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A mechanical degree of freedom is an independent coordinate needed to specify a system's configuration locally. A point particle moving freely in three-dimensional space has three such coordinates. If it is constrained to a sphere of fixed radius, two angles suffice; a simple pendulum confined to a vertical plane requires only its angle.

Counting coordinates requires identifying independent mechanical constraints. For $N$ particles subject to $k$ independent regular holonomic constraints, the configuration has $3N-k$ degrees of freedom locally. Repeating a dependent condition does not remove another coordinate. A second-order equation of motion usually also requires an initial velocity for each coordinate, so the dimension of the corresponding phase space is twice the number of configuration degrees of freedom. Nonholonomic constraints require a separate analysis of allowed velocities.

Particle dynamics predicts the motion of bodies whose internal size and rotation can be neglected for the question being studied. A point particle has a position and mass; applied forces determine its acceleration through Newton's second law. For fixed mass in an inertial frame, the basic problem iswith an initial position and velocity.

A ball's projectile motion can be approximated this way when its size is small compared with its path and spin has negligible influence. Air drag introduces a velocity-dependent force; contact with a surface may introduce mechanical constraints. The same method extends to interacting particles, where momentum conservation and conservation of energy can simplify the coupled equations of motion.

A particle is an entity described as a localized constituent in an appropriate physical model. In classical mechanics, a point particle has a position and trajectory while its internal size is neglected. This approximation can describe an extended object when the omitted structure does not affect the question being studied.

In quantum mechanics, a quantum particle has a state described by a wave function or a more general state description, with probabilistic measurement outcomes. Quantum field theory describes particle states as excitations of fields. A proton can be treated as a particle in one experiment while its quark and gluon structure matters in another. The particle description therefore specifies what is being resolved, rather than asserting that every particle is a structureless classical point.

A point particle is an idealization whose position is represented by a single spatial point and whose internal size and structure are omitted. It can still carry parameters such as mass, electric charge or spin.

The approximation is useful when the relevant length scales are much larger than the object's size and internal deformation is unimportant. A planet can be treated as a point mass for some orbital calculations even though it is extended. Short-distance fields and self-interaction can expose the limits of the idealization.