An Unusual Variational Problem Connected with Mather’s Theory for Monotone Twist Mappings

1994 ◽  
pp. 81-89
Author(s):  
Jürgen Moser
Keyword(s):  
Variational Problem ◽  
Twist Mappings

scholarly journals Monotone twist mappings and the calculus of variations

1986 ◽  
Vol 6 (3) ◽  
pp. 401-413 ◽  
Author(s):  
Jürgen Moser
Keyword(s):  
Variational Problem ◽  
Calculus Of Variations ◽  
Twist Mapping ◽  
Image Position ◽  
Twist Mappings ◽  
Area Preserving

AbstractIt is shown that a smooth area-preserving monotone twist mapping ϕ of an annulus A can be interpolated by a flow ϕt which is generated by a t-dependent Hamiltonian in ℝ × A having the period 1 in t and satisfying a Legendre condition. In other words, any such monotone twist mapping can be viewed as a section mapping for the extremals of variational problem on a torus:where F has period 1 in t and x and satisfies the Legendre condition Fẋẋ>0.

Dynamics of a Contact Structure Along a Vector Field of its Kernel

2008 ◽  
Vol 8 (1) ◽  
Author(s):  
Abbas Bahri ◽  
Yongzhong Xu
Keyword(s):  
Vector Field ◽  
Variational Problem ◽  
Contact Structure ◽  
Contact Form ◽  
Dual Form

AbstractIn this paper we prove that in order to define the homology of [3], the hypothesis that there exists a vector field in the kernel of the contact form which defines a dual form with the same orientation is not essential. The technique is quantitative: as we introduce a large amount of rotation near the zeroes of the vector field in the kernel, we track down the modification of the variational problem and provide bounds on a key quantity (denoted by τ).

scholarly journals Overlap Problems on the Circle

2013 ◽  
Vol 45 (03) ◽  
pp. 773-790
Author(s):  
S. Juneja ◽  
M. Mandjes
Keyword(s):  
Variational Problem ◽  
Decay Rate ◽  
Positive Constant ◽  
The Other ◽  
The Impact ◽  
Independent And Identically Distributed

Consider a circle with perimeter N > 1 on which k < N segments of length 1 are sampled in an independent and identically distributed manner. In this paper we study the probability π (k,N) that these k segments do not overlap; the density φ(·) of the position of the disks on the circle is arbitrary (that is, it is not necessarily assumed uniform). Two scaling regimes are considered. In the first we set k≡ a√N, and it turns out that the probability of interest converges (N→ ∞) to an explicitly given positive constant that reflects the impact of the density φ(·). In the other regime k scales as aN, and the nonoverlap probability decays essentially exponentially; we give the associated decay rate as the solution to a variational problem. Several additional ramifications are presented.

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