P09fl9 - Lecture 9 Lagrange I(23 Sep 09 0 review 21 Sep HW A Review constraints 1 Restrictions on the N-particle motion so that there are fewer

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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.

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P09fl9 - Lecture 9 Lagrange I(23 Sep 09 0 review 21 Sep HW A Review constraints 1 Restrictions on the N-particle motion so that there are fewer

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