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...

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

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

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

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...

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

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

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

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

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

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

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

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

(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)...

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...

{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...

introduce the generalized coordinates as {qk} = (r, θ). Then Hildebrand shows in polar coordinates with the qk = (r, θ) the "generalized momenta" are:...

explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. The opposite...

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

n-tuple of the generalized momenta. A Legendre transformation is performed to change the variables in the total differential of the generalized coordinate...

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

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...

ISBN 978-0-486-65067-8. For a description of generalized coordinates, see Ahmed A. Shabana (2003). "Generalized coordinates and kinematic constraints". Dynamics...

where the gradient of the potential energy with respect to the generalized coordinates is zero. If a particle in equilibrium has zero velocity, that particle...

potential. This relationship is represented in terms of the set of generalized coordinates q i = { q 1 , q 2 , … q n } {\displaystyle q_{i}=\left\{q_{1},q_{2}...

_{r}={\ddot {q}}_{r}} is an arbitrary generalized acceleration, or the second time derivative of the generalized coordinates q r {\displaystyle q_{r}} , and...