Generalized coordinates

Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian mechanics. The name is a holdover from a period when Cartesian coordinates were the standard system. A system with $n$ degrees of freedom can be fully described by the generalized coordinates

$\lbrace \mathbf {q_{1}} ,\mathbf {q_{2}} ,...,\mathbf {q_{n}} \rbrace$

The system's state may be fully described by such a set of generalized coordinates iff all

$n$ $\lbrace \mathbf {q_{i}} \rbrace$

are independent coordinates. This affords great flexibility in dealing with complex systems in the most convenient (not necessarily inertial) coordinates.

A system of $m$ particles may have up to $3m$ degrees of freedom, and therefore generalized coordinates - one for each dimension of motion of each particle, but will typically have many fewer. A system of $m$ rigid bodies may have up to $6m$ generalized coordinates, including 3 axes of rotation for each body.

Examples

A double-pendulum constrained to move in the plane of the page may be described by the four Cartesian coordinates $\lbrace x_{1},y_{1},x_{2},y_{2}\rbrace$, but the system only has two degrees of freedom, and a more efficient system would be to use

$\lbrace q_{1},q_{2}\rbrace =\lbrace \theta _{1},\theta _{2}\rbrace$, which are defined via the following relations:

$\lbrace x_{1},y_{1}\rbrace =\lbrace l_{1}\cos \theta _{1},l_{1}\sin \theta _{1}\rbrace$

$\lbrace x_{2},y_{2}\rbrace =\lbrace l_{2}\cos \theta _{2},l_{2}\sin \theta _{2}\rbrace$

A bead constrained to move on a wire has only one degree of freedom, and the generalized coordinate used to describe its motion is often:

$q_{1}=l$,

where $l$ is the distance along the wire from some reference point on the wire. Notice that a motion embedded in three dimensions has been reduced to only one dimension.

An object constrained to a surface has two degrees of freedom, even though its motion is again embedded in three dimensions. If the surface is a sphere, a good choice of coordinates would be:

$\lbrace q_{1},q_{2}\rbrace =\lbrace \theta ,\phi \rbrace$,

where $\theta$ and $\phi$ are the angle coordinates familiar from spherical coordinates. The $r$ coordinate has been effectively dropped, as a particle moving on an origin-centered sphere maintains a constant radius.

Generalized velocities and kinetic energy

Each generalized coordinate $q_{i}$ is associated with a generalized velocity ${\dot {q}}_{i}$, defined as:

${\dot {q}}_{i}={dq_{i} \over dt}$

The kinetic energy of a particle is

$T={\frac {m}{2}}\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)$.

In more general terms, for a system of $p$ particles with $n$ degrees of freedom, this may be written

$T=\sum _{i=1}^{p}{\frac {m_{i}}{2}}\left({\dot {x}}_{i}^{2}+{\dot {y}}_{i}^{2}+{\dot {z}}_{i}^{2}\right)$.

If the transformation equations between the Cartesian and generalized coordinates

$x_{i}=x_{i}\left(q_{1},q_{2},...,q_{n},t\right)$

$y_{i}=y_{i}\left(q_{1},q_{2},...,q_{n},t\right)$

$z_{i}=z_{i}\left(q_{1},q_{2},...,q_{n},t\right)$

are known, then these equations may be differentiated to provide the time-derivatives to use in the above kinetic energy equation:

${\dot {x}}_{i}={\frac {d}{dt}}x_{i}\left(q_{1},q_{2},...,q_{n},t\right)$

.

It is important to remember that the kinetic energy must be measured relative to inertial coordinates. If the above method is used, it means only that the Cartesian coordinates need to be inertial, even though the generalized coordinates need not be. This is another considerable convenience of the use of generalized coordinates.

Applications of generalized coordinates

Such coordinates are helpful principally in Lagrangian Dynamics, where the forms of the principal equations describing the motion of the system are unchanged by a shift to generalized coordinates from any other coordinate system.

The amount of virtual work done along any coordinate $q_{i}$ is given by:

$\delta \ W_{q_{i}}=F_{q_{i}}\cdot \delta \ q_{i}$,

where $F_{q_{i}}$ is the generalized force in the $q_{i}$ direction. While the generalized force is difficult to construct 'a priori', it may be quickly derived by determining the amount of work that would be done by all non-constraint forces if the system underwent a virtual displacement off $\delta \ q_{i}$, with all other generalized coordinates and time held fixed. This will take the form:

$\delta \ W_{q_{i}}=f\left(q_{1},q_{2},...,q_{n}\right)\cdot \delta \ q_{i}$,

and the generalized force may then be calculated:

$F_{q_{i}}={\frac {\delta \ W_{q_{i}}}{\delta \ q_{i}}}=f\left(q_{1},q_{2},...,q_{n}\right)$,

See also

• Lagrangian mechanics
• Degrees of freedom (physics and chemistry)
• Virtual work

References

• Wells, D.A. Schaum's Outline of Lagrangian Dynamics. McGraw-Hill, Inc. New York, 1967.