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 .
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.
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
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 :
If one understands the problem as three-dimensional, one also has to consider the constraint of the plane pendulum:
All other quantities of the movement such as speed or acceleration can also be expressed as a function of the generalized coordinate .
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 .
Individual evidence
- ^ Friedhelm Kuypers: Classical mechanics . 5th edition. VCH, 1997, ISBN 3-527-29269-1
- ^ Herbert Goldstein: Classical Mechanics . 2nd Edition. Addison-Wesley, 1980, ISBN 0-201-02918-9 (English).