Generalized coordinates
Parameters that uniquely define a system's configuration in configuration space.
In analytical mechanics, generalized coordinates are the parameters used to describe a system's configuration within its configuration space. These parameters must uniquely specify how the system is arranged relative to a reference setup. Their rates of change over time are called generalized velocities. The term "generalized" sets them apart from the usual Cartesian coordinates.
For instance, instead of using the x and y positions of a pendulum bob, you can describe its location with a single generalized coordinate: the angle it makes with the vertical. While many sets of generalized coordinates are possible for any physical system, they are typically chosen to make calculations easier, like solving the equations of motion. When the coordinates are independent of each other, the number of independent generalized coordinates equals the system's degrees of freedom. These coordinates are paired with generalized momenta to form canonical coordinates on phase space.
Generalized coordinates are often selected to give the smallest number of independent coordinates needed to define a system's configuration, which simplifies Lagrange's equations of motion. However, a useful set of generalized coordinates can sometimes be dependent, meaning they are linked by one or more constraint equations.
For a system of N particles in three-dimensional space, each particle's position is a set of three Cartesian coordinates. A holonomic constraint is an equation that ties together all three spatial coordinates of a particle, making them not fully independent. The constraint may change over time, so time appears explicitly in the equation. At any given moment, knowing two coordinates determines the third. Each such constraint counts as one constraint equation. If there are C constraints, there are C constraint equations, though not necessarily one per particle. With no constraints, there are no constraint equations.
Initially, the system's configuration is defined by 3N quantities, but C coordinates can be removed—one per constraint equation. This leaves n = 3N − C independent coordinates (or n = ND − C in D dimensions). The ideal is to use the minimum number of coordinates needed to define the entire system's configuration, taking advantage of the constraints. These quantities are called generalized coordinates, denoted qⱼ(t), and are conveniently collected into an n-tuple q(t) = (q₁, q₂, …, qₙ).
- 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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