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0. review 21 Sep HW
A. Review: constraints
1. Restrictions on the
N
particle motion so that there are fewer than
n
= 3
N
independent degrees of freedom.
2. There are forces present that maintain the constraints, but are speciﬁed
indirectly by their eﬀect.
3. The goal is to construct a formalism for system with
k
constraints that
has
n

k
degrees of freedom and no explicit constraining forces.
4. Notation: Roman subscript
j
for the original
n
Cartesian coordinates;
Greek subscript
σ
for the generalized degrees of freedom:
x
j
=
x
j
(
{
q
σ
}
,t
)
[This has an implicit restriction to holonomic (equality) constraints.]
5. Diﬀerential forms:
dx
j
=
n

k
X
σ
=1
∂x
j
∂q
σ
dq
σ
+
∂x
j
∂t
dt
˙
x
j
=
n

k
X
σ
=1
∂x
j
∂q
σ
˙
q
σ
+
∂x
j
∂t
= ˙
x
j
(
{
q
σ
}
,
{
˙
q
σ
}
,t
)
∂v
j
∂
˙
q
σ
=
∂
˙
x
j
∂
˙
q
σ
=
∂x
i
∂q
j
6. Virtual displacements:
δxj
is inﬁnitesimal, instantaneous (
δt
= 0) dis
placements consistent with the constraints.
δx
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This note was uploaded on 09/29/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at Wisconsin.
 Fall '09
 BRUCH
 Force

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