π N ) cos ( 2 k π N ) ) q ( t ) , deg ( q ( t ) q i ) = 1 , for  i = 1 , , N , deg ( q ( t ) q N + 1 ) = N 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ8_HTML.gif
(1.8)

We have the following theorem.

Theorem 1.1 For α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq20_HTML.gif, the minimizer of f ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq21_HTML.gif on Λ ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq22_HTML.gif ( i = 1 , 2 , 3 , 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq23_HTML.gif) exists and it is a non-collision periodic solution of (1.1) or (1.2)-(1.3) (please see Figures 1-4 for N = 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq24_HTML.gif).
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Fig1_HTML.jpg
Figure 1

q Λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq25_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Fig2_HTML.jpg
Figure 2

q Λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq26_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Fig3_HTML.jpg
Figure 3

q Λ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq27_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Fig4_HTML.jpg
Figure 4

q Λ 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq28_HTML.gif .

2 The proof of Theorem 1.1

We recall the following famous lemmas, which we need to prove Theorem 1.1.

Lemma 2.1 [9]

If x W 1 , 2 ( R / Z , R 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq29_HTML.gif, α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq20_HTML.gif, a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq30_HTML.gif, and there exists t 0 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq31_HTML.gif such that x ( t 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq32_HTML.gif, then 0 1 [ 1 2 | x ˙ ( t ) | 2 + a | x ( t ) | α ] d t = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq33_HTML.gif.

If x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq34_HTML.gif in W 1 , 2 ( R / Z , R 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq35_HTML.gif and t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq36_HTML.gif, s.t. x ( t 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq32_HTML.gif, α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq20_HTML.gif, then 0 1 1 | x n ( t ) | α d t + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq37_HTML.gif.

Lemma 2.2 (Palais’s symmetry principle [17])

Let σ be an orthogonal representation of a finite or compact group G on a real Hilbert space H, and let f : H R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq38_HTML.gif be such that for σ G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq39_HTML.gif, f ( σ x ) = f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq40_HTML.gif. Set H G = { x H : σ x = x , σ G } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq41_HTML.gif. Then the critical point of f in H G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq42_HTML.gif is also a critical point of f in H.

Lemma 2.3 [5]

If X is a reflexive Banach space, M is a weakly closed subset of X, and f : M R { + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq43_HTML.gif, f + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq44_HTML.gif is weakly lower semi-continuous and coercive, then f attains its infimum on M.

Lemma 2.4 (Poincare-Wirtinger inequality)

Let q W 1 , 2 ( R / ZT , R d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq45_HTML.gif and 0 T q ( t ) d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq46_HTML.gif, then 0 T | q ˙ ( t ) | 2 d t ( 2 π T ) 2 0 T q ( t ) 2 d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq47_HTML.gif. And the inequality takes the equality if and only if q ( t ) = α cos 2 π T t + β sin 2 π T t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq48_HTML.gif, α , β R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq49_HTML.gif.

We now prove Theorem 1.1.

Proof By the symmetry of Λ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq50_HTML.gif, we know for x Λ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq51_HTML.gif,
0 T q ( t ) d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ9_HTML.gif
(2.1)
If q n ( t ) q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq52_HTML.gif in Λ ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq22_HTML.gif, then by Sobolev’s compact embedding theorem, we have q n ( t ) q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq53_HTML.gif in C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq54_HTML.gif.
  1. (i)

    If q ( t ) Λ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq55_HTML.gif, then lim n + 0 1 U ( q n ( t ) ) d t = 0 1 U ( q n ( t ) ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq56_HTML.gif. Since 0 1 q n d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq57_HTML.gif, 1 2 0 1 | q ˙ n | 2 d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq58_HTML.gif can be regarded as the square of an equivalent norm for W 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq59_HTML.gif, so it is weakly lower semi-continuous, so lim ̲ f ( q n ( t ) ) f ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq60_HTML.gif.

     
  2. (ii)

    If q ( t ) Λ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq61_HTML.gif, then by Lemma 2.1, f ( q ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq62_HTML.gif, we have 0 1 U ( q n ( t ) ) d t + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq63_HTML.gif. So, lim ̲ n + f ( q n ) = + f ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq64_HTML.gif. Hence f is w.l.s.c.

     
Using (2.1), we know that f ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq21_HTML.gif is coercive on Λ ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq22_HTML.gif. Lemma 2.3 guarantees that f ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq21_HTML.gif attains its infimum on Λ ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq22_HTML.gif. Let the minimizer be q ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq65_HTML.gif, then
f ( q ˜ ) = inf q Λ ¯ i f ( q ) < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ10_HTML.gif
(2.2)
If q ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq65_HTML.gif is a collision periodic solution, then there exist t 0 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq31_HTML.gif and j { 1 , 2 , , N , N + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq66_HTML.gif such that q ˜ ( t 0 ) = q j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq67_HTML.gif. Let x ( t ) = q ˜ ( t ) q j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq68_HTML.gif and note x ( t 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq32_HTML.gif. By Lemma 2.1, we have
f ( q ˜ ) = 0 1 [ 1 2 | q ˜ ˙ ( t ) | 2 + m j | q ˜ ( t ) q j | α + i j N + 1 m i | q ˜ ( t ) q i | α ] d t 0 1 [ 1 2 | x ˙ ( t ) | 2 + m j | x ( t ) | α ] d t = + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_Equ11_HTML.gif
(2.3)

which contradicts the inequality in (2.2). By Lemma 2.2, q ˜ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq69_HTML.gif is the critical point of f in W 1 , 2 ( R / Z , R 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq70_HTML.gif; therefore, q ˜ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-129/MediaObjects/13661_2013_Article_388_IEq69_HTML.gif is a non-collision periodic solution.

This completes the proof of Theorem 1.1. □

Declarations

Acknowledgements

The authors sincerely thank the referees for their many helpful comments and suggestions and also express their sincere gratitude to Professor Zhang Shiqing for his discussions and corrections. This work is supported by NSF of China and Youth Fund of Mianyang Normal University.

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Mianyang Normal University
(2)
Department of Mathematics, Sichuan University
(3)
School of Science, Southwest University of Science and Technology

References

  1. Euler M: De motu coproris ad duo centra virium fixa attracti. Nov. Commun. Acad. Sci. Imp. Petrop. 1766, 10: 207–242.
  2. Euler M: De motu coproris ad duo centra virium fixa attracti. Nov. Commun. Acad. Sci. Imp. Petrop. 1767, 11: 152–184.
  3. Euler M: Probleme un corps etant attire en raison reciproque quarree des distances vers deux points fixes donnes trouver les cas ou la courbe decrite par ce corps sera algebrique. Hist. Acad. R. Sci. Bell. Lett. Berlin 1767, 2: 228–249.
  4. Ambrosetti A, Coti Zelati V: Critical points with lack of compactness and applications to singular dynamical systems. Ann. Mat. Pura Appl. 1987, 149: 237–259. 10.1007/BF01773936MathSciNetView ArticleMATH
  5. Ambrosetti A, Coti Zelati V: Periodic Solutions for Singular Lagrangian Systems. Springer, Boston; 1993.View ArticleMATH
  6. Bahri A, Rabinowitz PH: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 1989, 82: 412–428. 10.1016/0022-1236(89)90078-5MathSciNetView ArticleMATH
  7. Benci V, Giannoni G: Periodic solutions of prescribed energy for a class of Hamiltonian system with singular potentials. J. Differ. Equ. 1989, 82: 60–70. 10.1016/0022-0396(89)90167-8MathSciNetView ArticleMATH
  8. Degiovanni M, Giannoni F: Dynamical systems with Newtonian type potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1988, 15: 467–494.MathSciNetMATH
  9. Gordon W: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 1975, 204: 113–135.View ArticleMathSciNetMATH
  10. Gordon W: A minimizing property of Keplerian orbits. Am. J. Math. 1977, 99: 961–971. doi:10.2307/2373993 10.2307/2373993View ArticleMathSciNetMATH
  11. Rabinowitz PH: A note on periodic solutions of prescribed energy for singular Hamiltonian systems. J. Comput. Appl. Math. 1994, 52: 147–154. 10.1016/0377-0427(94)90354-9MathSciNetView ArticleMATH
  12. Siegel C, Moser J: Lectures on Celestial Mechanics. Springer, Berlin; 1971.View ArticleMATH
  13. Wang XR, He S: Lagrangian actions on 3-body problems with two fixed centers. Bound. Value Probl. 2012., 2012: Article ID 28
  14. Tanaka K: A prescribed energy problem for a singular Hamiltonian system with weak force. J. Funct. Anal. 1993, 113: 351–390. 10.1006/jfan.1993.1054MathSciNetView ArticleMATH
  15. Tanaka K: A prescribed energy problem for conservative singular Hamiltonian system. Arch. Ration. Mech. Anal. 1994, 128: 127–164. 10.1007/BF00375024View ArticleMathSciNetMATH
  16. Zhang SQ: Multiple geometrically distinct closed noncollision orbits of fixed energy for N -body type problems with strong force potentials. Proc. Am. Math. Soc. 1996, 124: 3039–3046. 10.1090/S0002-9939-96-03751-3View ArticleMathSciNetMATH
  17. Palais R: The principle of symmetric criticality. Commun. Math. Phys. 1979, 69: 19–30. 10.1007/BF01941322MathSciNetView ArticleMATH

Copyright

© Zhao et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.