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Generalized coordinates

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Wikipedia

By deriving equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any coordinate system that is ultimately specified." The name is a holdover from a period when Cartesian coordinates were the standard system.

Independent generalized coordinates

For any particular problem, it is advantageous to choose generalized coordinates such that they are independent, as is done in Lagrangian mechanics, because this eliminates the variables that would be required to express constraints on and among the coordinates. However, when dealing with nonholonomic constraints or when trying to find the force due to any constraint—holonomic or not, dependent generalized coordinates must be employed. Sometimes independent generalized coordinates are called internal coordinates because they are mutually independent, otherwise unconstrained, and together give the position of the system.

A system with $m$ degrees of freedom and n particles whose positions are designated with three dimensional vectors, $lbrace mathbf {r}_i rbrace$ , implies the existence of $3 n-m$ scalar constraint equations on those position variables. Such a system can be fully described by the scalar generalized coordinates, $lbrace q_1, q_2, ..., q_mrbrace$ , and the time, $t$ , iff all $m$ $lbrace q_j rbrace$ are independent coordinates. For the system, the transformation from old coordinates to generalized coordinates may be represented as follows:

mathbf{r}_1=mathbf{r}_1(q_1, q_2, ..., q_m, t)

mathbf{r}_2=mathbf{r}_2(q_1, q_2, ..., q_m, t)

, ...

mathbf{r}_n=mathbf{r}_n(q_1, q_2, ..., q_m, t)

This transformation affords the flexibility in dealing with complex systems to use the most convenient and not necessarily inertial coordinates. These equations are used to construct differentials when considering virtual displacements and generalized forces.

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_2rbrace

, but the system only has two degrees of freedom, and a more efficient system would be to use

lbrace q_1,  q_2 rbrace = lbracetheta_1,theta_2 rbrace

, which are defined via the following relations:

lbrace x_1, y_1 rbrace = lbrace l_1sintheta_1,   l_1costheta_1 rbrace

lbrace x_2, y_2 rbrace = lbrace l_1sintheta_1+l_2sintheta_2,   l_1costheta_1+l_2costheta_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 a 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 of $delta q_i$ , 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 )

References

Greenwood, Donald T. (1987). Principles of Dynamics. 2nd edition, Prentice Hall. ISBN 0-13-709981-9.
Wells, D. A. (1967). Schaum's Outline of Lagrangian Dynamics. New York: McGraw-Hill.