MECHANICAL NONLINEAR SYSTEMS LAGRANGIAN AND HAMILTONIAN FORMULATIONS Lagrangian formulation 1 Ek*(f,q) = 2 ft I(q) f q f generalized coordinates (displacement) generalized velocity (flow)
Ek*(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 Ek * Ek * dt f q = e
Mod. Sim. Dyn. Sys.
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General form: df I(q) 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 -1 dt = I(q) (e + C(f,q)) Note: The inertia tensor (matrix) must be inverted to find a statedetermined form.
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Lagranges equation may include conservative generalized forces. e = econservative + enon-conservative Ep(q) econservative = q Ep(q) potential energy function d Ek* Ek* Ep(q) dt f q + q = enon-conservative Potential energy is not a function of generalized velocity. d L L f q = enon-conservative dt L(f,q) is the Lagrangian state function L(f,q) = Ek*(f,q) Ep(q) In the usual notation, Ek*(f,q) is written as T(f,q) and Ep(q) is written as V(q), hence L(f,q) = T(f,q) V(q) i.e., L=TV
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Hamiltonian formulation Interaction between a capacitor and an inertia is the archetypal Hamiltonian system:
dp/dt = H(p,q)/q q p dq/dt = H(p,q)/p generalized coordinates generalized momenta H(p,q) is the Hamiltonian state function. H(p,q) = Ek(p,q) + Ep(q) In this case H(p,q) is equal to the total energy in the system.
Mod. Sim. Dyn. Sys.
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Lagrangian and Hamiltonian forms are related by a Legendre transformation. Lagrange used kinetic co-energy Hamilton used kinetic energy H(p,q) = ptf L(f,q) L(f,q) = ptf H(p,q) Differentiate with respect to generalized momentum. L(f,q)/p = 0 = f H(p,q)/p thus dq/dt = H(p,q)/p Generalized momentum is the gradient of kinetic co-energy with respect to flow (velocity) p = Ek*(f,q)/f = L(f,q)/f Lagrange's equation: dp/dt L(f,q)/q = e L(f,q)/q = H(p,q)/q thus dp/dt = H(p,q)/q + e These are Hamiltons equations dq/dt = H(p,q)/p dp/dt = H(p,q)/q + e
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EXAMPLE: MULTIPLE DEGREE OF FREEDOM MECHANISM Generalized coordinates: q Generalized momentum p = Ek*(f,q)/f = I(q) f Kinetic co-energy is a positive definite quadratic form,. The inertia tensor is symmetric and real. Kinetic energy 1 Ek(p,q) = ptf Ek*(f,q) = ptf 2 ftI(q)f 1 Ek(p,q) = ptI(q)-1p 2 ptI(q)-1I(q)I(q)-1p 1 Ek(p,q) = 2 ptI(q)-1p Note: Kinetic energy is always a quadratic form in generalized momentum.
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Hamiltons equations 1 dp/dt = [2 ptI(q)-1p]/q + e dq/dt = I(q)-1p Note: The Hamiltonian form may include arbitrary generalized forces (or torques) including dissipative or source terms. The most general form: dp/dt = H(p,q)/q + e(p,q,t) dq/dt = H(p,q)/p Sim. f(p,q,t)
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Hamiltons formulation applies to other energy domains EXAMPLE: SIMPLE ELECTRIC CIRCUIT.
The energy storage elements may be distinguished from the rest of the system:
Generalized coordinates: charge, q, and flux linkage, Hamiltonian: H(q,) = q2/2C + 2/2L Again, the Hamiltonian is the total system energy.
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Gradients: H/q = q/C H/ = /L Without the source or resistors, the conservative Hamiltonian equations would be eL = d/dt = H/q = q/C iC = dq/dt = H/ = /L With the source and resistors, the non-conservative Hamiltonian equations are d/dt = H/q + E0 eR2 dq/dt = H/ iR1
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CANONICAL TRANSFORMATIONS Choice of (state) variables is important in physical system modeling particularly important for nonlinear mechanical systems Geometry is fundamental. The choice of coordinates used to represent the geometry and kinematics of a system has a profound effect on the structure and complexity of its describing equations. Transformations of state variables are used extensively to analyze linear state determined systems. e.g., physical variables to diagonal form Structure of the state equations is preserved while mathematical convenience is gained e.g., in diagonalized form each equation is decoupled from the rest facilitates analysis, e.g. modal analysis proportionality between the rate and state vectors is preserved
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DOES THIS APPLY TO THE NONLINEAR CASE? The Hamiltonian form permits an analogous approach using canonical transformations. Any change of variables that preserves the value of the Hamiltonian preserves the structure of the Hamiltons equations is a canonical transformation.
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EXAMPLE: The old displacements may be functions of both the new displacements and/or the new momenta. A simple canonical transformation p = q* q = p* old equations (conservative terms only) dp/dt = H(p,q)/q dq/dt = H(p,q)/p define H*(p*,q*) = H(p,q) the value of the Hamiltonian is preserved H*/p* = (H/q) (q/p*) = H/q H*/q* = (H/p) (p/q*) = H/p time differentiate the new coordinates dq*/dt = dp/dt = H/q = H*/p* dp*/dt = dq/dt = H(p,q)/p = H*/q* new equations (conservative terms only) dq*/dt = H*/p* dp*/dt = H*/q* the structure of Hamiltons equations is preserved
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new equations with non-conservative terms dp*/dt = H(p*,q*)/q* + e*(p*,q*,t) dq*/dt = H(p*,q*)/p* f*(p*,q*,t) The forcing terms must be transformed as follows fk* = (ejpj/pk* + ejqj/pk*) j ek* = (fjpj/qk* + ejqj/qk*) j NOTE: Displacement and momentum may be exchanged canonical transformation may destroy the physical meaning of variables Parallel to transformation of a linear system to decoupled form the original real-valued physical variables become complex valued
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