Non-Involutive Constrained Systems and Hamilton-Jacobi Formalism

In this work we discuss the natural appearance of the Generalized Brackets in systems with non-involutive (equivalent to second class) constraints in the Hamilton-Jacobi formalism. We show how a consistent geometric interpretation of the integrability conditions leads to the reduction of degrees of freedom of these systems and, as consequence, naturally defines a dynamics in a reduced phase space.Comment: 12 page

Nilpotent Classical Mechanics

The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates $\eta$. Necessary geometrical notions and elements of generalized differential $\eta$-calculus are introduced. The so called $s-$geometry, in a special case when it is orthogonally related to a traceless symmetric form, shows some resemblances to the symplectic geometry. As an example of an $\eta$-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the $R$-symmetry known for the Graded Superfield Oscillator (GSO) is present also here for the supersymmetric $\eta$-system. The generalized Poisson bracket for $(\eta,p)$-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.Comment: 23 pages, no figures. Corrected version. 2 references adde

Hamilton-Jacobi approach to Berezinian singular systems

In this work we present a formal generalization of the Hamilton-Jacobi formalism, recently developed for singular systems, to include the case of Lagrangians containing variables which are elements of Berezin algebra. We derive the Hamilton-Jacobi equation for such systems, analizing the singular case in order to obtain the equations of motion as total differential equations and study the integrability conditions for such equations. An example is solved using both Hamilton-Jacobi and Dirac's Hamiltonian formalisms and the results are compared.Comment: LaTex, 30 pages, no figure

Elementary solution to the time-independent quantum navigation problem

A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realizes a required quantum process or task, under the influence of a prevailing ‘background’ Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of timeindependent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalizations to higher-dimensional systems are discussed

Verifiable conditions of ℓ1\ell_1-recovery of sparse signals with sign restrictions

We propose necessary and sufficient conditions for a sensing matrix to be "s-semigood" -- to allow for exact $\ell_1$-recovery of sparse signals with at most $s$ nonzero entries under sign restrictions on part of the entries. We express the error bounds for imperfect $\ell_1$-recovery in terms of the characteristics underlying these conditions. Furthermore, we demonstrate that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-semigood. We concentrate on the properties of proposed verifiable sufficient conditions of $s$-semigoodness and describe their limits of performance

Direct approach to the problem of strong local minima in Calculus of Variations

The paper introduces a general strategy for identifying strong local minimizers of variational functionals. It is based on the idea that any variation of the integral functional can be evaluated directly in terms of the appropriate parameterized measures. We demonstrate our approach on a problem of W^{1,infinity} weak-* local minima--a slight weakening of the classical notion of strong local minima. We obtain the first quasiconvexity-based set of sufficient conditions for W^{1,infinity} weak-* local minima.Comment: 26 pages, no figure

Hamilton-Jacobi Approach for First Order Actions and Theories with Higher Derivatives

In this work we analyze systems described by Lagrangians with higher order derivatives in the context of the Hamilton-Jacobi formalism for first order actions. Two different approaches are studied here: the first one is analogous to the description of theories with higher derivatives in the hamiltonian formalism according to [Sov. Phys. Journ. 26 (1983) 730; the second treats the case where degenerate coordinate are present, in an analogy to reference [Nucl. Phys. B 630 (2002) 509]. Several examples are analyzed where a comparison between both approaches is made

Hamilton-Jacobi equations and Brane associated Lagrangians

This article seeks to relate a recent proposal for the association of a covariant Field Theory with a string or brane Lagrangian to the Hamilton-Jacobi formalism for strings and branes. It turns out that since in this special case, the Hamiltonian depends only upon the momenta of the Jacobi fields and not the fields themselves, it is the same as a Lagrangian, subject to a constancy constraint. We find that the associated Lagrangians for strings or branes have a covariant description in terms of the square root of the same Lagrangian. If the Hamilton-Jacobi function is zero, rather than a constant, then it is in in one dimension lower, reminiscent of the `holographic' idea. In the second part of the paper, we discuss properties of these Lagrangians, which lead to what we have called `Universal Field Equations', characteristic of covariant equations of motion.Comment: 23 pages,LaTeX2e, clarified text, generalised proof in appendi

General Relativity in two dimensions: a Hamilton-Jacobi constraint analysis

We will analyze the constraint structure of the Einstein-Hilbert first-order action in two dimensions using the Hamilton-Jacobi approach. We will be able to find a set of involutive, as well as a set of non-involutive constraints. Using generalized brackets we will show how to assure integrability of the theory, to eliminate the set of non-involutive constraints, and to build the field equations