Generalized coordinates
Parameters that uniquely define a system's configuration in configuration space.
Generalized coordinates are a set of parameters used in analytical mechanics to represent the configuration of a system in a configuration space. They must uniquely define the configuration of the system relative to a reference configuration, and their time derivatives are the generalized velocities. The adjective 'generalized' distinguishes these parameters from the traditional use of the term 'coordinate' to refer to Cartesian coordinates.
- field
- Analytical mechanics
- known_for
- Representing system configuration via independent parameters, simplifying equations of motion
- related_concept
- Generalized momenta
- key_property
- Number of independent generalized coordinates equals degrees of freedom
Lore & Background
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, meaning they are related by one or more constraint equations. For a system of N particles in 3D real coordinate space, the position vector of each particle can be written as a 3-tuple in Cartesian coordinates, and a holonomic constraint is a constraint equation of the form f(r_k, t) = 0 that connects all three spatial coordinates of that particle together, so they are not independent.
Reader's Guide
Generalized coordinates are significant because they allow the configuration of a physical system to be described with the minimum number of independent variables, equal to the number of degrees of freedom. This reduction simplifies the solution of equations of motion, particularly in Lagrange's formulation. The coordinates can be lengths along straight lines, arc lengths along curves, or angles, not necessarily Cartesian coordinates. They are paired with generalized momenta to provide canonical coordinates on phase space. Although many choices exist for generalized coordinates, they are generally selected to simplify calculations. The number of independent generalized coordinates is defined by the number of degrees of freedom, which for a system of N particles in 3D with C constraints is n = 3N − C. This framework is essential for analyzing constrained mechanical systems.
Did You Know?
- Generalized coordinates are paired with generalized momenta to provide canonical coordinates on phase space.
- An example of a generalized coordinate is describing the position of a pendulum using the angle relative to vertical, rather than by x and y positions.
- The number of independent generalized coordinates equals the number of degrees of freedom of the system.
- Generalized coordinates may be dependent if they are related by one or more constraint equations.
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