- #1

- 10,123

- 137

**Classical Systems With Variable Mass And Other Geometric Systems**

1. INTRODUCTION

1a) The material particle/system

The fundamental object in classical mechanics is the material particle.

A material particle has a position and a mass, and can be subject to forces.

A material system is a physical system consisting of the SAME set of material particles over time.

1b) Classical laws valid for the material particle:

MASS CONSERVATION

The classical material particle has the same mass throughout time, that is, the material particle is subject to the law of mass conservation.

Clearly, since a material system consists of the same material particles throughout time, the total mass of the material system can't change with time either.

NEWTON'S LAWS OF MOTION:

The material particle can be subject to forces, and its acceleration [tex]\vec{a}[/tex] is related to the forces [tex]\vec{F}[/tex] it is subject to through Newton's 2.law of motion:

[tex]\vec{F}=m\vec{a}[/tex]

Since the material particle is presumed to have constant mass, a classically equivalent form of Newton's 2.law is:

[tex]\vec{F}=\frac{d\vec{p}}{dt},\vec{p}=m\vec{v},\vec{a}=\frac{d\vec{v}}{dt}[/tex]

where [tex]\vec{v}[/tex] is the particle's velocity, and [tex]\vec{p}[/tex] the particle's momentum.

1c) Galilean invariance

Note in particular that since the value of the particle's mass is presumed independent of different observers' relations to each other (i.e, mass is seen as an "absolute", or invariant quantity), Newton's 2.law is a Galilean invariant in either of its formulations, since, by the law of mass conservation, [tex]\frac{dm}{dt}=0[/tex]

2. GEOMETRIC PHYSICAL SYSTEMS

2a) MOTIVATION

It is an unnecessarily restrictive physics which only wants to deal with the behaviour of strictly material systems, since there are lots of other systems which might be of interest to study.

For example, consider a fluid flowing through a bent tube section.

It might be convenient to try and find a way to calculate what force the fluid exerts upon the tube section we're looking at throughout time, but because it is not the same fluid elements which is in contact with/in the vicinity of the tube section throughout time (the fluid flows through it and beyond), the isolated system of tube section+fluid contained in it is NOT a material system!

Also, a rocket moves forward by expelling exhaust backwards, so if we're interested in describing the motion of the rocket as a function of time, we're effectively wanting to describe the behaviour of a system which is continually losing mass through fuel ejection.

That is, the rocket+remaining fuel-system is not a material system.

Such naturally occurring systems are called geometric systems, and our aim is to formulate the proper form of the laws of motion for such systems.

To be continued..

Last edited by a moderator: