On Virtual Displacement and Virtual Work in Lagrangian Dynamics

The confusion and ambiguity encountered by students, in understanding virtual displacement and virtual work, is discussed in this article. A definition of virtual displacement is presented that allows one to express them explicitly for holonomic (velocity independent), non-holonomic (velocity dependent), scleronomous (time independent) and rheonomous (time dependent) constraints. It is observed that for holonomic, scleronomous constraints, the virtual displacements are the displacements allowed by the constraints. However, this is not so for a general class of constraints. For simple physical systems, it is shown that, the work done by the constraint forces on virtual displacements is zero. This motivates Lagrange's extension of d'Alembert's principle to system of particles in constrained motion. However a similar zero work principle does not hold for the allowed displacements. It is also demonstrated that d'Alembert's principle of zero virtual work is necessary for the solvability of a constrained mechanical problem. We identify this special class of constraints, physically realized and solvable, as {\it the ideal constraints}. The concept of virtual displacement and the principle of zero virtual work by constraint forces are central to both Lagrange's method of undetermined multipliers, and Lagrange's equations in generalized coordinates.Comment: 12 pages, 10 figures. This article is based on an earlier article physics/0410123. It includes new figures, equations and logical conten

Conservation of energy and momenta in nonholonomic systems with affine constraints

We characterize the conditions for the conservation of the energy and of the components of the momentum maps of lifted actions, and of their `gauge-like' generalizations, in time-independent nonholonomic mechanical systems with affine constraints. These conditions involve geometrical and mechanical properties of the system, and are codified in the so-called reaction-annihilator distribution

Progress in Classical and Quantum Variational Principles

We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schr\"{o}dinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems

Nonholonomic Constraints with Fractional Derivatives

We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle.Comment: 18 page

Atlas of two-dimensional irreversible conservative lagrangian mechanical systems with a second quadratic integral

This paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant, quadratic in velocities, other than the Hamiltonian. The configuration space is in general a 2D Riemannian or pseudo-Riemannian manifold and the determination of its geometry is a part of the process of solution. Forces acting on the system include a part derived from a scalar potential and a part derived from a vector potential, associated with terms linear in velocities in the Lagrangian function of the system. The last cause time-irreversibility of the system. We construct 41 multi-parameter integrable systems of the type described in the title mostly on Riemannian manifolds. They are mostly new and cover all previously known systems as special cases, corresponding to special values of the parameters. Those include all known cases of motion of a particle in the plane and all known cases in the dynamics of rigid body. In the last field we introduce a new integrable case related to Steklov's case of motion of a body in a liquid. Several new cases of motion in the plane, on the sphere and on the pseudo-sphere or in the hyperbolic plane are found as special cases. Prospective applications in mathematics and physics are also pointed out.Comment: Paper to be published in "Journal of Mathematical Physics", Vol. 48, issue 7, July 200

Autonomous zinc-finger nuclease pairs for targeted chromosomal deletion

Zinc-finger nucleases (ZFNs) have been successfully used for rational genome engineering in a variety of cell types and organisms. ZFNs consist of a non-specific FokI endonuclease domain and a specific zinc-finger DNA-binding domain. Because the catalytic domain must dimerize to become active, two ZFN subunits are typically assembled at the cleavage site. The generation of obligate heterodimeric ZFNs was shown to significantly reduce ZFN-associated cytotoxicity in single-site genome editing strategies. To further expand the application range of ZFNs, we employed a combination of in silico protein modeling, in vitro cleavage assays, and in vivo recombination assays to identify autonomous ZFN pairs that lack cross-reactivity between each other. In the context of ZFNs designed to recognize two adjacent sites in the human HOXB13 locus, we demonstrate that two autonomous ZFN pairs can be directed simultaneously to two different sites to induce a chromosomal deletion in ∼10% of alleles. Notably, the autonomous ZFN pair induced a targeted chromosomal deletion with the same efficacy as previously published obligate heterodimeric ZFNs but with significantly less toxicity. These results demonstrate that autonomous ZFNs will prove useful in targeted genome engineering approaches wherever an application requires the expression of two distinct ZFN pairs

Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler's third law: a numerical test

We test numerically the recently proposed linear relationship between the scale-invariant period Ts.i. = T|E| 3/2, and the topology of an orbit, on several hundred planar Newtonian periodic three-body orbits. Here T is the period of an orbit, E is its energy, so that Ts.i. is the scale-invariant period, or, equivalently, the period at unit energy |E| = 1. All of these orbits have vanishing angular momentum and pass through a linear, equidistant configuration at least once. Such orbits are classified in ten algebraically well-defined sequences. Orbits in each sequence follow an approximate linear dependence of Ts.i., albeit with slightly different slopes and intercepts. The orbit with the shortest period in its sequence is called the ‘progenitor’: six distinct orbits are the progenitors of these ten sequences. We have studied linear stability of these orbits, with the result that 21 orbits are linearly stable, which includes all of the progenitors. This is consistent with the Birkhoff–Lewis theorem, which implies existence of infinitely many periodic orbits for each stable progenitor, and in this way explains the existence and ensures infinite extension of each sequence