Newton's Second Law in A Multidimensional Space
Let's consider particles with masses in the regular three-dimensional Euclidean space. Let be their radius-vectors in some inertial coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them
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(1)
The three-dimensional radius-vectors can be built into a single -dimensional radius-vector. Similarly, three-dimensional velocity vectors can be built into a single -dimensional velocity vector:
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(2)
In terms of the multidimensional vectors (2) the equations (1) are written as
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(3)
i. e they take the form of Newton's second law applied to a single particle with the unit mass .
Definition. The equations (3) are called the equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector . The space whose points are marked by the pair of vectors is called the phase space of the dynamical system (3).
Read more about this topic: Newtonian Dynamics
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