Geometric constrained variational calculus. II: The second variation (Part I)

2016 ◽  
Vol 13 (01) ◽  
pp. 1550132 ◽  
Author(s):  
Enrico Massa ◽  
Danilo Bruno ◽  
Gianvittorio Luria ◽  
Enrico Pagani
Keyword(s):  
Gauge Transformation ◽  
Sufficient Conditions ◽  
Variational Calculus ◽  
Second Variation ◽  
Action Functional ◽  
Covariant Representation ◽  
Jacobi Fields ◽  
Necessary And Sufficient ◽  
The Second Variation ◽  
Variation Part

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

Geometric constrained variational calculus. III: The second variation (Part II)

2016 ◽  
Vol 13 (04) ◽  
pp. 1650038
Author(s):  
Enrico Massa ◽  
Gianvittorio Luria ◽  
Enrico Pagani
Keyword(s):  
Sufficient Conditions ◽  
Variational Calculus ◽  
Second Variation ◽  
Action Functional ◽  
Covariant Representation ◽  
Jacobi Fields ◽  
Gauge Transformations ◽  
Alternative Approach ◽  
The Second Variation ◽  
Variation Part

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.

scholarly journals On the notion of Jacobi fields in constrained calculus of variations

2016 ◽  
Vol 24 (2) ◽  
pp. 91-113
Author(s):  
Enrico Massa ◽  
Enrico Pagani
Keyword(s):  
Variational Calculus ◽  
Second Variation ◽  
Action Functional ◽  
Jacobi Fields ◽  
Gauge Transformations ◽  
Groups Of Diffeomorphisms ◽  
Relevant Role ◽  
Fixed End ◽  
The Second Variation

Abstract In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of local gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' strengths [16]. In dis- cussing the positivity of the second variation, a relevant role is played by the Jacobi fields, defined as infinitesimal generators of 1-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this point is presented. An alternative characterization of the Jacobi fields as solutions of a suitable accessory variational problem is established.

scholarly journals Second order infinitesimal bending of curves

Filomat ◽  
2017 ◽  
Vol 31 (13) ◽  
pp. 4127-4137 ◽  
Author(s):  
Marija Najdanovic ◽  
Ljubica Velimirovic
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Second Order ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Explicit Formulas ◽  
Infinitesimal Bending ◽  
Necessary And Sufficient ◽  
The Second Variation

We investigate a second order infinitesimal bending of curves in a three-dimensional Euclidean space in this paper. We give the necessary and sufficient conditions for the vector fields to be infinitesimal bending fields of the corresponding order, as well as explicit formulas which determine these fields. We examine the first and the second variation of some geometric magnitudes which describe a curve, specially a change of the curvature. Two illustrative examples (a circle and a helix) are studied not only analytically but also by drawing curves using computer program Mathematica.

scholarly journals Cálculo variacional e aplicações

2020 ◽  
Vol 42 ◽  
pp. 50
Author(s):  
Jardel Carpes Meurer ◽  
Lucas Tavares Cardoso ◽  
Glauber Rodrigues de Quadros
Keyword(s):  
Sufficient Conditions ◽  
Variational Calculus ◽  
Necessary And Sufficient Conditions ◽  
Stationary Points ◽  
Second Variation ◽  
Rigorous Treatment ◽  
Necessary And Sufficient ◽  
First And Second Variation ◽  
Lagrange Mechanics

This paper consists of a brief review and introduction to the main concepts of Classic Variational Calculus. Starting from thedefinitions of the concepts of first and second variation of a functional, we present a mathematically rigorous treatment for theVariational Calculus, establishing necessary and sufficient conditions for obtaining extrema. In this context, the notion of conjugatepoints is introduced, which is fundamental for the classification of weak extrema. Some simple and enlightening examples are dealtwith throughout the paper. Strong extrema are characterized and sufficient conditions for their occurrence are given. The paperconcludes with a brief application to Lagrange mechanics, showing the existence of actions whose stationary points are saddlepoints instead of minima.

scholarly journals A generalisation of the Morse inequalities

2001 ◽  
Vol 70 (3) ◽  
pp. 351-386
Author(s):  
Mohan Bhupal
Keyword(s):  
Critical Point ◽  
Maslov Index ◽  
Closed Manifold ◽  
Zero Section ◽  
Second Variation ◽  
Action Functional ◽  
Jet Bundle ◽  
Morse Inequalities ◽  
The Second Variation ◽  
Nondegenerate Critical Point

AbstractIn this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embdeddings, by discretising the action functional. We compute the second variation of a generating funciton obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generlisation of the Morse inequalities thus refining a theorem of Chekanov.

scholarly journals Singular extremals in L1-optimal control problems: sufficient optimality conditions

2020 ◽  
Vol 26 ◽  
pp. 99
Author(s):  
Francesca C. Chittaro ◽  
Laura Poggiolini
Keyword(s):  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Sufficient Optimality Conditions ◽  
Local Optimality ◽  
Second Variation ◽  
Absolute Value ◽  
Singular Extremals ◽  
Hamiltonian Methods ◽  
The Second Variation

In this paper we are concerned with generalised L1-minimisation problems, i.e. Bolza problems involving the absolute value of the control with a control-affine dynamics. We establish sufficient conditions for the strong local optimality of extremals given by the concatenation of bang, singular and inactive (zero) arcs. The sufficiency of such conditions is proved by means of Hamiltonian methods. As a by-product of the result, we provide an explicit invariant formula for the second variation along the singular arc.

scholarly journals The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang–Mills theory on a Minkowskian background

2020 ◽  
Vol 17 (08) ◽  
pp. 2050114
Author(s):  
Luca Accornero ◽  
Marcella Palese
Keyword(s):  
Field Theory ◽  
Geometric Structure ◽  
Arbitrary Order ◽  
Higher Order ◽  
Second Variation ◽  
Jacobi Fields ◽  
Yang Mills ◽  
Conserved Current ◽  
The Second Variation ◽  
Higher Order Lagrangian

We characterize the second variation of an higher order Lagrangian by a Jacobi morphism and by currents strictly related to the geometric structure of the variational problem. We discuss the relation between the Jacobi morphism and the Hessian at an arbitrary order. Furthermore, we prove that a pair of Jacobi fields always generates a (weakly) conserved current. An explicit example is provided for a Yang–Mills theory on a Minkowskian background.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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