54 research outputs found
Nonholonomic Constraints and Voronec's Equations
Is it allowed, in the context of the Lagrange multiplier formalism, to assume
that nonholonomic constraints are already in effect while setting up Lagrange's
function? This procedure is successfully applied in a recent book [L. N. Hand
and J. D Finch, {\it Analytical Mechanics}] to the problem of the rolling
penny, but it does not work in general, as we show by means of a
counterexample. It turns out that in many cases the use of nonholonomic
constraints in the process of construction of the Lagrangian is allowed, but
the correct equations of motion are the little known Voronec's equations.Comment: Translation of the paper "Vinculos Nao-Holonomos e Equacoes de
Voronec", to be published in Portuguese in Revista Brasileira de Ensino de
Fisic
Radiation-Dominated Quantum Friedmann Models
Radiation-filled Friedmann-Robertson-Walker universes are quantized according
to the Arnowitt-Deser-Misner formalism in the conformal-time gauge. Unlike
previous treatments of this problem, here both closed and open models are
studied, only square-integrable wave functions are allowed, and the boundary
conditions to ensure self-adjointness of the Hamiltonian operator are
consistent with the space of admissible wave functions. It turns out that the
tunneling boundary condition on the universal wave function is in conflict with
self-adjointness of the Hamiltonian. The evolution of wave packets obeying
different boundary conditions is studied and it is generally proven that all
models are nonsingular. Given an initial condition on the probability density
under which the classical regime prevails, it is found that a closed universe
is certain to have an infinite radius, a density parameter
becoming a prediction of the theory. Quantum stationary geometries are shown to
exist for the closed universe model, but oscillating coherent states are
forbidden by the boundary conditions that enforce self-adjointness of the
Hamiltonian operator.Comment: 18 pages, LaTex, to appear in J. Math. Phy
Failure of intuition in elementary rigid body dynamics
Suppose a projectile collides perpendicularly with a stationary rigid rod on
a smooth horizontal table. We show that, contrary to what one naturally
expects, it is not always the case that the rod acquires maximum angular
velocity when struck at an extremity. The treatment is intended for first year
university students of Physics or Engineering, and could form the basis of a
tutorial discussion of conservation laws in rigid body dynamics.Comment: Four pages; to appear in European Journal of Physic
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