The inverse problem for flat kinetic minus potencial Lagrangians.
The inverse problem in the calculus of variations: new developments
We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for . We then examine a new class of solutions in arbitrary dimension and give some non-trivial examples in dimension 3.
The Legendre transformation
The symmetry reduction of variational integrals
The Routh reduction of cyclic variables in the Lagrange function and the Jacobi-Maupertuis principle of constant energy systems are generalized. The article deals with one-dimensional variational integral subject to differential constraints, the Lagrange variational problem, that admits the Lie group of symmetries. Reduction to the orbit space is investigated in the absolute sense relieved of all accidental structures. In particular, the widest possible coordinate-free approach to the underdetermined...
The symmetry reduction of variational integrals, complement
Some open problems appearing in the primary article on the symmetry reduction are solved. A new and quite simple coordinate-free definition of Poincaré-Cartan forms and the substance of divergence symmetries (quasisymmetries) are clarified. The unbeliavable uniqueness and therefore the global existence of Poincaré-Cartan forms without any uncertain multipliers for the Lagrange variational problems are worth extra mentioning.
Totally singular Lagrangians and affine Hamiltonians.
Variational calculus in several variables : a hamiltonian approach
Variational calculus on sub-Riemannian manifolds.
Variational formulations I: Statics of mechanical systems
Two improvements of variational formulations of mechanics are proposed. The first consists in a modification of the definition of equilibrium. The second consists in adding elements of control by external devices. In the present note the proposed improvements are applied to variational principles of statics. Numerous examples are given.
Variational metrics on and the geometry of nonconservative mechanics