A Generalization of Chetaev's Principle for a Class of Higher Order Non-holonomic Constraints

The constraint distribution in non-holonomic mechanics has a double role. On one hand, it is a kinematic constraint, that is, it is a restriction on the motion itself. On the other hand, it is also a restriction on the allowed variations when using D'Alembert's Principle to derive the equations of motion. We will show that many systems of physical interest where D'Alembert's Principle does not apply can be conveniently modeled within the general idea of the Principle of Virtual Work by the introduction of both kinematic constraints and variational constraints as being independent entities. This includes, for example, elastic rolling bodies and pneumatic tires. Also, D'Alembert's Principle and Chetaev's Principle fall into this scheme. We emphasize the geometric point of view, avoiding the use of local coordinates, which is the appropriate setting for dealing with questions of global nature, like reduction.Comment: 27 pages. Journal of Mathematical Physics (to zappear

Spacetime Covariant Form of Ashtekar's Constraints

The Lagrangian formulation of classical field theories and in particular general relativity leads to a coordinate-free, fully covariant analysis of these constrained systems. This paper applies multisymplectic techniques to obtain the analysis of Palatini and self-dual gravity theories as constrained systems, which have been studied so far in the Hamiltonian formalism. The constraint equations are derived while paying attention to boundary terms, and the Hamiltonian constraint turns out to be linear in the multimomenta. The equivalence with Ashtekar's formalism is also established. The whole constraint analysis, however, remains covariant in that the multimomentum map is evaluated on {\it any} spacelike hypersurface. This study is motivated by the non-perturbative quantization program of general relativity.Comment: 22 pages, plain Tex, no figures, accepted for publication in Nuovo Cimento

Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings

The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew's triples. The results are also extended to the case of discrete dynamics and nonholonomic mechanics. Interesting applications to geometrical integration of Hamiltonian systems are obtained.Comment: 33 page

A RESEARCH ON THE CONCEPTUAL ORGANISATION OF PHYSICS CURRICULUM AND STANDARDS.

We outline the most significant features of a research conducted in the last years within three National Research Projects. First, we will discuss some shared, strategic research-choices and guidelines, validated by research results. Then, by a bottom-up approach, we will focus our attention on problems related to long-term Physics’ curricular planning and to a correlated schematic Standards’ definition. We propose to structure the phenomenological side of the curriculum according to eight content areas. Some structuring concepts and processes are made explicit as teaching/learning strategies (corresponding to ways of thinking, ways to look-at, modes of action, ways to talk about, etc). Such “universal categorisation tools” should be introduced from the very beginning of the talking-about and of the acting-on aspects of the physical world, and progressively explicated, formalised, reciprocally structured. Several levels of giving shape to (formalising) physical phenomena should thus become gradually accessible, keeping in mind that all the ones up to a given level are actually necessary for meaningful understanding