Lagrangian multipliers for coast-arcs of optimum space trajectories

Some properties of generalized canonical systems - special dynamical systems described by a Hamiltonian function linear in the adjoint variables - are applied in determining the solution of the two-dimensional coast-arc problem in an inverse-square gravity field. A complete closed-form solution for Lagrangian multipliers - adjoint variables - is obtained by means of such properties for elliptic, circular, parabolic and hyperbolic motions. Classic orbital elements are taken as constants of integration of this solution in the case of elliptic, parabolic and hyperbolic motions. For circular motion, a set of nonsingular orbital elements is introduced as constants of integration in order to eliminate the singularity of the solution.

Morphology and anatomy in Heliconia angusta Vell. and H. velloziana L. Emygd. (Zingiberales: Heliconiaceae) from the Atlantic forest of southeastern Brazil

Morphological and anatomical features of roots, stems, leaves, and scapes were studied in Heliconia angusta and H. velloziana from the Atlantic forest in the southeastern of Brazil. Morphologically H. angusta and H. velloziana show differences in their sizes, blade shapes, number and shape of inflorescence bracts. On the other hand, they have common anatomical characteristics such as: roots with air-canals in the cortex; rhizomes with isolated fiber bundles, collateral vascular bundles, and uniseriate endodermis and pericycle; leaves presenting air-canals and collateral vascular bundles forming arcs, and thin-walled epidermal cells; scapes with collateral vascular and fiber bundles in the cortex. The distribution of the fiber bundles in the leaves and in the scapes was different for each species, having a taxonomical value, H. velloziana presenting continuous fiber bundles. Air-canals in roots and leaves with narrow mesophyll might be related to the moist understorey of the Atlantic forest habitats.