Higher-order differential equations and higher-order lagrangian mechanics

1986 ◽  
Vol 99 (3) ◽  
pp. 565-587 ◽  
Author(s):  
M. Crampin ◽  
W. Sarlet ◽  
F. Cantrijn
Keyword(s):  
Order Differential Equation ◽  
Higher Order ◽  
Point Of View ◽  
Lagrangian Mechanics ◽  
First Order ◽  
The Subject ◽  
Tangent Bundles ◽  
Order Tangent ◽  
Higher Order Differential Equations ◽  
Number Of Authors

The study of higher-order mechanics, by various geometrical methods, in the framework of the theory of higher-order tangent bundles or jet spaces, has been undertaken by a number of authors recently: for example, Tulczyjew [16, 17], Rodrigues [14, 15] de León [8], Krupka and Musilova [11, and references therein]. In this article we wish to complement these studies by approaching the subject from a new point of view, one which we developed for second-order differential equation fields and first-order Lagrangian mechanics in [19]. In particular, our aim is to show that many of the results we obtained there may be extended to the higher-order case.

Higher-Order Multilevel Solvers for the EHL Line and Point Contact Problem

1994 ◽  
Vol 116 (4) ◽  
pp. 741-750 ◽  
Author(s):  
C. H. Venner
Keyword(s):  
Contact Problem ◽  
Computing Time ◽  
Point Contact ◽  
Higher Order ◽  
Second Order ◽  
Point Of View ◽  
First Order ◽  
Fundamental Point ◽  
Numerical Solver ◽  
Circular Contact

This paper addresses the development of efficient numerical solvers for EHL problems from a rather fundamental point of view. A work-accuracy exchange criterion is derived, that can be interpreted as setting a limit to the price paid in terms of computing time for a solution of a given accuracy. The criterion can serve as a guideline when reviewing or selecting a numerical solver and a discretization. Earlier developed multilevel solvers for the EHL line and circular contact problem are tested against this criterion. This test shows that, to satisfy the criterion a second-order accurate solver is needed for the point contact problem whereas the solver developed earlier used a first-order discretization. This situation arises more often in numerical analysis, i.e., a higher order discretization is desired when a lower order solver already exists. It is explained how in such a case the multigrid methodology provides an easy and straightforward way to obtain the desired higher order of approximation. This higher order is obtained at almost negligible extra work and without loss of stability. The approach was tested out by raising an existing first order multilevel solver for the EHL line contact problem to second order. Subsequently, it was used to obtain a second-order solver for the EHL circular contact problem. Results for both the line and circular contact problem are presented.

The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

scholarly journals Oscillation of even-order neutral differential equations via comparison principles

2014 ◽  
Vol 30 (3) ◽  
pp. 293-300
Author(s):  
J. DZURINA ◽  
◽  
B. BACULIKOVA ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Comparison Principles ◽  
Neutral Differential Equations ◽  
First Order ◽  
Even Order ◽  
Delay Differential

In the paper we offer oscillation criteria for even-order neutral differential equations, where z(t) = x(t) + p(t)x(τ(t)). Establishing a generalization of Philos and Staikos lemma, we introduce new comparison principles for reducing the examination of the properties of the higher order differential equation onto oscillation of the first order delay differential equations. The results obtained are easily verifiable.

scholarly journals New Comparison Theorems for the Nth Order Neutral Differential Equations with Delay Inequalities

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 454 ◽  
Author(s):  
Osama Moaaz ◽  
Shigeru Furuichi ◽  
Ali Muhib
Keyword(s):  
Differential Equations ◽  
Higher Order ◽  
Comparison Theorems ◽  
Differential Inequalities ◽  
Neutral Differential Equations ◽  
First Order ◽  
A New Technique ◽  
Equations With Delay ◽  
Higher Order Differential Equations ◽  
First Order Differential Equations

In this work, we present a new technique for the oscillatory properties of solutions of higher-order differential equations. We set new sufficient criteria for oscillation via comparison with higher-order differential inequalities. Moreover, we use the comparison with first-order differential equations. Finally, we provide an example to illustrate the importance of the results.

GEOMETRIC STUDY OF HAMILTON'S VARIATIONAL PRINCIPLE

1991 ◽  
Vol 03 (04) ◽  
pp. 379-401 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
CARLOS LÓPEZ
Keyword(s):  
Variational Calculus ◽  
Higher Order ◽  
Lagrangian Mechanics ◽  
Lagrange Equations ◽  
Order Case ◽  
Helmholtz Conditions ◽  
Higher Order Tangent Bundle ◽  
Order Tangent ◽  
Geometric Study

Higher order tangent bundle geometry and sections along maps are used in Geometrical Mechanics in order to develop an intrinsic variational calculus. The role of variational derivative as the bundle operator associated to exterior differential on the set of trajectories is remarked. Euler-Lagrange equations and Poincaré-Cartan form are rederived in this way. Helmholtz conditions for the inverse problem of Lagrangian Mechanics are geometrically obtained for the general higher order case.

Higher-order differential equations with a slowly decaying coefficient containing a periodic factor

1995 ◽  
Vol 117 (1) ◽  
pp. 175-184 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Asymptotic Solution ◽  
Asymptotic Form ◽  
Asymptotic Methods ◽  
Mean Value ◽  
Higher Order ◽  
Order Equation ◽  
Differential Systems ◽  
Image Position ◽  
Higher Order Differential Equations ◽  
Number Of Authors

Let p(x) be periodic with mean value c and let α > 0. Then the investigation of the asymptotic solution of the second-order equationcan be split into four cases: (i) α > 2, (ii) α > 1 and c = 0, (iii) 0 < α ≤ 1 and c = 0, (iv) 0 < α ≤ 2 and c ≠ 0. The first two cases are straightforward because then there are solutions y1 and y2 of (1·1) such that y1(x) ~ x and y2(x) → 1 as x → ∞ [5, theorems 2·7·2 and 2·7·5; 7, p. 375; 8]. Thus the coefficient of y in (1·1) represents a perturbation which has no effect on the dominant asymptotic behaviour of solutions. Case (iii) is less simple because here the nature of p(x) does influence the asymptotic form of solutions and, indeed, when α < 1 the solutions of (1·1) are oscillatory. This case has been fully covered by a number of authors: Atkinson, Eastham and McLeod[2, §3], Cassell[3, 4], Eastham[5, §4·13], Mahoney[10], Willett[13] and Wong[14, 15]. In addition, corresponding results for a higher-order self-adjoint analogue of (1·1) have been obtained recently by Al-Hammadi and Eastham [1, 6] as an application of general asymptotic methods for differential systems.

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