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Lagrangian

Lagrangian. The general coordinate transformation to velocity. q m = q m ( x 1 , … , x 3 N , t ) x r i = x i ( q 1 , … , q f , t ) Velocity is considered independent of position. Differentials dq m do not depend on q m . Generalized Velocity. time fixed. time varying.

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Lagrangian

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  1. Lagrangian

  2. The general coordinate transformation to velocity. qm = qm(x1, … , x3N, t) xri = xi(q1, … , qf, t) Velocity is considered independent of position. Differentials dqm do not depend on qm. Generalized Velocity time fixed time varying general identity

  3. Force acting over a small displacement is the work. Transform the force to generalized coordinates. Rewrite in terms of the generalized force components, Qm, Qt Generalized Force

  4. Constraint Forces • All the Qm are applied forces. • No dependence on constraint coordinates • Not forces of constraint • Constraint forces do no work • Forces of constraint are often unknown. • Newtonian problem is complicated when they must be found.

  5. Conservative force derives from a potential V. Generalized force derives from the same potential. Conservative Forces

  6. Work Transformed A helpful identity

  7. Work Compared • Work in terms of the kinetic energy must equal the work in terms of the force. • Each component of the equation can be considered separately. trivial identity; ma=F

  8. Lagrangian Function • Purely conservative force • Depends only on position • If non-conservative force, leave on the right • Lagrangian: L = T-V • Lagrange’s equations • Second order dif eqn • For f equations, 2fconstants next

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