Configuration space

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.

The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector r=(x, y, z), and therefore its configuration space is R³. If the particle is constrained to lie on a sphere, then its configuration space is the subset of coordinates in R³ that define points on the sphere S².

For n particles the configuration space is R³ⁿ, or possibly the subspace where no two positions are equal.

An important problem in physics considers the set of all trajectories of a particle between two points, which is a configuration space that is also known as a function space M. In quantum mechanics one formulation uses histories, or trajectories, as configurations.

The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted ${\displaystyle \mathbb {R} ^{3}\times \mathrm {SO} (3)}$ where ${\displaystyle \mathbb {R} ^{3}}$ represents the coordinates of the origin of the frame attached to the body, and ${\displaystyle \mathrm {SO} (3)}$ represents the rotation matrices that define the orientation of this frame relative to a ground frame. A configuration of the rigid body is defined by six parameters, three from ${\displaystyle \mathbb {R} ^{3}}$ and three from ${\displaystyle \mathrm {SO} (3)}$, and is said to have six degrees of freedom.

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