共查询到20条相似文献,搜索用时 31 毫秒
1.
A. Ghose Choudhury Partha Guha Barun Khanra 《Journal of Mathematical Analysis and Applications》2009,360(2):45-664
We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé–Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of second-order ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian. 相似文献
2.
B. Dragovich 《Theoretical and Mathematical Physics》2010,163(3):768-773
We study the construction of Lagrangians that can be considered the Lagrangians of the p-adic sector of an adelic open scalar
string. Such Lagrangians are closely related to the Lagrangian for a single p-adic string and contain the Riemann zeta function
with the d’Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach that combines
all p-adic Lagrangians. This new Lagrangian is attractive because it is an analytic function of the d’Alembertian. Investigating
the field theory with the Riemann zeta function is also interesting in itself. 相似文献
3.
We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Carathéodory form are identical. 相似文献
4.
We present a class of new augmented Lagrangian functions with the essential property that each member is concave quadratic when viewed as a function of the multiplier. This leads to an improved duality theory and to a related class of exact penalty functions. In addition, a relationship between Newton steps for the classical Lagrangian and the new Lagrangians is established.This work was supported in part by ARO Grant No. DAAG29-77-G-0125. 相似文献
5.
Yng-Ing Lee 《Calculus of Variations and Partial Differential Equations》2012,45(1-2):231-251
Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They are natural generalizations of special Lagrangians or Lagrangian and minimal submanifolds. In this paper, we obtain a local condition that gives the existence of a smooth family of Hamiltonian stationary Lagrangian tori in K?hler manifolds. This criterion involves a weighted sum of holomorphic sectional curvatures. It can be considered as a complex analogue of the scalar curvature when the weighting are the same. The problem is also studied by Butscher and Corvino (Hamiltonian stationary tori in Kahler manifolds, 2008). 相似文献
6.
7.
In this work we show that, if L is a natural Lagrangian system such that the k-jet of the potential energy ensures it does not have a minimum at the equilibrium and such that its Hessian has rank at least n−2, then there is an asymptotic trajectory to the associated equilibrium point and so the equilibrium is unstable. This applies, in particular, to analytic potentials with a saddle point and a Hessian with at most 2 null eigenvalues.The result is proven for Lagrangians in a specific form, and we show that the class of Lagrangians we are interested can be taken into this specific form by a subtle change of spatial coordinates. We also consider the extension of this results to systems subjected to gyroscopic forces. 相似文献
8.
I. Naeem 《Journal of Mathematical Analysis and Applications》2008,342(1):70-82
We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians. 相似文献
9.
Nail H. Ibragimov 《Journal of Mathematical Analysis and Applications》2006,318(2):742-757
Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations. 相似文献
10.
We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space can be regarded as another effective system with commuting spatial coordinates. Because the path integral for quadratic Lagrangians is exactly solvable and a general formula for the probability amplitude exists, we restrict our research to this class of Lagrangians. We find a general relation between quadratic Lagrangians in their commutative and noncommutative regimes and present the corresponding noncommutative path integral. This method is illustrated with two quantum mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator. 相似文献
11.
André Neves 《Inventiones Mathematicae》2007,168(3):449-484
We study singularities of Lagrangian mean curvature flow in ℂ
n
when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are
unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonian
isotopic to a plane that, nevertheless, develop finite time singularities under mean curvature flow.
We then prove two theorems regarding the tangent flow at a singularity when the initial condition is a zero-Maslov class Lagrangian.
The first one (Theorem A) states that that the rescaled flow at a singularity converges weakly to a finite union of area-minimizing
Lagrangian cones. The second theorem (Theorem B) states that, under the additional assumptions that the initial condition
is an almost-calibrated and rational Lagrangian, connected components of the rescaled flow converges to a single area-minimizing Lagrangian cone, as opposed to a possible non-area-minimizing union of area-minimizing Lagrangian cones.
The latter condition is dense for Lagrangians with finitely generated H
1(L,ℤ). 相似文献
12.
This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function. 相似文献
13.
The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy
of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian.
The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton-Jacobi
equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend
on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method.
We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method
determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate
these systematic methods for constructing variational integrators with numerical examples. 相似文献
14.
B. G. Dragovich 《Theoretical and Mathematical Physics》2010,164(3):1151-1155
We consider the construction of Lagrangians that might be suitable for describing the entire p-adic sector of an adelic open
scalar string. These Lagrangians are constructed using the Lagrangian for p-adic strings with an arbitrary prime number p.
They contain space-time nonlocality because of the d’Alembertian in the argument of the Riemann zeta function. We present
a brief review and some new results. 相似文献
15.
16.
Roman A. Polyak 《Mathematical Programming》2002,92(2):197-235
We introduce an alternative to the smoothing technique approach for constrained optimization. As it turns out for any given
smoothing function there exists a modification with particular properties. We use the modification for Nonlinear Rescaling
(NR) the constraints of a given constrained optimization problem into an equivalent set of constraints.?The constraints transformation
is scaled by a vector of positive parameters. The Lagrangian for the equivalent problems is to the correspondent Smoothing
Penalty functions as Augmented Lagrangian to the Classical Penalty function or MBFs to the Barrier Functions. Moreover the
Lagrangians for the equivalent problems combine the best properties of Quadratic and Nonquadratic Augmented Lagrangians and
at the same time are free from their main drawbacks.?Sequential unconstrained minimization of the Lagrangian for the equivalent
problem in primal space followed by both Lagrange multipliers and scaling parameters update leads to a new class of NR multipliers
methods, which are equivalent to the Interior Quadratic Prox methods for the dual problem.?We proved convergence and estimate
the rate of convergence of the NR multipliers method under very mild assumptions on the input data. We also estimate the rate
of convergence under various assumptions on the input data.?In particular, under the standard second order optimality conditions
the NR method converges with Q-linear rate without unbounded increase of the scaling parameters, which correspond to the active
constraints.?We also established global quadratic convergence of the NR methods for Linear Programming with unique dual solution.?We
provide numerical results, which strongly support the theory.
Received: September 2000 / Accepted: October 2001?Published online April 12, 2002 相似文献
17.
R. R. Metsaev 《Theoretical and Mathematical Physics》2016,187(2):730-742
In the framework of the BRST-BV approach to the formulation of relativistic mechanics, we consider massless and massive fields of arbitrary spin propagating in a flat space and massless fields propagating in the AdS space. For such fields, we obtain BRST-BV Lagrangians invariant under gauge transformations. The Lagrangians and gauge transformations are constructed in terms of traceless gauge fields and traceless parameters of the gauge transformations. We consider the fields in the AdS space using the Poincaré parameterization of this space, which leads to a simple form of the BRST-BV Lagrangian. We show that in the Siegel gauge, the Lagrangian of the massless AdS fields leads to a decoupling of the equations of motion, and this substantially simplifies the study of the AdS/CFT correspondence. In a conformal algebra basis, we find a realization of the relativistic symmetries of fields and antifields in the AdS space. 相似文献
18.
J. B. Van den Berg R. C. Vandervorst 《Transactions of the American Mathematical Society》2002,354(4):1393-1420
We consider a special class of Lagrangians that play a fundamental role in the theory of second order Lagrangian systems: Twist systems. This subclass of Lagrangian systems is defined via a convenient monotonicity property that such systems share. This monotonicity property (Twist property) allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels. This reduction has some similarities with the method of broken geodesics for the geodesic variational problem on Riemannian manifolds. On the other hand, the monotonicity property can be related to the existence of local Twist maps in the associated Hamiltonian flow.
The finite dimensional reduction gives rise to a second order monotone recurrence relation. We study these recurrence relations to find simple closed characteristics for the Lagrangian system. More complicated closed characteristics will be dealt with in future work. Furthermore, we give conditions on the Lagrangian that guarantee the Twist property.
19.
We study the Hamilton-Jacobi equation for undiscounted exit time
control problems with general nonnegative Lagrangians using the
dynamic programming approach. We prove theorems characterizing the
value function as the unique bounded-from-below viscosity solution
of the Hamilton-Jacobi equation that is null on the target. The
result applies to problems with the property that all trajectories
satisfying a certain integral condition must stay in a bounded
set. We allow problems for which the Lagrangian is not uniformly
bounded below by positive constants, in which the hypotheses of
the known uniqueness results for Hamilton-Jacobi equations are not
satisfied. We apply our theorems to eikonal equations from
geometric optics, shape-from-shading equations from image
processing, and variants of the Fuller Problem. 相似文献
20.
Jake P. Solomon 《Geometric And Functional Analysis》2014,24(2):670-689
A Lagrangian submanifold in an almost Calabi–Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an infinite-dimensional analog of the non-compact symmetric space dual to the unitary group, and thus has non-positive curvature. 相似文献