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Encyclopedia > Generalized coordinates

Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. 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 Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ...

$lbracemathbf{q_1},mathbf{q_2}, ..., mathbf{q_n} rbrace$

The system's state may be fully described by such a set of generalized coordinates iff all â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

n

$lbracemathbf{{q_i}}rbrace$

are independent coordinates. This affords great flexibility in dealing with complex systems in the most convenient (not necessarily inertial) coordinates. In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ...

A system of

m

particles may have up to

3 m

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

6 m

generalized coordinates, including 3 axes of rotation for each body.

1 Examples
2 Generalized velocities and kinetic energy
3 Applications of generalized coordinates
4 See also
5 References

Examples

A double-pendulum constrained to move in the plane of the page may be described by the four Cartesian coordinates ${x 1, y 1, x 2, y 2}$ , but the system only has two degrees of freedom, and a more efficient system would be to use A double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ...

{q 1, q 2} = {θ 1,θ 2}

, which are defined via the following relations:

{x 1, y 1} = {l 1 cosθ 1, l 1 sinθ 1}

{x 2, y 2} = {l 1 cosθ 1, l 1 sinθ 1}

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:

{q 1, q 2} = {θ,φ}

where $θ$ and $φ$ are the angle coordiates familiar from spherical coordinates. The $r$ coordinate has been effectively dropped, as a particle moving on an origin-centered sphere maintains a constant radius.
This article describes some of the common coordinate systems that appear in elementary mathematics. ...

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. In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ...

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. Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ...

The amount of virtual work done along any coordinate $q i$ is given by: Virtual work describes a variational approach to solving physics problems. ...

$delta W_{q_i} = F_{q_i} cdot delta q_i$ ,

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: The concept of a virtual displacement is meaningful only when discussing a physical system subject to contraints on its motion. ...

$delta W_{q_i} = f left ( q_1, q_2, ..., q_n right ) cdot delta q_i$ ,

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

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

Categories: Lagrangian mechanics | Dynamical systems

Results from FactBites:

Modelica (2215 words)
	Via the dynamic dummy derivative method the generalized coordinates on position and velocity level from one of the 7 joints are dynamically selected as states during simulation.
	For this mechanism, the generalized coordinates of joint j1 (i.e., the rotation angle of the revolute joint and its derivative) can always be used as states.
	When the generalized coordinates of revolute joint "innerJoint" (lower left part in figure) are used as states, then frame_a and frame_b of the jointUPS joint can be calculated.

Talk:Lagrangian mechanics - Wikipedia, the free encyclopedia (430 words)
	So the generalized reminds you that your coordinates may need to be a totally non-traditional system, like the length along a wire bent into a weird shape (perhaps a bead is moving along the wire).
	The link generalized coordinates links to this page (Lagrange mechanics), I believe it would be nice to have either a larger discussion of generalized coordinates on this page or perhaps its own article.
	Generalized coordinates now have their own article, which is not 100% yet, but is a good start.

More results at FactBites »

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Generalized velocities and kinetic energy

Applications of generalized coordinates

See also

References

Examples