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hamilton_and_lag - NONLINEAR MECHANICAL SYSTEMS LAGRANGIAN...

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NONLINEAR MECHANICAL SYSTEMS L AGRANGIAN AND H AMILTONIAN FORMULATIONS Lagrangian formulation E k * (f,q) = 1 2 f t I(q) f q generalized coordinates (displacement) f generalized velocity (flow) E k * (f,q) kinetic co-energy I(q) a configuration-dependent inertia tensor (matrix) Note: kinetic co-energy is a quadratic form in flow (generalized velocity). (Euler-)Lagrange equation: d dt E k * f E k * q = e Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 1
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General form: I(q) df dt – C(f,q) = e e generalized force (effort) C(f,q) contains coriolis and centrifugal “forces” Explicit state-determined form: dq dt = f df dt = I(q) -1 ( ) e + C(f,q) Note: The inertia tensor (matrix) must be inverted to find a state- determined form. Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 2
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Lagrange’s equation may include conservative generalized forces. e = e conservative + e non-conservative e conservative = – Ep(q) q E p (q) potential energy function d dt E k * f E k * q + E p (q) q = e non-conservative Potential energy is not a function of generalized velocity. d dt L f L q = e non-conservative L(f,q) is the Lagrangian state function L(f,q) = E k * (f,q) – E p (q) In the usual notation, E k * (f,q) is written as T(f,q) and E p (q) is written as V(q), hence L(f,q) = T(f,q) – V(q) i.e., L = T – V Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 3
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Hamiltonian formulation Interaction between a capacitor and an inertia is the archetypal Hamiltonian system: dp/dt = – H(p,q)/ q dq/dt = H(p,q)/ p q
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