Short-Term Lagrangian Points and Research Using Elementary Methods
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Since Euler and Lagrange have calculated and proved that the three-body problem in the celestial movement has the so-called Lagrangian point, the human study of the Lagrange point has not stopped. Lagrange points have a pivotal strategic position in aerospace engineering, so the calculation of Lagrange points also has important scientific significance. In this paper, we establish a simplified model of the circular motion of the binary system, and use the elementary method to roughly calculate the position of the five Lagrangian points of the Earth-Month system, and through computational studies, we believe that there are only four Lagers in the binary system with equal or similar mass. Lange point.Keywords:
Lagrangian point
Position (finance)
Euler–Lagrange equation
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Hamiltonian systems arise in a wide variety of idealized physical systems and Hamilton’s equations often must be solved numerically. In general, traditional finite difference methods for numerically integrating ordinary differential equations do not take into account the special structure of Hamilton’s equations. One would like the numerical solution to preserve some or all of the properties of the continuous time solution. Variational integrators, also called discrete Euler-Lagrange equations in this thesis, provide a way to do that. This thesis is about differentiating, i.e linearizing Euler-Lagrange equations and discrete Euler-Lagrange equations by differentiating the Lagrangian or discrete Lagrangian. The original motivation for differentiating the Lagrangian was to develop higher order variational integrators. However it was found that the resulting intgerators are not higher order. Nonetheless, the pursuit of this goal turned up several interesting results : i. Differentiating a Lagrangian along the acceleration, i.e along (q, q) results in the EulerLagrange equation 0 = 0 (Lemma 2, page 20). ii. Differentiating a Lagrangian along a general direction (u, u), determining the EulerLagrange equation and then setting u = q results in an Euler-Lagrange equation that is the time derivative of the original Euler-Lagrange equation corresponding to the original Lagrangian (Lemma 3, page 21). iii. A similar operation on the discrete Lagrangian gives a “discrete time derivative” of the discrete Euler-Lagrange equation corresponding to the original discrete Lagrangian (Lemma 4, page 23). iv. The operations described in the previous two items are part of a commutative diagram (Theorem 1, page 28). v. Numerical measurements and analytical calculation of order of accuracy for specific examples using this approach show that higher order methods are not obtained in general (Chapter 5, especially Section 5.4 and 5.5). vi. The manifold consisting of triples that satisfy a discrete Euler Lagrange equation is a smooth manifold (Proposition 4, page 35). vii. The algorithm on TQ × TQ is symplectic and the algorithm that updates triples of points on Q is presymplectic (Theorem 2, page 36). I also give a derivation of a symplectic form on TQ× TQ where Q is the configuration manifold. Most proofs given in this thesis are done using local coordinates. A possible direction for further study is to give intrinsic proofs in order to gain a deeper understanding of the geometric meaning of the operations of this thesis.
Variational integrator
Euler–Lagrange equation
Linearization
Lemma (botany)
Lagrange polynomial
Lagrangian mechanics
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If a Lagrangian defining a variational problem has order k then its Euler-Lagrange equations generically have order 2k.This paper considers the case where the Euler-Lagrange equations have order strictly less than 2k, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial.The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such k-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way.
Euler–Lagrange equation
Lagrange polynomial
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The Veselov approach provides a discrete formulation of the Euler–Lagrange equation. To get this, a discrete Lagrangian version of a continuous one is considered and then a variational process is used. This problem has been studied in many papers by different authors, according to references and therein citations. This type of discretization can be useful in the case when the continuous Euler–Lagrange equation is given in a semispray form, which is difficult to solve effectively (as for example in the many-body problem). Our aim is to consider a given continuous Lagrangian and to construct directly discrete approximations of the corresponding Euler–Lagrange equation. This is done without considering a discrete Lagrangian and a variational process, nor by using a difference equation of geodesics. Some numerical examples are included in order to compare the performance of the proposed approximations versus the classical Veselov approach.
Euler–Lagrange equation
Constraint algorithm
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Recently, there has been considerable interest in efficient formulations of manipulator dynamics. The inefficiency of the classic Lagrangian formulation is well known, leading several researchers to a new formulation based on the Newton-Euler equations. This formulation is highly ef ficient, but there may be some confusion as to the source of this efficiency. This paper shows that there is in fact no fundamental difference in computational efficiency between Lagrangian and Newton-Euler formulations. The efficiency of the above-mentioned Newton-Euler formula tion is due to two factors: the recursive structure of the computation and the representation chosen for the rota tional dynamics. Both of these factors can be achieved in a Lagrangian formulation. Recursive Lagrangian dy namics has been discussed previously by Hollerbach. This paper compares the representations that have been used and shows that with a proper choice the Lagrangian formulation is indeed equivalent to the Newton-Euler formulation.
Lagrangian mechanics
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This article mainly uses Lagrangian multiplier method wich is the important way of solving the Extremal Problem under the restraint condition to solve the distance question of space analytic geometry and proves some important conclusions.
Augmented Lagrangian method
Lagrangian relaxation
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Abstract : This paper develops new forms for the Lagrangian Multipliers used in studies of constrained systems, as well as variants of the Euler Lagrange equations. These formulas facilitate the computation of the multipliers and solution of the Euler Lagrange equations. In addition, the link between virtual displacement and the multipliers Euler Lagrange is elucidated. Keywords: Lagrangian multipliers; Constrained dynamics; Multipliers; Euler Lagrange equations.
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