1 / 9

Constraints

Constraints. Euclidean space E 3 N System of N particles: x r i r = 1 , N i = 1, 3 3 N coordinates. Motion is specified by second-order differential equations. Initial position Initial velocity. Newtonian Variables. Dynamical variables need not be Cartesian.

audra-lyons
Download Presentation

Constraints

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Constraints

  2. Euclidean space E3N System of N particles: xri r = 1, N i = 1, 3 3Ncoordinates Motion is specified by second-order differential equations. Initial position Initial velocity Newtonian Variables

  3. Dynamical variables need not be Cartesian. Introduce holonomic constraints, qj. k < 3N j = 1, k f = 3N – k m = 1, f The constraints reduce the number of degrees of freedom f. Degrees of Freedom

  4. Rigid Body • A rigid body has no more than 6 degrees of freedom. • For three masses rigidly attached, f= 6. • Assume N masses have f= 6, so k= 3N – 6. • Add one mass, three rigid attachments constrain it in space to all others. • For N+1 masses, k’= 3N – 6 + 3. • f = 3(N+1) – k’ = 6.

  5. The constraint for the block is moving but scleronomic Scleronomic constraints are time-independent. Static constraints Dynamic constraints if time is not explicit. Rheonomic constraints are time-dependent. Explicit dependency m Types of Constraints x X M q

  6. A set of dynamical variables used to describe the motion are generalized coordinates. Some are used in constraints A virtual displacement represents an infinitessimal change in coordinate. Generalized Coordinates

  7. Configuration Space • The space of coordinates needed to describe the system is the configuration space. • It is a manifold Q. • For N particles Q can be as large as E3N. • The number is reduced by constraints. • Generalized coordinates often reflect Q.

  8. Configuration Q = Sphere S2 Conical pendulum Q = Torus S1S1 Double plane pendulum Pendulum Configuration

  9. b b Winding Problem • Rotation through 2p can result in restoration of position. • Separate rotations don’t generally add up. • Internal rotations may require a different factor. 2p a a p p + a a a 2p + 2p a next

More Related