LINEARIZATION METHODS FOR VARIATIONAL INTEGRATORS AND EULER-LAGRANGE EQUATIONS
2
Citation
12
Reference
20
Related Paper
Citation Trend
Abstract:
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.Keywords:
Variational integrator
Euler–Lagrange equation
Linearization
Lemma (botany)
Lagrange polynomial
Lagrangian mechanics
Cite
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
Cite
Citations (2)
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
Cite
Citations (3)
Abstract We establish Euler-Lagrange equations for a problem of Calculus of Variations where the unknown variable contains a term of delay on a segment
Euler–Lagrange equation
Calculus of variations
Cite
Citations (2)
In this paper we examine the linear elliptic partial differential operators that appear as Euler-Lagrange systems of certain variational integrals. We give a sufficient condition for those systems to be deformable to the Laplace system.
Cite
Citations (0)
The proposed method transforming some of calculus of variation problems into Euler-Lagrange equations, the simplicity and effectiveness of this illustrated through some examples.
Calculus of variations
Variation (astronomy)
Cite
Citations (1)
A discrete Lagrange problem is defined as a discrete Lagrangian system endowed with a constraint submanifold in the space of 1-jets of the discretefibred manifold that configures the system. After defining the concepts of admissible section and infinitesimal admissible variation, the objective ofthese problems is to find admissible sections that are critical for the Lagrangian of the system with respect to the infinitesimal admissiblevariations. For admissible sections satisfying a certain regularity condition, we prove that critical sections are the solutions of an extendedunconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). Next, we define theconcept of Cartan 1-form, establish a Noether theory for symmetries and introduce a notion of 'constrained variational integrator' that wecharacterize through a Cartan equation ensuring its symplecticity. Under a certain regularity condition of the problem of Lagrange, we prove theexistence and uniqueness of this kind of integrators in the neighborhood of a critical section, showing then that such integrators can be constructedfrom a generating function of the second class in the sense of symplectic geometry. Finally, the whole theory is illustrated with three elementaryexamples.
Variational integrator
Constraint algorithm
Luke's variational principle
Cite
Citations (3)
In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler-Lagrange equations.They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton-Jacobi theory and theory of integration of variational equations.In this paper, Lepage equivalents of second-order Euler-Lagrange quasilinear PDE's are characterised explicitly.A closed (n + 1)-form uniquely determined by the Euler-Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations.
Cite
Citations (5)
Calculus of variations
Cite
Citations (31)
We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplecticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existence in finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodic perturbation. The numerical results are satisfactory.
Variational integrator
Symplectic integrator
Cite
Citations (5)
Overdetermined system
Algebraic equation
Euler–Lagrange equation
Euler method
Cite
Citations (28)