Generalized coordinates
Classical mechanics 


Core topics

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.^{1} The generalized velocities are the time derivatives of the generalized coordinates of the system.
An example of a generalized coordinate is the angle that locates a point moving on a circle. The adjective "generalized" distinguishes these parameters from the traditional use of the term coordinate to refer to Cartesian coordinates: for example, describing the location of the point on the circle using x and y coordinates.
Although there may be many choices for generalized coordinates for a physical system, parameters are usually selected which are convenient for the specification of the configuration of the system and which make the solution of its equations of motion easier. 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.^{2} ^{3}
Contents
Constraint equations
Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of Lagrange's equations of motion. However, it can also occur that a useful set of generalized coordinates may be dependent, which means that they are related by one or more constraint equations.
Holonomic constraints
If the constraints introduce relations between the generalized coordinates q_{i}, i=1,..., n and time, of the form,
they are called holonomic.^{1} These constraint equations define a manifold in the space of generalized coordinates q_{i}, i=1,...,n, known as the configuration manifold of the system. The degree of freedom of the system is d=nk, which is the number of generalized coordinates minus the number of constraints.^{4}^{:260}
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.
Nonholonomic constraints
A mechanical system can involve constraints on both the generalized coordinates and their derivatives. Constraints of this type are known as nonholonomic. Firstorder nonholonomic constraints have the form
An example of such a constraint is a rolling wheel or knifeedge that constrains the direction of the velocity vector. Nonholonomic constraints can also involve nextorder derivatives such as generalized accelerations.
Example: Simple pendulum
The relationship between the use of generalized coordinates and Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of a simple pendulum.^{5}^{6}
Coordinates
A simple pendulum consists of a mass M hanging from a pivot point so that it is constrained to move on a circle of radius L. The position of the mass is defined by the coordinate vector r=(x, y) measured in the plane of the circle such that y is in the vertical direction. The coordinates x and y are related by the equation of the circle
that constrains the movement of M. This equation also provides a constraint on the velocity components,
Now introduce the parameter θ, that defines the angular position of M from the vertical direction. It can be used to define the coordinates x and y, such that
The use of θ to define the configuration of this system avoids the constraint provided by the equation of the circle.
Virtual work
Notice that the force of gravity acting on the mass m is formulated in the usual Cartesian coordinates,
where g is the acceleration of gravity.
The virtual work of gravity on the mass m as it follows the trajectory r is given by
The variation δr can be computed in terms of the coordinates x and y, or in terms of the parameter θ,
Thus, the virtual work is given by
Notice that the coefficient of δy is the ycomponent of the applied force. In the same way, the coefficient of δθ is known as the generalized force along generalized coordinate θ, given by
Kinetic energy
To complete the analysis consider the kinetic energy T of the mass, using the velocity,
so,
Lagrange's equations
Lagrange's equations for the pendulum in terms of the coordinates x and y are given by,
This yields the three equations
in the three unknowns, x, y and λ.
Using the parameter θ, Lagrange's equations take the form
which becomes,
or
This formulation yields one equation because there is a single parameter and no constraint equation.
This shows that the parameter θ is a generalized coordinate that can be used in the same way as the Cartesian coordinates x and y to analyze the pendulum.
Example: Double pendulum
The benefits of generalized coordinates become apparent with the analysis of a double pendulum. For the two masses m_{i}, i=1, 2, let r_{i}=(x_{i}, y_{i}), i=1, 2 define their two trajectories. These vectors satisfy the two constraint equations,
The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates x_{i}, y_{i} i=1, 2 and the two Lagrange multipliers λ_{i}, i=1, 2 that arise from the two constraint equations.
Coordinates
Now introduce the generalized coordinates θ_{i} i=1,2 that define the angular position of each mass of the double pendulum from the vertical direction. In this case, we have
The force of gravity acting on the masses is given by,
where g is the acceleration of gravity. Therefore, the virtual work of gravity on the two masses as they follow the trajectories r_{i}, i=1,2 is given by
The variations δr_{i} i=1, 2 can be computed to be
Virtual work
Thus, the virtual work is given by
and the generalized forces are
Kinetic energy
Compute the kinetic energy of this system to be
Lagrange's equations
Lagrange's equations yield two equations in the unknown generalized coordinates θ_{i} i=1, 2, given by^{7}
and
The use of the generalized coordinates θ_{i} i=1, 2 provides an alternative to the Cartesian formulation of the dynamics of the double pendulum.
Generalized coordinates and virtual work
The principle of virtual work states that if a system is in static equilibrium, the virtual work of the applied forces is zero for all virtual movements of the system from this state, that is, δW=0 for any variation δr.^{4} When formulated in terms of generalized coordinates, this is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is F_{i}=0.
Let the forces on the system be F_{j}, j=1, ..., m be applied to points with Cartesian coordinates r_{j}, j=1,..., m, then the virtual work generated by a virtual displacement from the equilibrium position is given by
where δr_{j}, j=1, ..., m denote the virtual displacements of each point in the body.
Now assume that each δr_{j} depends on the generalized coordinates q_{i}, i=1, ..., n, then
and
The n terms
are the generalized forces acting on the system. Kane^{8} shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,
where v_{j} is the velocity of the point of application of the force F_{j}.
In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is
See also
 Hamiltonian mechanics
 Virtual work
 Orthogonal coordinates
 Curvilinear coordinates
 FrenetSerret formulas
 Mass matrix
 Stiffness matrix
 Generalized forces
References
 ^ ^{a} ^{b} Jerry H. Ginsberg (2008). "§7.2.1 Selection of generalized coordinates". Engineering dynamics, Volume 10 (3rd ed.). Cambridge University Press. p. 397. ISBN 0521883032.
 ^ Farid M. L. Amirouche (2006). "§2.4: Generalized coordinates". Fundamentals of multibody dynamics: theory and applications. Springer. p. 46. ISBN 0817642366.
 ^ Florian Scheck (2010). "§5.1 Manifolds of generalized coordinates". Mechanics: From Newton's Laws to Deterministic Chaos (5th ed.). Springer. p. 286. ISBN 3642053696.
 ^ ^{a} ^{b} Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0030633664.
 ^ Greenwood, Donald T. (1987). Principles of Dynamics (2nd edition ed.). Prentice Hall. ISBN 0137099819.
 ^ Richard Fitzpatrick, Newtonian Dynamics, http://farside.ph.utexas.edu/teaching/336k/Newton/Newtonhtml.html.
 ^ Eric W. Weisstein, Double Pendulum, scienceworld.wolfram.com. 2007
 ^ T. R. Kane and D. A. Levinson, Dynamics: theory and applications, McGrawHill, New York, 1985
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