User:Prof McCarthy/generalized coordinates

In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration.

An example of a generalized coordinate is the angle that locates a point moving on a circle, in contrast to its x and y coordinates. The adjective "generalized" distinguishes these parameters from the traditional use of the term coordinate as measured against a specified line, such as Cartesian coordinates .

Parameters that are convenient for the specification of the configuration of a system are selected to be generalized coordinates. If these parameters are independent of one another, then number of independent generalized coordinates is defined by the number of degrees of freedom of the system.

The generalized velocities are the time derivatives of the generalized coordinates of the system.

Constraint equations

File:Generalized coordinates 1df.svg

Generalized coordinates for one degree of freedom (of a particle moving in a complicated path). Instead of using all three Cartesian coordinates x, y, z (or other standard coordinate systems), only one is needed and is completely arbitrary to define the position. Four possibilities are shown. Top: distances along some fixed line, bottom left: an angle relative to some baseline, bottom right: the arc length of the path the particle takes. All are defined relative to a zero position - again arbitrarily defined.

Generalized coordinates may be independent (or unconstrained), in which case they are equal in number to the degrees of freedom of the system, or they may be dependent (or constrained), related by constraints on and among the coordinates. The number of dependent coordinates is the sum of the number of degrees of freedom and the number of constraints. For example, the constraints might take the form of a set of configuration constraint equations:

$f_{i}(\{q_{n}\},\ t)=0\ ,$

where q_n is the n-th generalized coordinate and i denotes one of a set of constraint equations, taken here to vary with time t. The constraint equations limit the values available to the set of q_n, and thereby exclude certain configurations of the system.

It can be advantageous to choose independent generalized coordinates, as is done in Lagrangian mechanics, because this eliminates the need for constraint equations. However, in some situations, it is not possible to identify an unconstrained set. For example, 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.

Top: one degree of freedom, bottom: two degrees of freedom, left: an open curve F (parameterized by t) and surface F, right: a closed curve C and closed surface S. The equations shown are the constraint equations. Generalized coordinates are chosen and defined with respect to these curves (one per degree of freedom), and simplify the analysis since even complicated curves are described by the minimum number of coordinates required.

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 $3n-m$ scalar constraint equations on those position variables. Such a system can be fully described by the scalar generalized coordinates, $\lbrace q_{1},q_{2},...,q_{m}\rbrace$, and the time, $t$, if and only if 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

Double pendulum

A double pendulum

A double pendulum constrained to move in a plane may be described by the four Cartesian coordinates {x₁, y₁, x₂, y₂}, 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:

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

$\lbrace x_{2},y_{2}\rbrace =\lbrace L_{1}\sin \theta _{1}+L_{2}\sin \theta _{2},L_{1}\cos \theta _{1}+L_{2}\cos \theta _{2}\rbrace$

Bead on a wire

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.

Motion on a surface

A point mass constrained to a surface has two degrees of freedom, even though its motion is 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 θ and φ 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 mechanics, 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:

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

See also

• Hamiltonian mechanics
• Virtual work
• Orthogonal coordinates
• Curvilinear coordinates
• Frenet-Serret formulas
• Mass matrix
• Stiffness matrix
• Generalized forces

References

• Greenwood, Donald T. (1987). Principles of Dynamics (2nd edition ed.). Prentice Hall. ISBN 0-13-709981-9. {{cite book}}: |edition= has extra text (help)
• Wells, D. A. (1967). Schaum's Outline of Lagrangian Dynamics. New York: McGraw-Hill.