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| {{Classical mechanics|cTopic=Formulations}}
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| In [[analytical mechanics]], specifically the study of the [[rigid body dynamics]] of [[multibody system]]s, the term '''generalized coordinates''' refers to the parameters that describe the [[Configuration space|configuration]] of the [[physical system|system]] relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration.<ref name=Ginsberg>{{cite book |title=Engineering dynamics, Volume 10 |author= Jerry H. Ginsberg |url=http://books.google.com/books?id=je0W8N5oXd4C&pg=PA397 |page=397 |chapter=§7.2.1 Selection of generalized coordinates |isbn=0-521-88303-2 |year=2008 |publisher=Cambridge University Press |edition=3rd}} </ref> The '''generalized velocities''' are the time [[derivative]]s of the generalized coordinates of the system.
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| An example of a generalized coordinate is the angle that locates a point moving on a circle. The adjective "generalized" distinguishes these parameters from the traditional use of the term coordinate to refer to [[Cartesian coordinates]]: for example, describing the location of the point on the circle using x and y coordinates.
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| Although there may be many choices for generalized coordinates for a physical system, parameters are usually selected which are convenient for the specification of the configuration of the system and which make the solution of its [[equations of motion]] easier. If these parameters are independent of one another, then number of independent generalized coordinates is defined by the number of [[Degrees of freedom (mechanics)|degrees of freedom]] of the system.<ref name=Amirouche> {{cite book |title=Fundamentals of multibody dynamics: theory and applications |author=Farid M. L. Amirouche |url=http://books.google.com/books?id=_nlEcQYldeIC&pg=PA46 |page=46 |chapter=§2.4: Generalized coordinates |publisher=Springer |isbn=0-8176-4236-6 |year=2006}}</ref> <ref name= Scheck>{{cite book |title=Mechanics: From Newton's Laws to Deterministic Chaos |author=Florian Scheck |url=http://books.google.com/books?id=yUDo7VptDgIC&pg=PA286 |page=286 |chapter=§5.1 Manifolds of generalized coordinates |isbn=3-642-05369-6 |publisher=Springer |edition =5th |year=2010}}</ref>
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| ==Constraint equations==
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| [[File:Generalized coordinates 1df.svg|right|350px|"350px"|thumb|Generalized coordinates for one degree of freedom (of a particle moving in a complicated path). Instead of using all three [[Cartesian coordinates]] ''x, y, z'' (or other standard [[coordinate systems]]), only one is needed and is completely arbitrary to define the position. Four possibilities are shown. '''Top:''' distances along some fixed line, '''bottom left:''' an angle relative to some baseline, '''bottom right:''' the [[arc length]] of the path the particle takes. All are defined relative to a zero position - again arbitrarily defined.]]
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| Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of [[Lagrangian|Lagrange's equations]] of motion. However, it can also occur that a useful set of generalized coordinates may be ''dependent'', which means that they are related by one or more [[constraint (mathematics)|constraint]] equations.
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| ===Holonomic constraints===
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| If the constraints introduce relations between the generalized coordinates ''q<sub>i</sub>'', i=1,..., n and time, of the form,
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| :<math>f_j(q_1,..., q_n, t) = 0, j=1,..., k,</math>
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| are called ''holonomic''.<ref name=Ginsberg/> These constraint equations define a manifold in the space of generalized coordinates ''q<sub>i</sub>'', i=1,...,n, known as the [[configuration space|configuration manifold]] of the system. The degree of freedom of the system is d=n-k, which is the number of generalized coordinates minus the number of constraints.<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|260}}
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| It can be advantageous to choose independent generalized coordinates, as is done in [[Lagrangian mechanics]], because this eliminates the need for constraint equations. However, in some situations, it is not possible to identify an unconstrained set. For example, when dealing with [[nonholonomic system|nonholonomic]] constraints or when trying to find the force due to any constraint—holonomic or not, dependent generalized coordinates must be employed. Sometimes independent generalized coordinates are called internal coordinates because they are mutually independent, otherwise unconstrained, and together give the position of the system.
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| [[File:Generalized coordinates 1 and 2 df.svg|right|350px|"350px"|thumb|'''Top:''' one degree of freedom, '''bottom:''' two degrees of freedom, '''left:''' an open [[curve]] ''F'' ([[parameter]]ized by ''t'') and [[surface]] ''F'', '''right:''' a [[closed curve]] ''C'' and [[closed surface]] ''S''. The equations shown are the ''constraint equations''. Generalized coordinates are chosen and defined with respect to these curves (one per degree of freedom), and simplify the analysis since even complicated curves are described by the ''minimum'' number of coordinates required.]]
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| ===Non-holonomic constraints===
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| A mechanical system can involve constraints on both the generalized coordinates and their derivatives. Constraints of this type are known as non-holonomic, and have the form
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| :<math>g_j(q_1,... , q_n, \dot{q}_1,... , \dot{q}_n, t) = 0, j=1,.... , k.</math>
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| An example of a non-holonomic constraint is a rolling wheel or knife-edge that constrains the direction of the velocity vector.
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| ==Example: Simple pendulum==
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| [[File:Pendulum.gif|thumb|Dynamic model of a simple pendulum.]]
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| The relationship between the use of generalized coordinates and Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of a simple pendulum.<ref>{{cite book
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| | last = Greenwood
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| | first = Donald T.
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| | year = 1987
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| | title = Principles of Dynamics
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| | edition = 2nd edition
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| | publisher = Prentice Hall
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| | isbn = 0-13-709981-9
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| }}</ref><ref>Richard Fitzpatrick, Newtonian Dynamics, [http://farside.ph.utexas.edu/teaching/336k/Newton/node90.html http://farside.ph.utexas.edu/teaching/336k/Newton/Newtonhtml.html].</ref>
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| ===Coordinates===
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| A simple pendulum consists of a mass M hanging from a pivot point so that it is constrained to move on a circle of radius L. The position of the mass is defined by the coordinate vector '''r'''=(x, y) measured in the plane of the circle such that y is in the vertical direction. The coordinates x and y are related by the equation of the circle
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| :<math>f(x, y) = x^2+y^2 - L^2=0,</math>
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| that constrains the movement of M. This equation also provides a constraint on the velocity components,
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| :<math> \dot{f}(x, y)=2x\dot{x} + 2y\dot{y} = 0.</math>
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| Now introduce the parameter θ, that defines the angular position of M from the vertical direction. It can be used to define the coordinates x and y, such that
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| :<math> \mathbf{r}=(x, y) = (L\sin\theta, -L\cos\theta).</math>
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| The use of θ to define the configuration of this system avoids the constraint provided by the equation of the circle.
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| ===Virtual work===
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| Notice that the force of gravity acting on the mass m is formulated in the usual Cartesian coordinates,
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| :<math> \mathbf{F}=(0,-mg),</math>
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| where g is the acceleration of gravity.
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| The virtual work of gravity on the mass m as it follows the trajectory '''r''' is given by
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| :<math> \delta W = \mathbf{F}\cdot\delta \mathbf{r}.</math>
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| The variation δ'''r''' can be computed in terms of the coordinates x and y, or in terms of the parameter θ,
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| :<math> \delta \mathbf{r} =(\delta x, \delta y) = (L\cos\theta, L\sin\theta)\delta\theta.</math>
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| Thus, the virtual work is given by
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| :<math>\delta W = -mg\delta y = -mgL\sin\theta\delta\theta.</math>
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| Notice that the coefficient of δy is the y-component of the applied force. In the same way, the coefficient of δθ is known as the [[generalized force]] along generalized coordinate θ, given by
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| :<math> F_{\theta} = -mgL\sin\theta.</math>
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| ===Kinetic energy===
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| To complete the analysis consider the kinetic energy T of the mass, using the velocity,
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| :<math> \mathbf{v}=(\dot{x}, \dot{y}) = (L\cos\theta, L\sin\theta)\dot{\theta},</math>
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| so,
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| :<math> T= \frac{1}{2} m\mathbf{v}\cdot\mathbf{v} = \frac{1}{2} m (\dot{x}^2+\dot{y}^2) = \frac{1}{2} m L^2\dot{\theta}^2.</math>
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| ===Lagrange's equations===
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| Lagrange's equations for the pendulum in terms of the coordinates x and y are given by,
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| :<math> \frac{d}{dt}\frac{\partial T}{\partial \dot{x}} - \frac{\partial T}{\partial x} = F_{x} + \lambda \frac{\partial f}{\partial x},\quad \frac{d}{dt}\frac{\partial T}{\partial \dot{y}} - \frac{\partial T}{\partial y} = F_{y} + \lambda \frac{\partial f}{\partial y}. </math>
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| This yields the three equations
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| :<math>m\ddot{x} = \lambda(2x),\quad m\ddot{y} = -mg + \lambda(2y),\quad x^2+y^2 - L^2=0,</math>
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| in the three unknowns, x, y and λ.
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| Using the parameter θ, Lagrange's equations take the form
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| :<math>\frac{d}{dt}\frac{\partial T}{\partial \dot{\theta}} - \frac{\partial T}{\partial \theta} = F_{\theta},</math>
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| which becomes,
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| :<math> mL^2\ddot{\theta} = -mgL\sin\theta,</math>
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| or
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| :<math> \ddot{\theta} + \frac{g}{L}\sin\theta=0.</math>
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| This formulation yields one equation because there is a single parameter and no constraint equation.
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| This shows that the parameter θ is a generalized coordinate that can be used in the same way as the Cartesian coordinates x and y to analyze the pendulum.
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| ==Example: Double pendulum==
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| [[File:Double-Pendulum.svg|thumb|right|A double pendulum]]
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| The benefits of generalized coordinates become apparent with the analysis of a double pendulum.
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| For the two masses m<sub>i</sub>, i=1, 2, let '''r<sub>i</sub>'''=(x<sub>i</sub>, y<sub>i</sub>), i=1, 2 define their two trajectories. These vectors satisfy the two constraint equations,
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| :<math>f_1 (x_1, y_1, x_2, y_2) = \mathbf{r}_1\cdot \mathbf{r}_1 - L_1^2 = 0, \quad f_2 (x_1, y_1, x_2, y_2) = (\mathbf{r}_2-\mathbf{r}_1) \cdot (\mathbf{r}_2-\mathbf{r}_1) - L_2^2 = 0.</math>
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| The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates x<sub>i</sub>, y<sub>i</sub> i=1, 2 and the two Lagrange multipliers λ<sub>i</sub>, i=1, 2 that arise from the two constraint equations.
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| ===Coordinates===
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| Now introduce the generalized coordinates θ<sub>i</sub> i=1,2 that define the angular position of each mass of the double pendulum from the vertical direction. In this case, we have
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| :<math>\mathbf{r}_1 = (L_1\sin\theta_1, -L_1\cos\theta_1), \quad \mathbf{r}_2 = (L_1\sin\theta_1, -L_1\cos\theta_1) + (L_2\sin\theta_2, -L_2\cos\theta_2).</math>
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| The force of gravity acting on the masses is given by,
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| :<math>\mathbf{F}_1=(0,-m_1 g),\quad \mathbf{F}_2=(0,-m_2 g)</math>
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| where g is the acceleration of gravity. Therefore, the virtual work of gravity on the two masses as they follow the trajectories '''r'''<sub>i</sub>, i=1,2 is given by
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| :<math> \delta W = \mathbf{F}_1\cdot\delta \mathbf{r}_1 + \mathbf{F}_2\cdot\delta \mathbf{r}_2.</math>
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| The variations δ'''r'''<sub>i</sub> i=1, 2 can be computed to be
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| :<math> \delta \mathbf{r}_1 = (L_1\cos\theta_1, L_1\sin\theta_1)\delta\theta_1, \quad \delta \mathbf{r}_2 = (L_1\cos\theta_1, L_1\sin\theta_1)\delta\theta_1 +(L_2\cos\theta_2, L_2\sin\theta_2)\delta\theta_2</math>
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| ===Virtual work===
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| Thus, the virtual work is given by
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| :<math>\delta W = -(m_1+m_2)gL_1\sin\theta_1\delta\theta_1 - m_2gL_2\sin\theta_2\delta\theta_2,</math>
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| and the generalized forces are
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| :<math>F_{\theta_1} = -(m_1+m_2)gL\sin\theta_1,\quad F_{\theta_2} = -m_2gL\sin\theta_2.</math>
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| ===Kinetic energy===
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| Compute the kinetic energy of this system to be
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| :<math> T= \frac{1}{2}m_1 \mathbf{v}_1\cdot\mathbf{v}_1 + \frac{1}{2}m_2 \mathbf{v}_2\cdot\mathbf{v}_2 = \frac{1}{2}(m_1+m_2)L_1^2\dot{\theta}_1^2 + \frac{1}{2}m_2L_2^2\dot{\theta}_2^2 + m_2L_1L_2 \cos(\theta_2-\theta_1)\dot{\theta}_1\dot{\theta}_2.</math>
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| ===Lagrange's equations===
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| Lagrange's equations yield two equations in the unknown generalized coordinates θ<sub>i</sub> i=1, 2, given by<ref>Eric W. Weisstein, [http://scienceworld.wolfram.com/physics/DoublePendulum.html Double Pendulum], scienceworld.wolfram.com. 2007</ref>
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| :<math>(m_1+m_2)L_1\ddot{\theta}_1+m_2L_1L_2\ddot{\theta}_2\cos(\theta_2-\theta_1) + m_2L_1L_2\sin(\theta_2-\theta_1) = -(m_1+m_2)gL_1\sin\theta_1,</math>
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| and
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| :<math>m_2L_2\ddot{\theta}^2+m_2L_1L_2\ddot{\theta}_1\cos(\theta_2-\theta_1) - m_2L_1L_2\sin(\theta_2-\theta_1)=-m_2gL_2\sin\theta_2.</math>
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| The use of the generalized coordinates θ<sub>i</sub> i=1, 2 provides an alternative to the Cartesian formulation of the dynamics of the double pendulum.
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| <!--
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| Constraints on the velocity system with <math>m</math> [[degrees of freedom (physics and chemistry)|degrees of freedom]] and n particles whose positions are designated with three dimensional vectors, <math>\lbrace \mathbf {r}_i \rbrace</math>, implies the existence of <math>3 n-m</math> scalar constraint equations on those position variables. Such a system can be fully described by the scalar generalized coordinates, <math>\lbrace q_1, q_2, ..., q_m\rbrace</math>, and the time, <math>t</math>, if and only if all <math>m</math> <math>\lbrace q_j \rbrace</math> are independent coordinates. For the system, the transformation from old coordinates to generalized coordinates may be represented as follows:<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|260}}
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| :<math>\mathbf{r}_1=\mathbf{r}_1(q_1, q_2, ..., q_m, t)</math>,
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| :<math>\mathbf{r}_2=\mathbf{r}_2(q_1, q_2, ..., q_m, t)</math>, ...
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| :<math>\mathbf{r}_n=\mathbf{r}_n(q_1, q_2, ..., q_m, t)</math>.
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| This transformation affords the flexibility in dealing with complex systems to use the most convenient and not necessarily [[inertial]] coordinates. These equations are used to construct differentials when considering [[virtual displacement]]s and [[generalized forces]].
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| -->
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| <!--These examples do not help explain generalized coordinates
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| ==Examples== | |
| === Double pendulum ===
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| [[File:Double-Pendulum.svg|thumb|right|A double pendulum]]
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| A [[double pendulum]] constrained to move in a plane may be described by the four [[Cartesian coordinates]] {''x''<sub>1</sub>, ''y''<sub>1</sub>, ''x''<sub>2</sub>, ''y''<sub>2</sub>}, but the system only has two [[degrees of freedom (mechanics)|degrees of freedom]], and a more efficient system would be to use
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| :<math>\lbrace q_1, q_2 \rbrace = \lbrace\theta_1,\theta_2 \rbrace</math>, | |
| which are defined via the following relations:
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| :<math>\lbrace x_1, y_1 \rbrace = \lbrace L_1\sin\theta_1, L_1\cos\theta_1 \rbrace</math>
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| :<math>\lbrace x_2, y_2 \rbrace = \lbrace L_1\sin\theta_1+L_2\sin\theta_2, L_1\cos\theta_1+L_2\cos\theta_2 \rbrace</math>
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| === Example: Bead on a wire ===
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| A bead constrained to move on a wire has only one degree of freedom, and the generalized coordinate used to describe its motion is often
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| :<math>q_1= l</math>,
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| where ''l'' is the distance along the wire from some reference point on the wire. Notice that a motion embedded in three dimensions has been reduced to only one dimension.
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| === Motion on a surface ===
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| A point mass constrained to a surface has two degrees of freedom, even though its motion is embedded in three dimensions. If the surface is a sphere, a good choice of coordinates would be:
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| :<math>\lbrace q_1, q_2 \rbrace = \lbrace \theta, \phi \rbrace </math>,
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| where θ and φ are the angle coordinates familiar from [[spherical coordinates]]. The ''r'' coordinate has been effectively dropped, as a particle moving on a sphere maintains a constant radius.
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| -->
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| <!--
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| ==Generalized velocities and kinetic energy==
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| Each generalized coordinate <math>q_i</math> is associated with a generalized velocity <math>\dot q_i</math>, defined as:
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| :<math>\dot q_i={dq_i \over dt}</math>
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| The kinetic energy of a particle is
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| :<math>T = \frac {m}{2} \left ( \dot x^2 + \dot y^2 + \dot z^2 \right )</math>.
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| In more general terms, for a system of <math>p</math> particles with <math>n</math> degrees of freedom, this may be written
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| :<math>T =\sum_{i=1} ^p \frac {m_i}{2} \left ( \dot x_i^2 + \dot y_i^2 + \dot z_i^2 \right )</math>.
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| If the transformation equations between the Cartesian and generalized coordinates
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| :<math>x_i = x_i \left (q_1, q_2, ..., q_n, t \right )</math> | |
| :<math>y_i = y_i \left (q_1, q_2, ..., q_n, t \right )</math> | |
| :<math>z_i = z_i \left (q_1, q_2, ..., q_n, t \right )</math>
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| are known, then these equations may be differentiated to provide the time-derivatives to use in the above kinetic energy equation:
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| :<math>\dot x_i = \frac {d}{dt} x_i \left (q_1, q_2, ..., q_n, t \right ).</math>
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| It is important to remember that the kinetic energy must be measured relative to inertial coordinates. If the above method is used, it means only that the Cartesian coordinates need to be [[inertial]], even though the generalized coordinates need not be. This is another considerable convenience of the use of generalized coordinates.
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| -->
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| <!--this section is vague. It has been replaced by the following section
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| ==Applications of generalized coordinates==
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| Such coordinates are helpful principally in [[Lagrangian mechanics]], where the forms of the principal equations describing the motion of the system are unchanged by a shift to generalized coordinates from any other coordinate system.
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| The amount of [[virtual work]] done along any coordinate <math>q_i</math> is given by:
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| :<math>\delta W_{q_i} = F_{q_i} \cdot \delta q_i </math>, </center>
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| where <math>F_{q_i}</math> is the generalized force in the <math>q_i</math> direction. While the generalized force is difficult to construct 'a priori', it may be quickly derived by determining the amount of work that would be done by all non-constraint forces if the system underwent a [[virtual displacement]] of <math>\delta q_i </math>, with all other generalized coordinates and time held fixed. This will take the form:
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| :<math>\delta W_{q_i} = f \left ( q_1, q_2, ..., q_n \right ) \cdot \delta q_i </math>,
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| and the generalized force may then be calculated:
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| :<math>F_{q_i} = \frac {\delta W_{q_i}}{\delta q_i} = f \left ( q_1, q_2, ..., q_n \right ) </math>.
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| -->
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| ==Generalized coordinates and virtual work==
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| The ''principle of virtual work'' states that if a system is in static equilibrium, the virtual work of the applied forces is zero for all virtual movements of the system from this state, that is, δW=0 for any variation δ'''r'''.<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref> 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 ''F''<sub>i</sub>=0.
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| Let the forces on the system be '''F'''<sub>j</sub>, ''j=1, ..., m'' be applied to points with Cartesian coordinates '''r'''<sub>j</sub>, j=1,..., m, then the virtual work generated by a virtual displacement from the equilibrium position is given by
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| :<math>\delta W = \sum_{j=1}^m \mathbf{F}_j\cdot \delta\mathbf{r}_j.</math>
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| where δ'''r'''<sub>j</sub>, ''j=1, ..., m'' denote the virtual displacements of each point in the body.
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| Now assume that each δ'''r'''<sub>j</sub> depends on the generalized coordinates ''q''<sub>i</sub>, ''i=1, ..., n'', then
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| :<math> \delta \mathbf{r}_j = \frac{\partial \mathbf{r}_j}{\partial q_1} \delta{q}_1 + \ldots + \frac{\partial \mathbf{r}_j}{\partial q_n} \delta{q}_n,</math>
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| and
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| :<math> \delta W = \left(\sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_1}\right) \delta{q}_1 + \ldots + \left(\sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_n}\right) \delta{q}_n. </math>
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| The ''n'' terms
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| :<math> F_i = \sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_i},\quad i=1,\ldots, n,</math>
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| are the generalized forces acting on the system. Kane<ref>T. R. Kane and D. A. Levinson, Dynamics: theory and applications, McGraw-Hill, New York, 1985</ref> shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,
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| :<math> F_i = \sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{v}_j}{\partial \dot{q}_i},\quad i=1,\ldots, n,</math>
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| where '''v'''<sub>j</sub> is the velocity of the point of application of the force '''F'''<sub>j</sub>.
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| In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is
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| :<math> \delta W = 0 \quad \Rightarrow \quad F_i =0, i=1,\ldots, n.</math>
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| ==See also==
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| *[[Hamiltonian mechanics]]
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| *[[Virtual work]]
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| *[[Orthogonal coordinates]]
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| *[[Curvilinear coordinates]]
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| * [[Frenet-Serret formulas]]
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| *[[Mass matrix]]
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| *[[Stiffness matrix]]
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| *[[Generalized forces]]
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| ==References==
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| <references/>
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| <!--
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| *{{cite book
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| | last = Greenwood
| |
| | first = Donald T.
| |
| | year = 1987
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| | title = Principles of Dynamics
| |
| | edition = 2nd edition
| |
| | publisher = Prentice Hall
| |
| | isbn = 0-13-709981-9
| |
| }}
| |
| *{{cite book
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| | last = Wells
| |
| | first = D. A.
| |
| | year = 1967
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| | title = Schaum's Outline of Lagrangian Dynamics
| |
| | location = New York
| |
| | publisher = McGraw-Hill
| |
| | |
| }}
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| -->
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| [[Category:Lagrangian mechanics| ]]
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| [[Category:Dynamical systems]]
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| [[Category:Rigid bodies]]
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