Generalized Coordinates
See also: Analytical mechanicsNewton'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 ...
... 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 ...
... 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 ...
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