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Topic: Generalized coordinates

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	Modelica
		Via the dynamic dummy derivative method the generalized coordinates on position and velocity level from one of the 7 joints are dynamically selected as states during simulation.
		For this mechanism, the generalized coordinates of joint j1 (i.e., the rotation angle of the revolute joint and its derivative) can always be used as states.
		When the generalized coordinates of revolute joint "innerJoint" (lower left part in figure) are used as states, then frame_a and frame_b of the jointUPS joint can be calculated.
	www.dynasim.se /ModelicaStandardLibrary/help/Modelica_Mechanics_MultiBody_UsersGuide_Tutorial_LoopStructures.html (2215 words)

	Talk:Lagrangian mechanics - Wikipedia, the free encyclopedia
		So the generalized reminds you that your coordinates may need to be a totally non-traditional system, like the length along a wire bent into a weird shape (perhaps a bead is moving along the wire).
		The link generalized coordinates links to this page (Lagrange mechanics), I believe it would be nice to have either a larger discussion of generalized coordinates on this page or perhaps its own article.
		Generalized coordinates now have their own article, which is not 100% yet, but is a good start.
	en.wikipedia.org /wiki/Talk:Lagrangian_mechanics (430 words)

	Hamiltonian mechanics - Wikipedia, the free encyclopedia
		In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta.
		In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum.
		For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.
	en.wikipedia.org /wiki/Hamiltonian_mechanics (1197 words)

	Lagrangian mechanics - Wikipedia, the free encyclopedia
		For example, for a simple pendulum of length l, a logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, for which the transformation equation would be
		The term "generalized coordinates" is really a leftover from the period when Cartesian coordinates were the default coordinate system.
		Since work is a physical scalar quantity, we should be able to rewrite this equation in terms of the generalized coordinates and velocities.
	en.wikipedia.org /wiki/Lagrangian_mechanics (926 words)

	Generalized Coordinates and Degrees of Freedom
		The set of coordinates or parameters, which uniquely describes the geometric position and/or orientation of body, or system of bodies, are called the generalized coordinates of the system.
		As previously stated, it is the minimum number of generalized coordinates that determines the number of degrees of freedom and each coordinate must be independent.
		Generally, the constraint of a pin reduces the number of degrees of freedom by two.
	www.volaeris.com /GeneralizedCoord.html (848 words)

	Generalized Coordinates (Site not responding. Last check: )
		Generalized coordinates are any set of coordinates that are used to describe the motion of a physical system.
		Cartesian coordinates and spherical polar coordinates are other examples of generalized coordinates.
		We may choose any convenient set of generalized coordinates for a particular problem.
	www.chm.uri.edu /urichm/chm531/cm/node5.html (278 words)

	Dictionary of Technical Terms for Aerospace Use - L
		Examples of such coordinates are: (a) the values of any properties of the fluid conserved in the motion; or (b) more generally, the positions in space of the parcels at some arbitrarily selected moment.
		Generally, and quite loosely, that part of the atmosphere in which most weather phenomena occur (i.e., the troposphere and lower stratosphere); hence, used in contrast to the common meaning for the upper atmosphere.
		That component of general precession caused by the combined effect of the sun and moon on the equatorial protuberance of the earth, producing a westward motion of the equinoxes along the ecliptic.
	roland.lerc.nasa.gov /~dglover/dictionary/l.html (7224 words)

	Coordinates and Generalized Coordinates
		Those coordinates are called generalized coordinates since they are not necessarily measured in length but might be angles etc. As coordinates are related to forces, velocities, momenta etc. a set of generalized quantities comes together with these new coordinates.
		Generalized quantities might therefore not have the same meaning as the quantities related to the cartesian coordinates, which we are used to.
		then all generalized coordinates are independent and the number of generalized coordinates is equal to the number of degrees of freedom of the system.
	mit.fnal.gov /~paus/8.21-IAP2001/notes/notes/node3.html (258 words)

	AMS Glossary
		Usually employed in problems involving a finite number of degrees of freedom, the generalized coordinates are chosen so as to take advantage of the constraints of the system in reducing the total number of coordinates.
		It is postulated that snow crystals are formed and grow in the generating cells and that the cells are maintained by convection induced by the release of latent heat accompanying the crystal growth.
		The shape of the snow trail below a generating cell depends on the fall speed of the snow and the vertical profile of the horizontal wind.
	amsglossary.allenpress.com /glossary/browse?s=g&p=8 (484 words)

	ENCE613 Course Log (Site not responding. Last check: )
		Established that the kinetic energy is, in general a function of the generalized velocities, generalized coordinates, and time.
		Completed the three element beam example problem by obtaining: (1) the mass matrix and the generalized force vector by means of the kinetic energy and virtual work expressions; and (2) the equations of motion by use of the Lagrange equation.
		Introduce the static equilibrium position as the datum for all generalized coordinates so as to be able to ignore all static loads in the all cases of small deflection vibrations.
	www.cee.umd.edu /stud/encecourlst/ence613log.html (1541 words)

	[No title]
		The generalized coordinates Qdof1 through Qdof4 describing the \ rotations are in the form of euler parameters {e1,e2,e3,e4}, where e4 is the \ cosine of the half-rotational angle.
		GInerFrc[All] returns the sum of the generalized inertia forces for all bodies, with respect to each of the generalized speeds."; ground::usage = "ground is the default Newtonian reference frame, which all \ other frames are defined relative to."; Hinge::usage = "A Hinge joint is a single degree-of-freedom rotatory joint.
		The generalized speeds, Udof1 and Udof2, are defined to be the \ time-derivatives of the respective generalized coordinates, Qdof1 and \ Qdof2."; VelCOM::usage = "VelCOM[body] returns a vector describing the velocity of the \ center of mass of body with respect to the ground reference frame.
	www-personal.umich.edu /~artkuo/DynamicsWorkbench/DynamicsWorkbench.m (3844 words)

	Jacobi, Karl Gustav Jacob (1804-1851)
		Along with William Hamilton, he developed an approach to mechanics based on generalized coordinates.
		In this method, the total energy of a mechanical system is represented as a function of generalized coordinates and corresponding generalized momenta; for example, in a double pendulum the two generalized coordinates could be two angles.
		After a collapse from overwork in 1843, Jacobi was allowed to stay in Berlin with a generous allowance from the King of Prussia.
	www.daviddarling.info /encyclopedia/J/Jacobi.html (457 words)

	EOM's: Generalized Forces and Coordinates (Site not responding. Last check: )
		The centroidal positions and the rotations of the rigid bodies in a system can be expressed in terms of a set of "generalized coordinates." This set of generalized coordinates can be made up of Cartesian coordinates (e.g., x-y-z), path coordinates and/or cylindrical coordinates.
		In the next lecture it will be discovered that we will be able to extract a set of N so-called "Lagrange's equations" from the power equation for the system provided that N is equal to the number of degrees of freedom in the system.
		In this lecture, we will develop explicit representations of the terms in the power equation in terms of the generalized coordinates (and their time derivatives) as well as introduce a set of forces/moments (the "generalized forces") that are associated with the set of generalized coordinates of the system.
	tools.ecn.purdue.edu /~me563/Lectures/EOMs/Generalized/lecture_page.html (135 words)

	Lagrangian Mechanics (Site not responding. Last check: )
		A particle is constraint to move in the x-y plane, the equation of constraint is z = 0, the constraint is holonomic.
		L is a function of the coordinates and the velocities
		So only if Lagrangian does not explicitly depend on time and the generalized coordinates do not explicitly depend on time, then H = T + U = E and the energy is a constant of motion.
	electron6.phys.utk.edu /phys594/Tools/mechanics/summary/lagrangian/lagrangian.htm (567 words)

	No Title
		Covered the general heuristics for the setup of mechanics problems: generalized coordinates, counting degrees of freedom, proper sets of coordinates, and using overcomplete sets of coordinates with Lagrange multipliers to get constraints.
		We have a "proper" set of generalized coordinates when r=s -- we have just enough coordinates to fully specify the system, and all are independent (no constraints relate them).
		For example, to find a tension that is holding r constant, reintroduce r as a coordinate, with the constraint g = r-c, and find the tension from the Lagrange multiplier method.
	www.emory.edu /PHYSICS/Faculty/Benson/361/notes/25/25.html (440 words)

	Tall Frame Analysis by Reduced Generalized Coordinates (Site not responding. Last check: )
		The displacements and rotations of joints on the exterior (corner) columns are independently expressed in terms of a set of functions with undetermined coefficients (generalized coordinates).
		Transformation matrix connecting the actual joint displacements to the generalized coordinates are then derived from the joint compatibility conditions.
		A 26-story, 3-bay frame was analyzed using 24 generalized coordinates and the errors are generally less than 5% and 10%, respectively, for the displacements and internal forces.
	www.pubs.asce.org /WWWdisplay.cgi?5013611 (131 words)

	Paul Flory - Wikipedia, the free encyclopedia
		In modeling the position vectors of atoms in macromolecules it is often necessary to convert from Cartesian coordinates (x,y,z) to generalized coordinates.
		Applying a vector conversion from the Cartesian coordinates to the generalized coordinates will describe the same three dimensional structure using the Flory convention.
		His contributions to the field of polymer science is best summarized in his classic 1953 text, Principles of Polymer Chemistry, where he provides a comprehensive account of experimental and theoretical results proven in his day.
	www.wikipedia.org /wiki/Paul_Flory (414 words)

	Appendix A
		The introduction of coordinate-dependent mass-metric factors into the Lagrangian/Hamiltonian of a system via a transformation to generalized coordinates leads to a considerable increase in complexity in the description of the true dynamics of the system.
		Although the AFED method requires that generalized coordinates be used, the dynamics only needs to generate the correct configurational averages, and, hence, it is not necessary to work with correct conjugate momenta and the true adiabatic dynamics.
		It should be noted that, under certain conditions, it may be straightforward to work in terms of a full canonical set of generalized coordinates, such as in the example of Sec.
	www.nyu.edu /classes/tuckerman/eccc7/adb_free/node11.html (345 words)

	[No title]
		D_{k,i} F_{i,j} = 0_{kj}, i=1..n, j=1..dof and k=1..m, where D_k(x_i)=0 are the constraints and x_i=F_i(q_j) are the coordinates of the cm of the bodies expressed in terms of the generalized coordinates q_j.
		A more general approach to transform the equations of motion in terms of generalized independent coordinates is the method of coordinate partitioning as proposed by R. Wehage, "Generalized Coordinate Partitioning in Dynamic Analysis of Mechanical Systems", Ph.D. Thesis, University of Iowa, December 1980.
		The new state (x,xdot)_n+1 is in general not on the constraint surface define by the constraints D(x)=0.
	www.tam.cornell.edu /~als93/TAM674.htm (6714 words)

	In-Focus (Site not responding. Last check: )
		Any number of possibilities exist for the choice of generalized coordinates to be used for a given problem.
		As mentioned earlier in the lecture, we need to guarantee that the choice of coordinates completely describe the position/orientation of each body and that the coordinates be independent.
		Also, determine the inertia coordinates and generalized forces for each set of generalized coordinates.
	tools.ecn.purdue.edu /~me563/Lectures/EOMs/Generalized/In_Focus/page.html (166 words)

	TOWARDS A FORMALLY STANDARDIZED #tex2html_wrap_inline35986#-FAMILY OF INTEGRAL/ INTEGRATION OPERATORS FOR DYNAMIC FIELD ...
		In the case of proportional damping, a change of coordinates from the physical to generalized coordinates leads to a set of uncoupled equations, one for each mode, that can be solved by relatively simple techniques.
		If the damping is not proportional, the same change of coordinates leads to a set of coupled equations for the determination of the generalized coordinates due to the fact that the damping matrix cannot be diagonalized by the change of coordinates.
		In the present exposition, a generalized methodology encompassing both time integral representations and the bridging of the relationships systematically leading to the so-called integration operators in time applicable to non-proportionally damped mechanical systems is overviewed.
	www-users.cs.umn.edu /~xiangmin/arc/node63.html (624 words)

	Lagrange's Equations in Generalized Coordinates (Site not responding. Last check: )
		Lagrange has shown that the form of Lagrange's equations is invariant to the particular set of generalized coordinates chosen.
		We do emphasize that the invariance to coordinate system is not a property of the equations of motion when expressed in the usual form of Newton's second law.
		We now illustrate how to use Lagrange's equations in generalized coordinates by applying the approach to the free motion of a particle confined to move on the perimeter of a ring as discussed previously.
	www.chm.uri.edu /urichm/chm531/cm/node6.html (197 words)

	Generalized coordinates (Site not responding. Last check: )
		The {qa} are called “generalized” coordinates and completely describe the state of the mechanical system being considered.
		A system of N interacting particles might require 3?N generalized coordinates, but usually far fewer coordinates are required because of constraints.
		The case of a bead constrained on a uniformly rotating, frictionless wire, is a system with only one generalized coordinate (r).
	wug.physics.uiuc.edu /courses/phys326/spring02/lectures/lagrange_1/sld004.htm (107 words)

	1.3 Generalized Coordinates and the Lagrangian (Site not responding. Last check: )
		So we transform to six position coordinates, three representing the motion of the center of mass (X, Y, Z) and three representing the motion of the particles about their center of mass.
		We know that we will need six coordinates for our two particles, but we don't have to specify exactly what they are until we actually sit down to work out a specific problem that uses them.
		However, their use is greater than even this, since as we will discuss, generalized coordinates, no matter what they are, obey the same set of rules.
	scholar.chem.nyu.edu /2600/classnotes2/node16.html (501 words)

	ICR and IHA in 2D/3D dynamic analysis (Site not responding. Last check: )
		Each of the N generalized coordinates is associated with a generalized force (the 'moments', if the coordinates are angles).
		Where dq is a small change in the N-vector q of generalized coordinates, dr is the resulting (3D-vector) change in position of the point of application of each force vector F. Q is the N-vector of generalized forces.
		Dynamic equations of motion can also be formulated in generalized coordinates: Q = M(q).q" I am not familiar with the method to find the inertia matrix M, which may depend in a complex way on q.
	isb.ri.ccf.org /biomch-l/archives/biomch-l-1991-06/00022.html (1030 words)

	Ph 425
		The number of generalized coordinates required is the number of degrees of freedom of the system.
		Expressing Hamilton’s principle in terms of generalized coordinates and repeating the arguments of Section 10.1 leads to the Lagrange equations of motion:
		Eq.(10.6.9) is the general equation of motion, S being a constant, related to the angular momentum p
	physics.pdx.edu /~egertonr/ph425/Ph425chX.htm (1372 words)

	1.3.1 Generalized Coordinates (Site not responding. Last check: )
		It is easy to see that it is not always possible since one might by accident chose generalized coordinates that are not independent of each other.
		The velocities in generalized coordinates are obtained from Equations 1.3.3 by differentiation.
		It would be wrong to think that generalized coordinates are useful only for converting from Cartesian coordinates to some other-well known coordinate system.
	scholar.chem.nyu.edu /2600/classnotes/node17.html (396 words)

	Generalized Coordinate Package for the MechanicalSystems Pack -- from Mathematica Information Center
		The Generalized Coordinate Package is an extension package for the Mathematica MechanicalSystems Pack, which is commercially available through WRI.
		The G.C. Package allows the location and orientation of some or all of the bodies in a MechanicalSystems Pack model to be represented by user-specified generalized coordinates, instead of the reference point coordinates that are normally used by the M.S. Pack.
		The G.C. Package can be used to generate a minimal set of equations of motion for any open loop system and, with some algebraic reduction, many closed loop systems.
	library.wolfram.com /infocenter/MathSource/1879 (278 words)

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