Analytical Mechanics: Variational Principles
Analytical Mechanics: Variational PrinciplesShinichi HiraiDept. Robotics, Ritsumeikan Univ.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles1 / 69
Agenda1Variational Principle in Statics2Variational Principle in Statics under Constraints3Variational Principle in Dynamics4Variational Principle in Dynamics under ConstraintsShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles2 / 69
StaticsVariation principle in staticsminimize I U Wunder constraintminimize I U Wsubject to R 0Solutionsanalytically solve δI 0numerical optimization (fminbnd or fmincon)Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles3 / 69
Example (simple pendulum)CθlymOxλmgsimple pendulum of length l and mass m suspended at point Cτ : external torque around C, θ: angle around CGiven τ , derive θ at equilibrium.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles4 / 69
Statics in variational formUWpotential energywork done by external forces/torquesVariational principle in staticsInternal energy I U W reaches to its minimum at equilibrium:I U W minimumShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles5 / 69
Statics in variational formSolutions:1Solveminimize I U Wanalytically2Solveminimize I U Wnumerically3SolveδI 0analyticallyShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles6 / 69
Example (simple pendulum)CθlymOxλmgU mgl(1 cos θ), W τ θI mgl(1 cos θ) τ θShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles7 / 69
Example (simple pendulum)Solveminimize I mgl(1 cos θ) τ θanalytically I mgl sin θ τ 0 θEquilibrium of moment around CShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles8 / 69
Example (simple pendulum)Solveminimize I mgl(1 cos θ) τ θ( π θ π)numerically Apply fminbnd to minimize a function numericallyShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles9 / 69
Example (simple pendulum)Sample Programsminimizing internal energyinternal energy of simple pendulmShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles10 / 69
Example (simple pendulum)Result internal energy simple pendulum minthetamin 0.5354Imin -0.0261Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles11 / 69
Example (simple pendulum)SolveδI 0analytically I mgl(1 cos θ) τ θI δI mgl(1 cos(θ δθ)) τ (θ δθ)Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles12 / 69
Example (simple pendulum)Note that cos(θ δθ) cos θ (sin θ)δθ:I mgl(1 cos θ) τ θI δI mgl(1 cos θ (sin θ)δθ) τ (θ δθ) δI mgl(sin θ)δθ τ δθ (mgl sin θ τ )δθ 0, δθ mgl sin θ τ 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles13 / 69
Example (pendulum in Cartesian coordinates)CR 0R 0lR 0ymOxλmgsimple pendulum of length l and mass m suspended at point C[ x, y ]T : position of mass[ fx , fy ]T : external force applied to massGiven [ fx , fy ]T , derive [ x, y ]T at equilibrium.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles14 / 69
Example (pendulum in Cartesian coordinates)CR 0R 0lR 0ymOxλmggeometric constraintdistance between C and mass l}1/2 {R x 2 (y l)2 l 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles15 / 69
Statics under single constraintUWRpotential energywork done by external forces/torquesgeometric constraintVariational principle in staticsInternal energy U W reaches to its minimum at equilibrium undergeometric constraint R 0:minimize U Wsubject to R 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles16 / 69
Statics under single constraintSolutions:1Solveminimize U Wsubject to R 0analytically2Solveminimize U Wsubject to R 0numericallyShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles17 / 69
Statics under single constraintSolveminimize U Wsubject to R 0analytically minimize I U W λRλ: Lagrange’s multiplier δI δ(U W λR) 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles18 / 69
Example (pendulum in Cartesian coordinates)CR 0R 0lR 0ymOxλmgU mgy , W fx x fy y{}1/2 lR x 2 (y l)2Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles19 / 69
Example (pendulum in Cartesian coordinates)I mgy (fx x fy y ) λ[{]}1/2x 2 (y l)2 lNote that δR Rx δx Ry δy , where{} 1/2 R x x 2 (y l)2 x{} 1/2 R (y l) x 2 (y l)2Ry y Rx δI mg δy fx δx fy δy λRx δx λRy δy ( fx λRx )δx (mg fy λRy )δy 0,Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles δx, δy20 / 69
Example (pendulum in Cartesian coordinates) fx λRx 0mg fy λRy 0[[0 mg0 mg]][ [grav. force,fxfyfxfy] λ[RxRy]][ [ext. force,RxλRy00]]constraint forcegradient vector ( to R 0)Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles21 / 69
Example (pendulum in Cartesian coordinates)three equations w.r.t. three unknowns x, y , and λ: fx λRx 0mg fy λRy 0R 0 we can determine position of mass [ x, y ]T and magnitude ofconstraint force λShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles22 / 69
Example (pendulum in Cartesian coordinates)NoteI U W λR U (W λR)CλRR -1θR 0lR 1constraint force contour R constantR 2ymOxλmgW λRmagnitude of a constraint forcedistance along the forceλRwork done by a constraint forcework done by external & constraint forcesShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles23 / 69
Statics under single constraintSolveminimize I U Wsubject to R 0numerically Apply fmincon to minimize a function numerically under constraintsNote: ”Optimization Toolbox” is needed to use fminconShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles24 / 69
Example (pendulum in Cartesian coordinates)Sample Programsminimizing internal energy (Cartesian)internal energy of simple pendulm (Cartesian)constraintsShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles25 / 69
Example (pendulum in Cartesian coordinates)Result: internal energy pendulum Cartesian minLocal minimum found that satisfies the constraints. stopping criteria details qmin 1.40013.4281Imin -0.4897Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles26 / 69
Statics under multiple constraintsUWR1 , R2potential energywork done by external forces/torquesgeometric constraintsVariational principle in staticsInternal energy U W reaches to its minimum at equilibrium undergeometric constraints R1 0 and R2 0:minimize U Wsubject to R1 0,R2 0δI δ(U W λ1 R1 λ2 R2 ) 0λ1 , λ2 : Lagrange’s multipliersShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles27 / 69
DynamicsLagrangianL T U WL T U W λR(under constraint)Lagrange equations of motion Ld L 0 qdt q̇Solutionsnumerical ODE solver (ode45)constraint stabilization method (CSM)Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles28 / 69
Example (simple pendulum)CθlymOxλmgsimple pendulum of length l and mass m suspended at point Cτ : external torque around C at time t, θ: angle around C at time tDerive the motion of the pendulum.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles29 / 69
Dynamics in variational formTUWkinetic energypotential energywork done by external forces/torquesLagrangianL T U WLagrange equation of motion Ld L 0 θdt θ̇Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles30 / 69
Example (simple pendulum)CθlymOxλmg1T (ml 2 )θ̇22U mgl(1 cos θ),W τθShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles31 / 69
Example (simple pendulum)Lagrangian1L (ml 2 )θ̇2 mgl(1 cos θ) τ θ2partial derivatives L mgl sin θ τ, θ L (ml 2 )θ̇ θ̇d L ml 2 θ̈dt θ̇Lagrange equation of motion mgl sin θ τ ml 2 θ̈ 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles32 / 69
Example (simple pendulum)Equation of the pendulum motionml 2 θ̈ mgl sin θ τ Canonical form of ordinary differential equationθ̇ ω1ω̇ 2 (τ mgl sin θ)mlcan be solved numerically by an ODE solverShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles33 / 69
Example (simple pendulum)Sample Programssolve the equation of motion of simple pendulumequation of motion of simple pendulumexternal torqueShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles34 / 69
Example (simple pendulum)ResultShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles35 / 69
Example (simple pendulum)ResultShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles35 / 69
Example (simple pendulum)ResultShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles35 / 69
Example (pendulum with viscous friction)Assumptionsviscous friction around supporting point C worksviscous friction causes a negative torque around Cmagnitude of the torque is proportional to angular velocityviscous friction torque b θ̇(b: positive constant)Replacing τ by τ b θ̇:(ml 2 )θ̈ (τ b θ̇) mgl sin θ θ̇ ω1ω̇ 2 (τ bω mgl sin θ)mlShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles36 / 69
Example (pendulum with viscous friction)Sample Programssolve the equation of motion of damped pendulumequation of motion of damped pendulumexternal torqueShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles37 / 69
Example (pendulum with viscous friction)ResultShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles38 / 69
Example (pendulum with viscous friction)ResultShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles38 / 69
Example (pendulum with viscous friction)ResultShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles38 / 69
Example (pendulum in Cartesian coordinates)CR 0R 0lR 0ymOxλmgsimple pendulum of length l and mass m suspended at point C[ x, y ]T : position of mass at time t[ fx , fy ]T : external force applied to mass at time tDerive the motion of the pendulum in Cartesian coordinates.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles39 / 69
Example (pendulum in Cartesian coordinates)CR 0R 0lR 0ymOxλmggeometric constraintdistance between C and mass l}1/2 {R x 2 (y l)2 l 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles40 / 69
Dynamics under single constraintTUWkinetic energypotential energywork done by external forces/torquesLagrangianL T U W λRLagrange equations of motiond L L 0 xdt ẋ Ld L 0 ydt ẏShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles41 / 69
Example (pendulum in Cartesian coordinates)CθlymOxλmg1T m{ẋ 2 ẏ 2 }2U mgy , W fx x fy y{}1/2 lR x 2 (y l)2Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles42 / 69
Example (pendulum in Cartesian coordinates)Lagrangian[{]}1/21 lL m{ẋ 2 ẏ 2 } mgy fx x fy y λ x 2 (y l)22partial derivatives L fx λRx , x L mg fy λRy , y L mẋ ẋ L mẏ ẏLagrange equations of motionfx λRx mẍ 0 mg fy λRy mÿ 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles43 / 69
Example (pendulum in Cartesian coordinates)Lagrange equations of motion[] [][] {[ ]} [ ]0fxRxẍ0 λ m mgfyRyÿ0gravitational external constraintinertialdynamic equilibrium among forcesShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles44 / 69
Example (pendulum in Cartesian coordinates)three equations w.r.t. three unknowns x, y , and λ:mẍ fx λRxmÿ mg fy λRyR 0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles45 / 69
Example (pendulum in Cartesian coordinates)three equations w.r.t. three unknowns x, y , and λ:mẍ fx λRxmÿ mg fy λRyR 0Mixture of differential and algebraic equations Difficult to solve the mixture of differential and algebraic equationsShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles45 / 69
Constraint stabilization method (CSM)Constraint stabilizationconvert algebraic eq. to its almost equivalent differential eq.algebraic eq. R 0 differential eq. R̈ 2αṘ α2 R 0(α: large positive constant)critical damping (converges to zero most quickly)Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles46 / 69
Constraint stabilization method (CSM)Dynamic equation of motion under geometric constraint: Ld L 0 qdt q̇algebraic eq. R 0differential eq. Ld L 0 qdt q̇differential eq. R̈ 2αṘ α2 R 0differential eq.can be solved numerically by an ODE solver.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles47 / 69
Computing equation for constraint stabilizationAssume R depends on x and y : R(x, y ) 0Differentiating R(x, y ) w.r.t time t:Ṙ R dx R dy Rx ẋ Ry ẏ x dt y dtDifferentiating Rx (x, y ) and Ry (x, y ) w.r.t time t: Rx x RyṘy xṘx dx Rx dy Rxx ẋ Rxy ẏdt y dtdx Ry dy Ryx ẋ Ryy ẏdt y dtSecond order time derivative:R̈ (Ṙx ẋ Rx ẍ) (Ṙy ẏ Ry ÿ ) (Rxx ẋ Rxy ẏ )ẋ Rx ẍ (Ryx ẋ Ryy ẏ )ẏ Ry ÿShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles48 / 69
Computing equation for constraint stabilizationSecond order time derivative:[ ][][ ][] ẍ[] Rxx RxyẋR̈ Rx Ry ẋ ẏÿRyx RyyẏEquation to stabilize constraint:[ ][][ ][] ẍ] Rxx Rxy[ẋ Rx Ry ẋ ẏÿRyx Ryyẏ 2α(Rx ẋ Ry ẏ ) α2 R [ Rx Ry][v̇xv̇y vx ẋ, vy ẏ] [vx vy][Rxx RxyRyx Ryy][vxvy] 2α(Rx vx Ry vy ) α2 RShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles49 / 69
Example (pendulum in Cartesian coordinates)Equation for stabilizing constraint R(x, y ) 0: Rx v̇x Ry v̇y C (x, y , vx , vy )whereC (x, y , vx , vy ) [vx vy][Rxx RxyRyx Ryy][vxvy] 2α(Rx vx Ry vy ) α2 RandP {x 2 (y l)2 } 1/2 ,Rx xP,Rxx P x 2 P 3 ,Ry (y l)PRyy P (y l)2 P 3Rxy Ryx x(y 1)P 3Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles50 / 69
Example (pendulum in Cartesian coordinates)Combining equations of motion and equation for constraintstabilization: ẋ vxẏ vy m Rxm Ry Rxv̇xfx Ry v̇y mg fy λC (x, y , vx , vy )five equations w.r.t. five unknown variables x, y , vx , vy and λgiven x, y , vx , vy ẋ, ẏ , v̇x , v̇yThis canonical ODE can be solved numerically by an ODE solver.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles51 / 69
Example (pendulum in Cartesian coordinates)Sample Programssolve the equation of motion of simple pendulum (Cartesian)equation of motion of simple pendulum (Cartesian)Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles52 / 69
Example (pendulum in Cartesian coordinates)t–x, yShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles53 / 69
Example (pendulum in Cartesian coordinates)t–vx , vyShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles53 / 69
Example (pendulum in Cartesian coordinates)x–yShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles53 / 69
Example (pendulum in Cartesian coordinates)t–computed θShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles53 / 69
Example (pendulum in Cartesian coordinates)t–constraint RShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles53 / 69
NoticeLagrangianL T U W λR T (U W λR) T ILagrangian is equal to the difference between kinetic energy andinternal energy under a constraintShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles54 / 69
SummaryVariational principlesstatics I U Wstatics under constraint I U W λRδI 0dynamics L T U Wdynamics under constraint L T U W λR Ld L 0 qdt q̇constraint stabilization methodShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles55 / 69
SummaryHow to solve a static problemSolve (nonlinear) equations originated from variationorNumerically minimize internal energyHow to solve a dynamic problemStep 1 Derive Lagrange equations of motion analyticallyStep 2 Solve the derived equations numericallyShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles56 / 69
ReportReport # 1due date : Nov. 2 (Mon)Simulate the dynamic motion of a pendulum under viscous frictiondescribed with Cartesian coordinates x and y . Apply constraintstabilization method to convert the constraint into its almostequivalent ODE, then apply any ODE solver to solve a set of ODEs(equations of motion and equation for constraint stabilization)numerically.Submit your report in pdf format to manaba RFile name shoud be:student number (11 digits) your name (without space).pdfFor example 12345678901HiraiShinichi.pdfShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles57 / 69
Appendix: Variational calculusSmall virtual deviation of variables or functions.y x2Let us change variable x to x δx, then variable y changes to y δyaccordingly.y δy (x δx)2 x 2 2x δx (δx)2 x 2 2x δxThusδy 2x δxShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles58 / 69
Appendix: Variational calculusSmall virtual deviation of variables or functions. TI {x(t)}2 dt0Let us change function x(t) to x(t) δx(t), then variable I changesto I δI accordingly. TI δI {x(t) δx(t)}2 dt0 T {x(t)}2 2x(t) δx(t) dt0Thus δI T2x(t) δx(t) dt0Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles59 / 69
Appendix: Variational calculusVariational operator δδθδf (θ)virtual deviation of variable θvirtual deviation of function f (θ)δf (θ) f ′ (θ)δθvirtual increment of variableincrement of functionθ θ δθf (θ) f (θ δθ) f (θ) f ′ (θ)δθf (θ) f (θ) δf (θ)simple examplesδ(5x) 5 δx δx 2 2x δxδ sin θ (cos θ) δθ,δ cos θ ( sin θ) δθShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles60 / 69
Appendix: Variational calculusVariational operator δδθδf (θ)virtual deviation of variable θvirtual deviation of function f (θ)δf (θ) f ′ (θ)δθvirtual increment of variableincrement of functionθ θ δθf (θ) f (θ δθ) f (θ) f ′ (θ)δθf (θ) f (θ) δf (θ)simple examplesδ(5x) 5 δx δx 2 2x δxδ sin θ (cos θ) δθ,δ cos θ ( sin θ) δθShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles60 / 69
Appendix: Variational calculusassume that θ depends on time tvirtual increment of function θ(t) θ(t) δθ(t)dθddθd (θ δθ) δθdt dt dt dt θ dt (θ δθ) dt θ dt δθ dtvariation of derivative and integraldθd δθ dt dtδ θ dt δθ dtδvariational operator and differential/integral operator can commuteShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles61 / 69
Appendix: Lagrange multiplier methodconverts minimization (maximization) under conditions intominimization (maximization) without conditions.minimize f (x)subject to g (x) 0 minimize I (x, λ) f (x) λg (x) f g I λ 0 x x x I g (x) 0 λShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles62 / 69
Appendix: Lagrange multiplier method (example)Length of each edge of a cube is given by x, y , and z.Determine x, y , and z that minimizes the surface of the cube whenthe cube volume is constantly specified by a3 :minimize S(x, y , z) 2xy 2yz 2zx subject to R(x, y , z) xyz a3 0Introducing Lagrange multiplier λ, the above conditional minimizationcan be converted into the following unconditional minimization:minimize I (x, y , z, λ) S(x, y , z) λR(x, y , z) 2xy 2yz 2zx λ(xyz a3 )Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles63 / 69
Appendix: Lagrange multiplier method (example)Calculating partial derivatives: I x I y I z I λ 2y 2z λyz 0(1) 2z 2x λzx 0(2) 2x 2y λxy 0(3) xyz a3 0(4)Calculating (1) · x (2) · y , we havez(x y ) 0,which directly yields x y . Similarly, we have y z and z x.Consequently, we concludes x y z a.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles64 / 69
Appendix: ODE solverLet us solve van del Pol equation:ẍ 2(1 x 2 )ẋ x 0Canonical form:ẋ vv̇ 2(1 x 2 )ẋ x[State variable vector:q xv]Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles65 / 69
Appendix: ODE solver (MATLAB)File van der Pol.m describes the canonical form:function dotq van der Pol (t,q)x q(1);v q(2);dotx v;dotv 2*(1-x 2)*v - x;dotq [dotx; dotv];endFile name van der Pol should conincide with function namevan der Pol.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles66 / 69
Appendix: ODE solver (MATLAB)File van der Pol solve.m solves van der Pol equation numerically:timestep 0.00:0.10:10.00;qinit [2.00;0.00];[time,q] ode45(@van der Pol,timestep,qinit);% line stylesolid broken -.chain -plot(time,q(:,1),’-’, time,q(:,2),’-.’);Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principlesdotted :67 / 69
Appendix: ODE solver (MATLAB) timetime 00.10000.20000.30000.4000 qq 22-0.3125The first and second columns corresponds to x and v .Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles68 / 69
Appendix: ODE solver (MATLAB)Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles69 / 69
Agenda 1 Variational Principle in Statics 2 Variational Principle in Statics under Constraints 3 Variational Principle in Dynamics 4 Variational Principle in Dynamics under Constraints Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles 2 / 69
II. VARIATIONAL PRINCIPLES IN CONTINUUM MECHANICS 4. Introduction 12 5. The Self-Adjointness Condition of Vainberg 18 6. A Variational Formulation of In viscid Fluid Mechanics . . 25 7. Variational Principles for Ross by Waves in a Shallow Basin and in the "13-P.lane" Model . 37 8. The Variational Formulation of a Plasma . 9.
Variational Form of a Continuum Mechanics Problem REMARK 1 The local or strong governing equations of the continuum mechanics are the Euler-Lagrange equation and natural boundary conditions. REMARK 2 The fundamental theorem of variational calculus guarantees that the solution given by the variational principle and the one given by the local
Action principles in Lagrangian/Hamiltonian formulations of electrodynamics Schwinger variational principles for transmission lines, waveguides, scattering specialized variational principles for lasers and undulators (e.g. Xie) Variational Principles are Perhaps Better Known in
of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected recent developments in hybrid and nonlinear variational principles. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations and natural boundary conditions.
On Arnold’s variational principles in fluid mechanics 5 where fi(x) and fl(x) are scalar functions, which, in the case of singly-connected domain D, are uniquely determined by the conditions r –u r –2u 0 in D; –u n –2u n 0 on @D: Variational principle. Now we shall show that the first variation of
Variational methods and complementary formulations in dynamics. Vol. 31. Springer Science & Business Media, 2013. (first edition 1994) Lanczos, Cornelius. The variational principles of mechanics. Courier Corporation, 2012. (first edition from 1949)
Stochastic Variational Inference. We develop a scal-able inference method for our model based on stochas-tic variational inference (SVI) (Hoffman et al., 2013), which combines variational inference with stochastic gra-dient estimation. Two key ingredients of our infer
for use in animal nutrition. Regulation (EC) No 178/2002 laying down the general principles and requirements of food law. Directive 2002/32/EC on undesirable substances in animal feed. Directive 82/475/EEC laying down the categories of feed materials which may be used for the purposes of labelling feedingstuffs for pet animals The Animal Feed (Hygiene, Sampling etc and Enforcement) (England .
