Narrow Results

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form $\frac{d}{dt}(\nabla \mathrm{\Phi }(\dot{u}(t)))+V_u(t,u(t))+\lambda G_u(t,u(t))=0$, where $\mathrm{\Phi }:{ℝ}^n\to [0,\infty )$ is a ${𝒢}$-function in the sense of Trudinger, $V,G:ℝ\times {ℝ}^n\to ℝ$ are $C^1$-smooth, $T$-periodic in the time variable $t$ and $\lambda$ is a real parameter. We prove the existence of a $T$-periodic solution $u_{\lambda }:ℝ\to {ℝ}^n$ for any sufficiently small $\left|\lambda \right|$, and show that the found solutions converge to a $T$-periodic solution of the unperturbed system if $\lambda$ tends to $0$. Let us stress that our theorem only makes a natural assumption on the potential $V$ and no additional assumption on $G$ except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges.

In this paper, we prove that the Lagrangian system on any Riemannian torus with C³-smooth even and τ-periodic potential in time possesses infinitely many different periodic contractible even solutions with integer multiple periods of τ. As a consequence, we get that the same conclusion holds for any τ > 0 and the Lagrangian system on any Riemannian torus with C³-smooth potential independent of time.

The Hamilton–Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron–monopole system.

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.