**Generalized Coordinates**

Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by *constraints*. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot. Many such constraints can be incorporated by changing the normal Cartesian coordinates to a set of *generalized coordinates* that may be fewer in number. Refined mathematical methods have been developed for solving mechanics problems in generalized coordinates. They introduce a *generalized momentum*, also known as the *canonical* or *conjugate momentum*, that extends the concepts of both linear momentum and angular momentum. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as *mechanical*, *kinetic* or *kinematic momentum*. The two main methods are described below.

Read more about this topic: Momentum

### Other articles related to "generalized coordinates, coordinates, generalized":

... second law, which can be written in matrix form where M is a mass matrix and q is the vector of

**generalized coordinates**that describe the particles' positions ... For example, the vector q may be a 3N Cartesian

**coordinates**of the particle positions rk, where k runs from 1 to N in the absence of constraints, M would be the 3Nx3N diagonal square matrix of the ... The vector f represents the

**generalized**forces and the scalar V(q) represents the potential energy, both of which are functions of the

**generalized coordinates**q ...

**Generalized Coordinates**

... In Lagrangian mechanics, a different set of

**coordinates**are used, called the

**generalized coordinates**... These are commonly denoted as with called the

**generalized**position and the

**generalized**velocity ... When a Hamiltonian is defined on the cotangent bundle, then the

**generalized coordinates**are related to the canonical

**coordinates**by means of the Hamilton–Jacobi equations ...

**Generalized Coordinates**and Virtual Work

... 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 Fi=0 ... m be applied to points with Cartesian

**coordinates**rj, j=1.. ... Now assume that each δrj depends on the

**generalized coordinates**qi, i=1.. ...

... of planar particle motion Lagrangian mechanics formulates mechanics in terms of

**generalized coordinates**{qk}, which can be as simple as the usual polar

**coordinates**or a ... Within this formulation the motion is described in terms of

**generalized**forces, using in place of Newton's laws the Euler–Lagrange equations ... Among the

**generalized**forces, those involving the square of the time derivatives {(dqk ⁄ dt )2} are sometimes called centrifugal forces ...

... For many systems, the kinetic energy is quadratic in the

**generalized**velocities although the mass tensor may be a complicated function of the

**generalized coordinates**... For such systems, a simple relation relates the kinetic energy, the

**generalized**momenta and the

**generalized**velocities provided that the potential energy does not involve the

**generalized**velocities ... By defining a normalized distance or metric in the space of

**generalized coordinates**one may immediately recognize the mass tensor as a metric tensor ...

### Famous quotes containing the word generalized:

“One is conscious of no brave and noble earnestness in it, of no *generalized* passion for intellectual and spiritual adventure, of no organized determination to think things out. What is there is a highly self-conscious and insipid correctness, a bloodless respectability submergence of matter in manner—in brief, what is there is the feeble, uninspiring quality of German painting and English music.”

—H.L. (Henry Lewis)