Nathaniel Grossman

Force inversion in floating plate dynamics

An axisymmetric floating plate of infinite extent is subjected to prescribed dynamics under the action of a point load. The problem of inverting the forcing is studied mathematically with a view to classifying its properties within the framework of integral equations. In the process, the floating plate problem is shown to be moderately ill-posed and the rate of change of the forcing is shown to initially be directly proportional to the acceleration of the plate. This latter result is incorporated into the numerical study that follows. Throughout the paper, a model problem is used as an analytical and numerical benchmark.

A C ¿ Lagrange Inversion Theorem

Naturally there are questions of existence and convergence associated with (2). We will discuss these in due course. The purpose of this note is to present a short and transparent derivation of (2) based on the Laplace transform under hypotheses less stringent than are usually imposed. Krantz and Parks [6] present a history of Lagranges inversion theorem. Several different derivations of (2) can be found in the modem literature:

Expansions for an Elliptic Orbit

We have studied motion in an elliptic orbit under the inverse square law of attraction to a center. Among the formulas we derived in §111.3 and (196) were these:

Gravitation and Closed Orbits

The solar system appears to be stable: Calculations into the future using Newton’s Law of Universal Gravitation show that the Sun and the planets will behave in the future much as they have done in the past. The Earth will move in its yearly, very nearly elliptical, Keplerian orbit around the Sun, as will the other planets in their years. No planet will suddenly take off for deep space or fall into the Sun, nor will two of the (major) planets collide.

Gravitational Properties of Solids

The potential energy at O of a particle of mass m at P is −Gm/r. The potential produced at O by a solid body with density function ρ is

Orbits under the Inverse Square Law

Kepler formulated his famous three laws of planetary motion during the first decades of the seventeenth century. It was an arduous task, proceeding strictly from observed data, the most accurate of which were contributed by Kepler’s mentor, Tycho Brahe.

Dynamical Properties of Rigid Bodies

Fix a set of cartesian axes in space. Suppose a finite system of particles is distributed in space, the ith particle having mass m i and position r i . The center of mass of the system is defined to be the point \(\bar{r}\), where

On Young's Inequality

\bar{r}=\frac{\Sigma m_i r_i}{\Sigma m_i}