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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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.

History

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
• 70B: Kinematics, see also 53A17
• 70C20: Statics
• 70E: Dynamics of a rigid body and of multibody systems
• 70F: Dynamics of a system of particles, including celestial mechanics
• 70G: General representations of dynamical systems, see also 58FXX
• 70H: Hamiltonian and Lagrangian mechanics, see also 58F05
• 70J: Linear vibration theory
• 70K: Nonlinear motions, see also 34CXX, 37-XX
• 70L05: Random vibrations, See also 74H50
• 70M20: Orbital mechanics
• 70P05: Variable mass, rockets
• 70Q05: Control of mechanical systems, See also 49-XX, 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

Software and tables

Other web sites with this focus

• Mathematical Physics Preprint Archive at the University of Texas, Austin
• Here are the AMS and Goettingen resource pages for area 70.
• The American Physical Society
• Mathematical Physics sites. (AIP, etc).
• The Net Advance of Physics

Selected topics at this site

• Computing possible motions of a robot arm.
• Location of the sun in the sky as seen from a point on earth
• Describe the motion of a Stewart platform (a triangle suspended by 6 pistons of changeable length joining the vertices to three stationary points on the ground.)
• Actually using a Stewart platform -- control system
• Details of control of a Stewart platform (parallel manipulator).

Really most of the files here are topics saved for the benefit of undergraduate students looking for example of mathematical modelling of physical phenomena.

• What really happens when you flip a coin -- couldn't it land on its edge?
• Is angular momentum really what keeps a bicycle up?
• Model the fall of an elevator.
• Modeling swinging cables.
• What's the real path of a billiard ball?
• What path do flying objects really follow? (artillery and fungus spores!)
• Shape of a rotating chain with top end fixed
• Can a square wheel roll smoothly (on the appropriate surface)?
• Sample numerical calculations showing non-unique positions possible for Stewart platform
• Acceleration is not a contravariant tensor