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# 70: Mechanics of particles and systems

## Introduction

Mechanics of particles and systems: studies dynamics of sets of particles or solid bodies, including rotating and vibrating bodies. Uses variational principles (energy-minimization) as well as differential equations.

Note that mathematically speaking, classical problems in celestial mechanics belong in this section, since bodies in space are treated as (very big!) particles.

Orbits

## Applications and related fields

For relativistic mechanics, See 83-XX 83A05 and 83C10; for statistical mechanics, See 82-XX

Clearly the study of the dynamics of smoothly moving systems requires the use of (systems of) differential equations.

Among the other branches of mathematics useful for these investigations, we mention geometric topology, which can provide a language for describing sets of movements, and commutative algebra, which can allow algebraic calculations of constraints on complex moving systems.

Some comments should be made about mathematical physics and engineering (subject headings 70-86) generally, but they don't easily fit the areas of the MSC. For now, they'll go here. For example, one should note that "field theory" used in these sections has nothing to do with "field theory" in abstract algebra.

## Subfields

• 70A05: Axiomatics, foundations
• 70C20: Statics
• 70D: Dynamics of a particle, see also 70HXX [dropped in MSC2000]
• 70E: Dynamics of a rigid body and of multibody systems
• 70F: Dynamics of a system of particles, including celestial mechanics
• 70M20: Orbital mechanics
• 70P05: Variable mass, rockets
• 70Q05: Control of mechanical systems, See also 49-XX, 73K50, 93CXX, 93DXX
• 70S: Classical field theories [See also 37Kxx, 37Lxx, 78-XX, 81Txx, 83-XX] [new in 2000]

Until the late 1970s the AMS used a major heading 69, General Applied Mathematics. Many of the reviews in that area could arguably be reclassed now into heading 70, Mechanics.

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Szebehely, Victor G.: "Adventures in celestial mechanics. A first course in the theory of orbits". University of Texas Press, Austin, TX, 1989. 175 pp. ISBN 0-292-75105-2 MR90f:70017

Jeffreys, Harold: "What is Hamilton's principle?" Quart. J. Mech. Appl. Math. 7, (1954). 335--337. MR16,533b