Hamiltonian equa-ons of mo-on and the extra conserved quan-ty for - PowerPoint PPT Presentation

Hamiltonian equa-ons of mo-on and the ‘extra’ conserved quan-ty for the Lanchester-Joukowski glider Charlie ________ Charles R. Doering Departments of Mathema/cs & Physics, and Center for the Study of Complex Systems University of Michigan, Ann Arbor MI 48109 USA

Hamiltonian equa-ons of mo-on and the ‘extra’ conserved quan-ty for the Lanchester-Joukowski glider Charlie ________ Charles R. Doering

Hamiltonian equa-ons of mo-on and the ‘extra’ conserved quan-ty for the Lanchester-Joukowski glider Charlie ________ Charles R. Doering Jane Wang Tony Bloch Vakhtang Poutkaradze Melvin Leok Dmitry Zenkov Phil Morrison and Darryl Dallas Holm himself!

Phugoid mo1on of a self-propelled or gliding aircra3 or underwater seacra3 is one of the basic modes of flight dynamics … The phugoid has a constant angle of a<ack with varying pitch due to repeated exchange of airspeed & alAtude: the vehicle periodically pitches up & climbs and subsequently pitches down & descends.

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F. W. Lanchester, Aerial Flight: Aerodone/cs (Constable, London, 1908) y = ƒ( V ) Lift V Drag mg cos θ θ θ mg mg sin θ x x = V cos θ ˙ y = V sin θ ˙ m d 2 x/dt 2 = − f ( V ) sin θ m d 2 y/dt 2 = f ( V ) cos θ − mg

F. W. Lanchester, Aerial Flight: Aerodone/cs (Constable, London, 1908) y = ƒ( V ) Lift V Drag mg cos θ θ θ mg mg sin θ x x = V cos θ ˙ y = V sin θ ˙ m d 2 x/dt 2 = − f ( V ) sin θ m d 2 y/dt 2 = f ( V ) cos θ − mg For stationary incompressible ideal irrotational flows when the circulation around the airfoil is proportional to the speed, Kutta-Jukowski ⇒ f ( V ) ∼ V 2 .

F. W. Lanchester, Aerial Flight: Aerodone/cs (Constable, London, 1908) y = ƒ( V ) Lift V Drag mg cos θ θ θ mg mg sin θ x The closed set of equations of motion for V and θ are ˙ V = − g sin θ (1) mV ˙ = f ( V ) − mg cos θ . (2) θ

F. W. Lanchester, Aerial Flight: Aerodone/cs (Constable, London, 1908) y = ƒ( V ) Lift V Drag mg cos θ θ θ mg mg sin θ x The closed set of equations of motion for V and θ are ˙ V = − g sin θ (1) mV ˙ = f ( V ) − mg cos θ . (2) θ The total mechanical energy is Z t E = 1 2 mV 2 + mgy = 1 2 mV ( t ) 2 + mg V ( t 0 ) sin ✓ ( t 0 ) dt 0 (3)

F. W. Lanchester, Aerial Flight: Aerodone/cs (Constable, London, 1908) y = ƒ( V ) Lift V Drag mg cos θ θ θ mg mg sin θ x The closed set of equations of motion for V and θ are ˙ V = − g sin θ (1) mV ˙ = f ( V ) − mg cos θ . (2) θ The total mechanical energy is Z t E = 1 2 mV 2 + mgy = 1 2 mV ( t ) 2 + mg V ( t 0 ) sin ✓ ( t 0 ) dt 0 (3) and (1) guarantees its conservation: dE ⇣ ⌘ ˙ dt = mV V + g sin ✓ = 0 . (4)

F. W. Lanchester, Aerial Flight: Aerodone/cs (Constable, London, 1908) y = ƒ( V ) Lift V Drag mg cos θ θ θ mg mg sin θ x The closed set of equations of motion for V and θ are ˙ V = − g sin θ (1) ⇣ ⌘ mV ˙ = f ( V ) − mg cos θ . (2) θ The curious ‘extra’ conserved quantity preserved by (1) & (2) is proportional to Z V f ( v ) dv − mgV cos θ (5)

F. W. Lanchester, Aerial Flight: Aerodone/cs (Constable, London, 1908) y = ƒ( V ) Lift V Drag mg cos θ θ θ mg mg sin θ x The closed set of equations of motion for V and θ are ˙ V = − g sin θ (1) ⇣ ⌘ mV ˙ = f ( V ) − mg cos θ . (2) θ The curious ‘extra’ conserved quantity preserved by (1) & (2) is proportional to Z V f ( v ) dv − mgV cos θ (5) = λ 3 V 3 − mgV cos θ when f ( V ) = λ V 2

˙ V = − g sin θ (1) mV ˙ = f ( V ) − mg cos θ . (2) θ . θ 2 π π 0 - π -2 π 0 V

Hamiltonian equa-ons of mo-on and the ‘extra’ conserved . quan-ty for the Lanchester-Joukowski glider

Hamiltonian equa-ons of mo-on and the ‘extra’ conserved . quan-ty for the Lanchester-Joukowski glider Nonlinear Stability and Control of Gliding Vehicles Pradeep Bhatta A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Mechanical and Aerospace Engineering (citing previous unpublished 2000 notes of Dong Chang and Naomi Leonard) September, 2006

. Point vortex-like Hamiltonian dynamics: q = ˙ x q = x = V cos θ ˙ p = − ˙ y = − V sin θ p = − ˙ y with the conserved quantity serving directly as the Hamiltonian, Z √ p 2 + q 2 H 0 ( p, q ) = 1 f ( v ) dv − g q m = λ 3 m ( p 2 + q 2 ) 3 / 2 − g q for f ( V ) = λ V 2 so that q = ∂ H 0 p = − ∂ H 0 ˙ and ˙ ∂ p ∂ q reproduce the original Cartesian equations of motion. … emo-onally distressing dynamical variables!

˙ V = − g sin θ (1) mV ˙ = f ( V ) − mg cos θ . (2) θ . Using the dynamical variables p ≡ V 2 q ≡ θ and the extra conserved is Z p f ( r 1 / 2 ) dr H 1 ( p, q ) = 1 − 2 gp 1 / 2 cos q r 1 / 2 m = 2 λ 3 mp 3 / 2 − 2 gp 1 / 2 cos q for f ( V ) = λ V 2

˙ V = − g sin θ (1) mV ˙ = f ( V ) − mg cos θ . (2) θ . Using the dynamical variables p ≡ V 2 q ≡ θ and the extra conserved is Z p f ( r 1 / 2 ) dr H 1 ( p, q ) = 1 − 2 gp 1 / 2 cos q r 1 / 2 m = 2 λ 3 mp 3 / 2 − 2 gp 1 / 2 cos q for f ( V ) = λ V 2 and upon changing variables equations (1) and (2) are p = − ∂ H 1 ˙ ∂ q = { p, H 1 } 1 q = ∂ H 1 ˙ ∂ p = { q, H 1 } 1 where { F, G } 1 = ∂ F ∂ G ∂ p − ∂ F ∂ G ∂ q is the old familiar Poisson bracket. ∂ q ∂ p

˙ V = − g sin θ (1) mV ˙ = f ( V ) − mg cos θ . (2) θ . Darryl noted that instead of the canonical Hamiltonian structure we can reconsider the original variables and q ≡ θ p ≡ V and the non-canonical Poisson bracket ✓ 1 ◆ ✓ ∂ G ◆ ✓ ∂ F ◆ ✓ 1 ◆ ∂ F ∂ G { F, G } 2 = − p ∂ q ∂ p ∂ p p ∂ q so that p = { p, H 2 } 2 ˙ and q = { q, H 2 } 2 ˙ with the conserved quantity serving directly as the Hamiltonian, Z p H 2 ( p, q ) = 1 f ( v ) dv − gp cos q m λ 3 mp 3 − g p cos q for f ( V ) = λ V 2 =

Hamiltonian equa-ons of mo-on and the ‘extra’ conserved quan-ty for the Lanchester-Joukowski glider • Okay … so what? What does it all mean? • Thermal glider ( e - ßH )? Quantum glider (e iHt )? • QuesAon for you all: is the ‘extra’ conserved quanAty the consequence of a symmetry? • If so, what symmetry? • Jane suggests that it is horizontal translaAon invariance ... but Darryl says “ It is what it is! ” • Anyway maybe this is a nice elementary physically moAvated model upon which to hone your tools!

Hamiltonian equa-ons of mo-on and the ‘extra’ conserved quan-ty for the Lanchester-Joukowski glider