1,721,048 research outputs found
Learning and Playing Mathematics
No full textIn this paper we report our experience of teaching mathematics by playing traditional games. The experience aim was to stimulate the students to actively participate, encouraging them to collaborate each other. Moreover we remark that this didactical games help students to develop a more conscious and personal approach to standard mathematics arguments and offer an unusual but very effective chance for self-evaluation
Reduction by lambda-symmetries and sigma-symmetries: a Frobenius approach
No full textDifferent kinds of reduction for ordinary differential equations, such as lambda-symmetry and sigma--symmetry reductions, are recovered as particular cases of Frobenius reduction theorem for distribution of vector fields. This general approach provides some hints to tackle the reconstruction problem and to solve it under suitable assumptions on the distribution involved in the reduction process
Variational Problems with Symmetries : A Pfaffian System Approach
No full textReduction methods for completely integrable Pfaffian Systems with symmetry are applied to variational problems, providing analogues of the Arnold–Liouville Theorem and Marsden–Weinstein reduction in the Lagrangian setting. A generalization of the
Mishchenko–Fomenko Theorem for non Abelian integrability is given in terms of solvable
structures
A geometric approach to constrained mechanical systems, symmetries and inverse problems
No full textWe introduce a geometrical framework for the description of constrained mechanical systems, and we analyse different kinds of symmetries and their relationships. We propose a new definition for non-holonomic Lagrangian mechanical systems, and we give a geometrical characterization for the Helmholtz conditions related to the inverse problem
Constitutive Characterizations of Non-Ideal Constraints in Classical Mechanics
No full textThe theory of two-sided, positional constraints with friction acting on a material point in Classical Mechanics is presented in the geometrical context of the jet-extensions of an affine bundle. This context allows one to study both rigid and non-rigid constraints. We prove that the geometry of the system and the forces acting on the point determine in a natural way the direction of the friction reaction. This allows one to introduce a wide class of constitutive characterizations of constraints with friction, that includes the most common ones
Liouville condition, Nambu mechanics and differential forms
No full textIntroducing a geometrical framewok for description of Nambu mechanics, we give conditions to guarantee that an ordinary differential equation (ODE) may be written in the Nambu for
Reduction of Exterior Differential Systems for ordinary variational problems
No full textThis paper deals with the reduction of ordinary variational problems
with Abelian and non Abelian symmetry group. It differs from the existing results
on geometric reduction in the approach that is completely Lagrangian and based on
Exterior Differential Systems. In this setting two procedures of reduction come out
naturally: the first one is the Lagrangian analogue of the Marsden-Weinstein reduction
holding in the symplectic case; the other one, especially in the non Abelian case, comes out naturally in our setting
Deformation of Lie derivative and mu-symmetries
No full textWe introduce, in the spirit of Witten''s
gauging of exterior differential, a deformed Lie derivative that
allows a geometrical interpretation of lambda and mu-symmetries
in complete analogy with standard symmetries. The case of
variational symmetries (both for ODEs and for PDEs) is also
considered in this approach, leading to lambda and
mu-conservation laws
Hyper-Hamiltonian dynamics
No full textWe introduce an extension of Hamiltonian dynamics, defined on hyper-Kahler manifolds, which we call ‘hyperHamiltonian dynamics’. We show that this has many of the attractive features of standard Hamiltonian dynamics. We also discuss the prototypical integrable hyper-Hamiltonian systems, i.e. quaternionic
oscillators
PDEs reduction and lambda-symmetries
No full textSo called lambda-symmetries were introduced by Muriel and Romero, and geometrically
characterized by Pucci and Saccomandi [8, 12], in the ODE case. We extend them to
the PDE framework. In this context the central object is a horizontal one-form mu, and we
speak of mu-prolongations of vector fields and mu-symmetries of PDEs. The latter are as good
as standard symmetries in providing symmetry reduction of PDEs (or systems thereof) and
explicit invariant solutions
- …
