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      Suppose we have a particle (or body) of mass m in space.  If we apply a force to the particle, we cause it to undergo motion (the particle accelerates) in the direction of the force.  Assuming we know the impressed force causing the acceleration, we can use Newton's Second Law of Motion    (F = ma) to write the equations of motion in the x, y, and z directions.  Suppose further that we specify certain conditions the particle's motion must satisfy throughout the duration of movement.  For instance, we could constrain the particle to move in a two-dimensional plane, say z = 2y.  This type of motion is called "constrained motion."  Constrained motion is very common in the real world; the planets are constrained to move in elliptical orbits about the sun and the bob of a clock pendulum is constrained by a metal rod to travel in a circular arc.

     Historically, many problems in dynamics have eluded the greatest mathematicians.  The three body problem, for example, went unsolved for centuries until Henri Poincaré produced a result in the late 19th Century.  

     What we hope to gain in our analysis of the motion of particles and bodies is a knowledge of the forces required to produce the specified constrained motion.  Knowledge of the constraint force, impressed force, and certain initial conditions (position and velocity) will allow us to completely describe the motion of the particle or body. 

Henri Poincaré (1854 - 1912)

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For information on Henri Poincaré and his work on the three body problem see Poincaré and the Three Body Problem by June Barrow-Green.  The book is the second volume in the History of Mathematics series by the American and London Mathematical Societies.