Lagrangian description of the variational equations

Abstract

A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required (prolonged) Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all the virtual displacements joining any two integral curves. Our main result establishes that both the Euler-Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the first prolongation of the original LagrangianAfter discussing certain features of the formulation, we introduce the so-called inherited constants of the motion and relate them to the Noether constants of the extended system.

A review on extension of Lagrangian-Hamiltonian mechanics

“…These principles helped in establishing connections and applications of these disciplines, and in devising diverse approximation techniques. Arizmendi et al (2003) developed a variant of the usual Lagrangian, which describes both the equations of motion and the variational equations of the system. The required Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all virtual displacements joining any two integral curves.…”

“…The required Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all virtual displacements joining any two integral curves. An extremal principal for obtaining the variational equations of a Lagrangian system is reviewed and formalized by Delgado et al (2004) by relating the new Lagrangian function (Arizmendi et al, 2003) needed in such scheme to a prolongation (Hassani, 1999;Olver, 1986) of the original Lagrangian. In their work, they considered an N-degree of freedom dynamical system described by an autonomous non-singular Lagrangian function , a = 1.2…N defined in the tangent bundle TQ of its configuration Manifold Q.…”

Extension of Lagrangian–Hamiltonian mechanics for continuous systems—investigation of dynamics of a one-dimensional internally damped rotor driven through a dissipative coupling

“…Some other useful results related with the symmetry aspects of the Lagrangian and Hamiltonian formalism are discussed in papers of Katzin and Levine [5], Sarlet and Cantrazin [6] and Simic [7]. Recently, Arizmendi et al [8] have defined a new Lagrangian in extended configuration space comprised of the original configuration of the system and all virtual displacements joining any two integral curves. In other papers, Damianou and Sophocleous [9,10] classified the symmetries (Lie point and Noether) of Hamiltonian systems with two and three degrees of freedom.…”

“…In (8), the first term is the kinetic energy of the system, the second term is the strain energy for small deflections where E is the Young modulus of the shaft material and I is the cross-sectional moment of inertia, the third term is contributed by the rotary moments with the angular velocity ∂ 2 u i (η, x)/∂η∂x, the fourth term is the umbra-potential due to external damping, the fifth term is contributed by the gyroscopic forces with constant angular velocity ω, and the sixth term is the umbra-potential due to the internal damping. All the terms are per unit length.…”

“…The umbra-Lagrangian density for a rotating shaft with circular cross-sectional incorporating external and internal damping and gyroscopic coupling is given in (8). This Lagrangian density admits SO(2) symmetry.…”