Generalized coordinate

The generalized (or generalized ) coordinates form in theoretical mechanics and technical mechanics a minimal set of independent coordinates for the unambiguous description of the spatial state of the system under consideration . They are chosen so that the mathematical formulation of movements that are subject to constraints is as simple as possible. Generalized coordinates often have the formula symbol as variables . ${\displaystyle q}$

For example, in the case of a mathematical pendulum, instead of the x and z coordinates of the mass point , it is sufficient to specify the deflection angle to clearly describe the position. The constant rope length is given by the binding equation. ${\displaystyle l={\text{const}}}$

The minimum number of generalized coordinates that are required to describe a system corresponds to the number of its degrees of freedom . The generalized coordinates span the configuration space . Important examples are the action angle coordinates , the Jacobi coordinates and the (all cyclic ) coordinates of the Hamilton-Jacobi formalism .

example

Thread pendulum: is the deflection from the equilibrium position and generalized coordinate
${\displaystyle \varphi }$

The mass of the plane mathematical pendulum in the xy-plane can only move on a circular path with a constant rope length (scleronomic-holonomic constraint) , the angle is the only degree of freedom of movement. The position of the pendulum mass can thus be clearly described by the generalized coordinate : ${\displaystyle l}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi }$

${\displaystyle {\vec {r}}_{\text{2D}}=l\,{\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}}$

If one understands the problem as three-dimensional, one also has to consider the constraint of  the plane pendulum: ${\displaystyle z=0}$

${\displaystyle {\vec {r}}_{\text{3D}}=l\,{\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}}}$

All other quantities of the movement such as speed or acceleration can also be expressed as a function of the generalized coordinate . ${\displaystyle \varphi }$

The equations of motion can always be solved for the second time derivative of the generalized coordinates; in the example, a differential equation of the second order is obtained for the angle . ${\displaystyle \varphi }$

Individual evidence

1. ^ Friedhelm Kuypers: Classical mechanics . 5th edition. VCH, 1997, ISBN 3-527-29269-1
2. ^ Herbert Goldstein: Classical Mechanics . 2nd Edition. Addison-Wesley, 1980, ISBN 0-201-02918-9 (English).