reference state. The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes...

point not having unique generalized barycentric coordinates except when P is a simplex. Dual to generalized barycentric coordinates are slack variables,...

details. As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the...

this vk, the kinetic energy in generalized coordinates depends on the generalized velocities, generalized coordinates, and time if the position vectors...

the generalized coordinates and velocities (q, q̇) with (q, p); the generalized coordinates and the generalized momenta conjugate to the generalized coordinates:...

mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces Fi...

Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates. Generalized coordinates are...

symbol Π. In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as p j = ∂ L...

coordinates, which we denote as w {\displaystyle \mathbf {w} } (the action angles, which are the generalized coordinates) and their new generalized momenta...

formulated in generalized coordinates of motion. Note that "generalized coordinates of motion" are related to—but distinct from—generalized coordinates as used...

choice of generalized coordinates. For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta...

the action-angle coordinates, called the "action" of the generalized coordinate qk, is defined by integrating a single generalized momentum around a...

are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient...

perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related...

the configuration of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the...

q_{N}{\left(\sigma \right)}\right)} described by N {\displaystyle N} generalized coordinates between two specified states at two specified parameters σA and...

= 1. Taking δθ as the ε parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by Q = n ×...

mechanics). Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations...

n } {\displaystyle \{u_{1},u_{2},u_{3},\ldots ,u_{n}\}} are n generalized coordinates that describe the system (in unconstrained configuration space)...

virtual displacements. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates q i {\displaystyle q_{i}} , i = 1...

Retrieved 2020-11-09. For a description of generalized coordinates, see Ahmed A. Shabana (2003). "Generalized coordinates and kinematic constraints". Dynamics...

(geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows. Let K be any division ring (skewfield)...

{q}}}}\right),} is the generalized force acting on this one degree of freedom system. If the mechanical system is defined by m generalized coordinates, qj, j = 1...

three-dimensional space by using a distance and two angles as its three coordinates. These are the radial distance r along the line connecting the point...

uses generalized coordinates and corresponding generalized velocities in configuration space, and Hamiltonian mechanics, which uses coordinates and corresponding...

has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field...

,p_{N})} are the particle momenta or the conjugate momenta of generalized coordinates, defined by the equation p k   = def   ∂ L ∂ q ˙ k , {\displaystyle...

states that the true evolution q(t) of a system described by N generalized coordinates q = (q1, q2, ..., qN) between two specified states q1 = q(t1) and...

freedom is defined by generalized coordinates q = (q1, q2, ... qN). There are generalized momenta conjugate to these coordinates, p = (p1, p2, ..., pN)...

involve the generalized velocities. By defining a normalized distance or metric d s {\displaystyle ds} in the space of generalized coordinates d s 2 = d...