Galileo

Interface Galileo in app designer facilitates presentation of the mathematics of celestial mechanics.

Calculus of Variations

Two types of variations

A short path is subject to transverse variation. The location of the disturbance is set by the blue bell curve to be between the endpoints of the action integral shown in white. The kinetic energy of the mass is shown in yellow. The potential energy is in red. The Lagrangian and its auto scale is in green. The auto scale Lagrangian will always be

visible no matter how small the disturbance is.

The variation is negated. This is most obvious on the path itself, and can be seen also in the changed Lagrangian_n graphics,and numerically in the tables.

Axial variation alone is imposed.The kinetic energy during the disturbance is different from that due to transverse disturbance. The action after disturbance is increased by .000355 J sec

The disturbance is negated, leaving the varied action unchanged. This shows the the path is stationary, if it is true for smaller variations as well.

The x coordinates of the mass during an axial variation.

The axial variation is negated, the change in action remains .000355 J sec.

Newton's Impulsive Forces

Newton considered the continuous force on a planet as a series of impulses. The actual motion of the planet is the composition of the inertial motion of the body with that caused by the impulsive force, which motions can be added as infinitesimals using the parallelogram rule. The calculator has several graphic screens which display graphically these differential constructions.

YT and WT (yellow triangle and white triangle) show the conservation of area, and therefore angular momentum, between impulsive central forces.

r

r