Helmholtz's inverse problem of the discrete calculus of variations (original) (raw)
The inverse problem of the calculus of variations for discrete systems
Journal of Physics A: Mathematical and Theoretical
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design and computer simulation of numerical integrators.
Discrete Calculus of Variations for Quadratic Lagrangians
Communications in Mathematical Analysis
The intent of this paper is to develop a framework for discrete calculus of variations with action densities involving a new class of discretization operators. We introduce first the generalized scale derivatives, study their regularity and state some Leibniz formulas. Then, we deduce the discrete Euler-Lagrange equations for critical points of sampled actions that we compare to existing versions. Next we investigate the case of general quadratic Lagrangians and provide two examples of such Lagrangians. At last, we find nontrivial properties for the discretization of a quadratic null Lagrangian.
A discrete algorithm to the calculus of variations
arXiv preprint arXiv:1003.0934, 2010
Abstract: A numerical study of an algorithm proposed by Gusein Guseinov, which determines approximations to the optimal solution of problems of calculus of variations using two discretizations and correspondent Euler-Lagrange equations, is investigated. The results we obtain to discretizations of the brachistochrone problem and Mania example with Lavrentiev's phenomenon are compared with the solutions found by other methods and solvers. We conclude that Guseinov's method presents better solutions in most of the ...
On the inverse problem of the calculus of variations
1981
We consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first-order form. We provide a way of constructing infinitely many Lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach. We present examples of first-order Lagrangians for systems for which no second-order Lagrangians exist. The Hamiltonian theory for first-order (degenerate) Lagrangians is constructed using Dirac's method for singular Lagrangians.
2010
We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, J 1 π. We prove that a system of time dependent SODE, identified with a semispray S, is Lagrangian if and only if a special class, Λ 1 S (J 1 π), of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class Λ 1 S (J 1 π) of semi-basic 1-forms. Each such class contains the Poincaré-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in Λ 1 S (J 1 π), which is not a Poincaré-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.
A Variational Principle for Discrete Integrable Systems
Symmetry Integrability and Geometry-methods and Applications, 2018
For multidimensionally consistent systems we can consider the Lagrangian as a form, closed on the multidimensional equations of motion. For 2-dimensional systems this allows us to define an action on a 2-dimensional surface embedded in a higher dimensional space of independent variables. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also variations of the surface in the space of independent variables. Here we give the resulting set of Euler-Lagrange equations firstly in 2 dimensions, and show how they can specify equations on a single quad in the lattice. We give the defining set of Euler-Lagrange equations also for 3-dimensional systems, and in general for n-dimensional systems. In this way the variational principle can be considered as supplying Lagrangians as solutions of a system of equations, as much as the equations of motion themselves.
Variational Approaches for Lagrangian Discrete Nonlinear Systems
Mathematics
In this paper, we study the multiple solutions for Lagrangian systems of discrete second-order boundary value systems involving the discrete p-Laplacian operator. The technical approaches are based on a local minimum theorem for differentiable functionals in a finite dimensional space and variational methods due to Bonanno. The existence of at least one solution, as well as three solutions for the given system are discussed and some examples and remarks have also been given to illustrate the main results.
The variational principles of mechanics are firmly rooted in the soil of that great century of Liberalism which starts with Descartes and ends with the French Revolution and which has witnessed the lives of Leibniz, Spinoza, Goethe, and Johann Sebastian Bach. It is the only period of cosmic thinking in the entire history of Europe since the time of the Greeks. 1
The delta-nabla calculus of variations
arXiv preprint arXiv:0912.0494, 2009
The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we propose a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.