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
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
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.
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
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
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.
Configuration Q = Sphere S2 Conical pendulum Q = Torus S1S1 Double plane pendulum Pendulum Configuration
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