Generalized coordinates
Encyclopedia
In the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration
of a system
relative to some reference configuration.
A restriction for a set of coordinates to serve as generalized coordinates is that they should uniquely define any possible configuration of the system relative to the reference configuration. Frequently the generalized coordinates are chosen to be independent of one another. The number of independent generalized coordinates is defined by the number of degrees of freedom of the system. The adjective "generalized" is a holdover from a period when Cartesian coordinates were the standard.
Apart from practical reasons, any set of generalized coordinates is as good as another. The physics of the system is independent of the choice. However, there are more and less practical choices, that is, coordinates that are more or less optimally adapted to the system and make the solution of its equations of motion easier or more difficult.
The generalized velocities are the time derivative
s of the generalized coordinates of the system.
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:
where qn 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 qn, 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.
A system with
degrees of freedom
and n particles whose positions are designated with three dimensional vectors,
, implies the existence of 
scalar constraint equations on those position variables. Such a system can be fully described by the scalar generalized coordinates, 
, and the time, 
, if and only if all 
are independent coordinates. For the system, the transformation from old coordinates to generalized coordinates may be represented as follows:
,
, ...
.
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.
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
,
which are defined via the following relations:
,
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.
,
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.
is associated with a generalized velocity 
, defined as:
The kinetic energy of a particle is
.
In more general terms, for a system of
particles with 
degrees of freedom, this may be written
.
If the transformation equations between the Cartesian and generalized coordinates
are known, then these equations may be differentiated to provide the time-derivatives to use in the above kinetic energy equation:
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.
, 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
is given by:
,
where
is the generalized force in the 
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 
, with all other generalized coordinates and time held fixed. This will take the form:
,
and the generalized force may then be calculated:
.
Configuration space
- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
of a system
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
relative to some reference configuration.
A restriction for a set of coordinates to serve as generalized coordinates is that they should uniquely define any possible configuration of the system relative to the reference configuration. Frequently the generalized coordinates are chosen to be independent of one another. The number of independent generalized coordinates is defined by the number of degrees of freedom of the system. The adjective "generalized" is a holdover from a period when Cartesian coordinates were the standard.
Apart from practical reasons, any set of generalized coordinates is as good as another. The physics of the system is independent of the choice. However, there are more and less practical choices, that is, coordinates that are more or less optimally adapted to the system and make the solution of its equations of motion easier or more difficult.
The generalized velocities are the time derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s of the generalized coordinates of the system.
Constraint equations
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 constraintsConstraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality 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:
where qn 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 qn, and thereby exclude certain configurations of the system.
It can be advantageous to choose independent generalized coordinates, as is done in Lagrangian mechanics
Lagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
, 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
Nonholonomic system
A nonholonomic system in physics and mathematics is a system whose state depends on the path taken to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the...
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
degrees of freedomDegrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
and n particles whose positions are designated with three dimensional vectors,
, implies the existence of scalar constraint equations on those position variables. Such a system can be fully described by the scalar generalized coordinates, , and the time, , if and only if all are independent coordinates. For the system, the transformation from old coordinates to generalized coordinates may be represented as follows:,, ....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.
Double pendulum
Double pendulum
In mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a set of...
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
,which are defined via the following relations:
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,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:,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 is associated with a generalized velocity , defined as:The kinetic energy of a particle is
.In more general terms, for a system of
particles with degrees of freedom, this may be written.If the transformation equations between the Cartesian and generalized coordinates
are known, then these equations may be differentiated to provide the time-derivatives to use in the above kinetic energy equation:
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 mechanicsLagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
, 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
Virtual work
Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the...
done along any coordinate
is given by:, where
is the generalized force in the 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 , with all other generalized coordinates and time held fixed. This will take the form:,and the generalized force may then be calculated:
.See also
- Hamiltonian mechanicsHamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
- Virtual workVirtual workVirtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the...
- Orthogonal coordinatesOrthogonal coordinatesIn mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...
- Curvilinear coordinatesCurvilinear coordinatesCurvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...
- Frenet-Serret formulasFrenet-Serret formulasIn vector calculus, the Frenet–Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R3...
- Mass matrixMass matrixIn computational mechanics, a mass matrix is a generalization of the concept of mass to generalized coordinates. For example, consider a two-body particle system in one dimension...
- Stiffness matrix