7
Basic Concepts: Degrees of Freedom
I Definition
N The minimum number of generalized coordinates required to
completely specify the configuration of a system
I
2 particles with distance constraint
N Constraint represents a surface in 6D space
N System is required to move on that surface in 6D space.The
constraint imposes a force along its gradient (normal), that
prohibits any motion along that direction
N This force is known as a Lagrange Multiplier (λ)
I
Generalization
N System characterized by n generalized coordinates q, and, m
constraints, Φ, has f = (n-m) dof
N It is possible to characterize system with (n-m) coordinates
N Or, with n coordinates q, and m variables λ, (n+m) variables!
I
Do the λ’s do any work?
8
Basic Concepts: Constraint Classification
I Holonomic Constraints: Φ(q,t) = 0
N Scleronomic Constraints: Φ(q) = 0
• No explicit dependence on time
• Constraint force λ does not do any work
N Rheonomic Constraints: Φ(q,t) = 0
• Explicit dependence on time
• Constraint force λ does work
I
Non-holonomic Constraints: ΣA
ij
(q) dq
j
+ A
it
(q) dt = 0
N Constraint can only be expressed in terms of differentials
N Cannot be integrated to a holonomic form
N Must be linear in differentials (Pfaffian form)
N Constraints on velocities! …
N Ball rolling without any slip along a curve
I
Do non-holonomic constraints decrease the DOF of a
system?
ij i it
A (q) q + A = 0
∑
&
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2012-09-11
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