Non-Noether symmetries in singular dynamical systems.
On classical dynamics of affinely-rigid bodies subject to the Kirchhoff-Love constraints.
On D’Alembert’s Principle
A formulation of the D’Alembert principle as the orthogonal projection of the acceleration onto an affine plane determined by nonlinear nonholonomic constraints is given. Consequences of this formulation for the equations of motion are discussed in the context of several examples, together with the attendant singular reduction theory.
On the linearization of second-order differential and difference equations.
On the mobility and efficiency of mechanical systems
It is shown that self-locomotion is possible for a body in Euclidian space, provided its dynamics corresponds to a non-quadratic Hamiltonian, and that the body contains at least 3 particles. The efficiency of the driver of such a system is defined. The existence of an optimal (most efficient) driver is proved.
Orbits of families of vector fields on subcartesian spaces
Orbits of complete families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows a description of the structure of the reduced phase space of a Hamiltonian system in terms of the reduced Poisson algebra. Moreover, one can give a global description of smooth geometric structures on a family of manifolds, which form a singular foliation of a subcartesian space, in terms of objects defined on the corresponding family of vector fields. Stratified...
Reconstruction phases for Hamiltonian systems on cotangent bundles.
Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric
For a riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the riemannian exponential...
Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric
For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential...
Reductions and conservation laws for BBM and modified BBM equations
In this paper, the classical Lie theory is applied to study the Benjamin-Bona-Mahony (BBM) and modified Benjamin-Bona-Mahony equations (MBBM) to obtain their symmetries, invariant solutions, symmetry reductions and differential invariants. By observation of the the adjoint representation of Mentioned symmetry groups on their Lie algebras, we find the primary classification (optimal system) of their group-invariant solutions which provides new exact solutions to BBM and MBBM equations. Finally, conservation...
Relations between constants of motion and conserved functions
We study relations between functions on the cotangent bundle of a spacetime which are constants of motion for geodesics and functions on the odd-dimensional phase space conserved by the Reeb vector fields of geometrical structures generated by the metric and an electromagnetic field.
Singularities of momentum maps of integrable Hamiltonian systems with two degrees of freedom.
Spontaneously broken symmetries
Symmetries and constants of the motion for dynamics in implicit form
Symmetries and integrals of motion in optimal control
Symmetries of a dynamical system represented by singular Lagrangians
Dynamical properties of singular Lagrangian systems differ from those of classical Lagrangians of the form . Even less is known about symmetries and conservation laws of such Lagrangians and of their corresponding actions. In this article we study symmetries and conservation laws of a concrete singular Lagrangian system interesting in physics. We solve the problem of determining all point symmetries of the Lagrangian and of its Euler-Lagrange form, i.e. of the action. It is known that every point...
Symmetries of second-order potential differential systems.
Symmetries of spin Calogero models.
The degenerate C. Neumann system I: symmetry reduction and convexity
The C. Neumann system describes a particle on the sphere S n under the influence of a potential that is a quadratic form. We study the case that the quadratic form has ℓ +1 distinct eigenvalues with multiplicity. Each group of m σ equal eigenvalues gives rise to an O(m σ)-symmetry in configuration space. The combined symmetry group G is a direct product of ℓ + 1 such factors, and its cotangent lift has an Ad*-equivariant momentum mapping. Regular reduction leads to the Rosochatius system on S ℓ,...
The symmetric tensor Lichnerowicz algebra and a novel associative Fourier-Jacobi algebra.