Generalized coordinates

The generalized coordinates are a term used in engineering mechanics. You are a reduced set of independent coordinates for unambiguous description of the location of points or bodies in space. They are chosen so that the mathematical formulation of movements under constraints is as simple as possible. For mathematical pendulums, such as the indication of the deflection angle to describe the circular motion is sufficient. The constant cable length is given by the binding equation.

Generally matches the number of degrees of freedom of a system with the number of the minimum required for the description of the generalized coordinates. The generalized coordinates describing the configuration space.

Example

The mass of the planar mathematical pendulum in the xy plane can move at constant rope length ( skleronom holonomic constraint ) only on a circular path. The angle is the only degree of freedom of movement. The position of the pendulum mass can thus be uniquely described by the generalized coordinate. Summing up the problem as a three-dimensionally on so the main constraint of the planar pendulum has additionally to consider z = 0:

You might as well understand the problem of the planar mathematical pendulum as a two-dimensional and omit the z- coordinate:

All other sizes of movement such as velocity or acceleration can also be expressed as a function of the generalized coordinate.

The equations of motion can be resolved are always after the second derivatives of the generalized coordinates. In the example, we obtain a second order differential equation for the angle.