ConstraintsNewtonian VariablesDegrees of FreedomRigid BodyTypes of ConstraintsGeneralized CoordinatesConfiguration SpacePendulum ConfigurationWinding ProblemConstraintsConstraintsNewtonian VariablesNewtonian VariablesEuclidean space Euclidean space EE33NNSystem of System of NN particles: particles: xxii•  = = 11, , NN• ii = = 1, 31, 3•33NN coordinatescoordinatesFxm0),( txiiiFxmMotion is specified by Motion is specified by second-order differential second-order differential equations.equations.•Initial positionInitial position•Initial velocityInitial velocity)0(ix)0(ixDegrees of FreedomDegrees of FreedomDynamical variables need Dynamical variables need not be Cartesian.not be Cartesian.Introduce holonomic Introduce holonomic constraints, constraints, qqjj..•kk < < 33NN•j j = 1= 1, k, k•ff = = 33NN – – kk•m m = 1= 1, f, fThe constraints reduce the The constraints reduce the number of degrees of number of degrees of freedom freedom ff..0),( tqj633fkN0)(,,,jijijiRrrRigid BodyRigid BodyA rigid body has no more than 6 degrees of freedom.A rigid body has no more than 6 degrees of freedom.•For three masses rigidly attached, For three masses rigidly attached, ff = 6= 6..•Assume Assume NN masses have masses have ff = 6= 6, so , so kk = 3= 3NN – 6 – 6..•Add one mass, three rigid attachments constrain it in space Add one mass, three rigid attachments constrain it in space to all others.to all others.•For For NN+1+1 masses, masses, k’k’ = 3= 3NN – 6 + 3 – 6 + 3..•ff = 3( = 3(NN+1) – +1) – k’k’ = 6 = 6..The constraint for the block The constraint for the block is moving but scleronomicis moving but scleronomicTypes of ConstraintsTypes of ConstraintsScleronomic constraints are Scleronomic constraints are time-independent.time-independent.•Static constraintsStatic constraints•Dynamic constraints if time Dynamic constraints if time is not explicit.is not explicit.Rheonomic constraints are Rheonomic constraints are time-dependent.time-dependent.•Explicit dependencyExplicit dependency1FmM1F2FXxGeneralized CoordinatesGeneralized CoordinatesA set of dynamical variables A set of dynamical variables used to describe the motion used to describe the motion are generalized coordinates.are generalized coordinates.•Some are used in Some are used in constraintsconstraintsA virtual displacement A virtual displacement represents an infinitessimal represents an infinitessimal change in coordinate.change in coordinate.mmiiqqxx),( txqqimmConfiguration SpaceConfiguration SpaceThe space of coordinates needed to describe the The space of coordinates needed to describe the system is the configuration space.system is the configuration space.•It is a manifold It is a manifold QQ..For For NN particles particles QQ can be as large as can be as large as EE33NN..•The number is reduced by constraints.The number is reduced by constraints.•Generalized coordinates often reflect Generalized coordinates often reflect QQ..Pendulum ConfigurationPendulum ConfigurationConfiguration Configuration QQ = Sphere = Sphere SS22•Conical pendulumConical pendulumQQ = Torus = Torus SS11  SS11•Double plane pendulumDouble plane pendulumWinding ProblemWinding ProblemRotation through 2Rotation through 2 can can result in restoration of result in restoration of position.position.Separate rotations don’t Separate rotations don’t generally add up.generally add up.Internal rotations may Internal rotations may require a different factor.require a different factor.+2a aaaa ba