Newtons equations in spaces of constant curvature Florin Diacu - PowerPoint PPT Presentation

Newton’s equations in spaces of constant curvature Florin Diacu Pacific Institute for the Mathematical Sciences and Department of Mathematics and Statistics University of Victoria CANADA Tor Vergata, Roma, Italia 12 May 2014 Florin Diacu The curved n -body problem

Goal: to present some results from F . Diacu. On the singularities of the curved n -body problem, Trans. Amer. Math. Soc. 363 , 4 (2011), 2249-2264. F . Diacu and E. Pérez-Chavela. Homographic solutions of the curved 3-body problem, J. Differential Equations 250 (2011), 340-366. F . Diacu. Polygonal homographic orbits of the curved n -body problem, Trans. Amer. Math. Soc. 364 , 5 (2012), 2783-2802. F . Diacu. Relative equilibria in the curved N -body problem , Atlantis Studies in Dynamical Systems, vol. I, Atlantis Press, 2012. F . Diacu. Relative equilibria in the 3-dimensional curved n -body problem, Memoirs Amer. Math. Soc. 228 , 1071 (2013), ISBN: 978-0-8218-9136-0. F . Diacu and S. Kordlou. Rotopulsating orbits of the curved N -body problem J. Differential Equations 255 (2013), 2709-2750. Florin Diacu The curved n -body problem

History of the problem 1830s Nikolai Lobachevsky and János Bolyai: 2-BP in H 3 1852 Lejeune Dirichlet: 2-BP in H 3 1860 Paul Joseph Serret: 2-BP in S 2 1870 Ernst Schering: 2-BP in H 3 1873 Rudolph Lipschitz: 2-BP in S 3 1885 Wilhelm Killing: 2-BP in H 3 1902 Heinrich Liebmann: 2-BP in S 2 and H 2 also proves an analogue of Bertrand’s theorem 1940 Erwin Schrödinger: quantum 2-BP in H 3 1945 Leopold Infeld and Alfred Schild: quantum 2-BP in H 3 1990s Russian school of celestial mechanics 2005 José Cariñena, Manuel Rañada, Mariano Santander: 2-BP in S 2 and H 2 Florin Diacu The curved n -body problem

Setting The space in which the motion of the bodies takes place is: κ = { ( w, x, y, z ) | w 2 + x 2 + y 2 + σz 2 = κ − 1 ( z > 0 if κ < 0) } , M 3 where σ is the signum function � +1 , for κ > 0 σ = − 1 , for κ < 0 Notice that 1 = S 3 and M 3 M 3 − 1 = H 3 Florin Diacu The curved n -body problem

Notations Consider m 1 , . . . , m n > 0 in R 4 for κ > 0 and M 3 , 1 (Minkowski space) for κ < 0 , with positions given by q i = ( w i , x i , y i , z i ) , i = 1 , n q = ( q 1 , . . . , q n ) is the configuration of the system ∇ q i := ( ∂ w i , ∂ x i , ∂ y i , σ∂ z i ) , ∇ := ( ∇ q 1 , . . . , ∇ q n ) is the gradient For a := ( a w , a x , a y , a z ) , b := ( b w , b x , b y , b z ) , a · b := ( a w b w + a x b x + a y b y + σa z b z ) is the inner product Florin Diacu The curved n -body problem

Potential For κ � = 0 , the force function is m i m j | κ | 1 / 2 κ q i · q j � U κ ( q ) = [ σ ( κ q i · q i )( κ q j · q j ) − σ ( κ q i · q j ) 2 ] 1 / 2 1 ≤ i<j ≤ n − U κ is the potential (a homogeneous function of degree 0). Euler’s formula for homogeneous functions: q i · ∇ q i U κ ( q ) = 0 , i = 1 , n. Florin Diacu The curved n -body problem

Equations of motion Using variational methods (constrained Lagrangian dynamics), we obtain the equations of motion: m i ¨ q i = ∇ q i U κ ( q ) − m i κ ( ˙ q i · ˙ q i ) q i , q i · q i = κ − 1 , q i · ˙ q i = 0 , κ � = 0 , i = 1 , n n m i m j | κ | 3 / 2 ( κ q j · q j )[( κ q i · q i ) q j − ( κ q i · q j ) q i ] � ∇ q i U κ ( q ) = , [ σ ( κ q i · q i )( κ q j · q j ) − σ ( κ q i · q j ) 2 ] 3 / 2 j =1 j � = i i = 1 , n Florin Diacu The curved n -body problem

Elimination of κ Coordinate and time-rescaling transformations q i = | κ | − 1 / 2 r i , i = 1 , n and τ = | κ | 3 / 4 t lead to the equations of motion n m j [ r j − σ ( r i · r j ) r i ] � r ′′ [ σ − σ ( r i · r j ) 2 ] 3 / 2 − σ ( r ′ i · r ′ i = i ) r i , i = 1 , n, j =1 ,j � = i where ′ = d dτ , r i · r i = | κ | q i · q i = | κ | κ − 1 = σ Florin Diacu The curved n -body problem

The positive case and the negative case Equations of motion in S 3 : n m j [ q j − ( q i · q j ) q i ] � ¨ q i = − ( ˙ q i · ˙ q i ) q i , [1 − ( q i · q j ) 2 ] 3 / 2 j =1 ,j � = i q i · q i = 1 , q i · ˙ q i = 0 , i = 1 , n Equations of motion in H 3 : n m j [ q j + ( q i · q j ) q i ] � q i = ¨ + ( ˙ q i · ˙ q i ) q i , [( q i · q j ) 2 − 1] 3 / 2 j =1 ,j � = i q i · q i = − 1 , q i · ˙ q i = 0 , i = 1 , n Florin Diacu The curved n -body problem

Hamiltonian form p := ( p 1 , . . . , p n ) , p i := m i ˙ q i , i = 1 , n, momenta � n T ( q , p ) = 1 i =1 m − 1 i ( p i · p i )( σ q i · q i ) , kinetic energy 2 H ( q , p ) = T ( q , p ) − U ( q ) , Hamiltonian function  q i = ∇ p i H ( q , p ) = m − 1 ˙ i p i ,   p i = −∇ q i H ( q , p ) = ∇ q i U ( q ) − σm − 1 ˙ i ( p i · p i ) q i ,  q i · q i = σ, q i · p i = 0 , i = 1 , n  Florin Diacu The curved n -body problem

The wedge product Consider the basis e w = (1 , 0 , 0 , 0) , e x = (0 , 1 , 0 , 0) , e y = (0 , 0 , 1 , 0) , e z = (0 , 0 , 0 , 1) The wedge product of u = ( u w , u x , u y , u z ) , v = ( v w , v x , v y , v z ) ∈ R 4 is defined as u ∧ v := ( u w v x − u x v w ) e w ∧ e x + ( u w v y − u y v w ) e w ∧ e y + ( u w v z − u z v w ) e w ∧ e z + ( u x v y − u y v x ) e x ∧ e y + ( u x v z − u z v x ) e x ∧ e z + ( u y v z − u z v y ) e y ∧ e z , where e w ∧ e x , e w ∧ e y , e w ∧ e z , e x ∧ e y , e x ∧ e z , e y ∧ e z represent the bivectors that form a canonical basis of the exterior Grassmann algebra over R 4 Florin Diacu The curved n -body problem

Integrals of the total angular momentum n � m i q i ∧ ˙ q i = c , i =1 where c = c wx e w ∧ e x + c wy e w ∧ e y + c wz e w ∧ e z + c xy e x ∧ e y + c xz e x ∧ e z + c yz e y ∧ e z , with the coefficients c wx , c wy , c wz , c xy , c xz , c yz ∈ R – on components, 6 integrals: n n � � m i ( w i ˙ x i − ˙ w i x i ) = c wx , m i ( w i ˙ y i − ˙ w i y i ) = c wy , i =1 i =1 n n � � m i ( w i ˙ z i − ˙ w i z i ) = c wz , m i ( x i ˙ y i − ˙ x i y i ) = c xy , i =1 i =1 n n � � m i ( x i ˙ z i − ˙ x i z i ) = c xz , m i ( y i ˙ z i − ˙ y i z i ) = c yz i =1 i =1 Florin Diacu The curved n -body problem

Isometries in S 3 In some suitable basis, rotations can be written as   cos θ − sin θ 0 0 sin θ cos θ 0 0   A =  , θ, φ ∈ [0 , 2 π )   0 0 cos φ − sin φ  0 0 sin φ cos φ – simple rotations (elliptic): lead to new solutions – double rotations (elliptic-elliptic): lead to new solutions Florin Diacu The curved n -body problem

Isometries in H 3 In some suitable basis, rotations can be written as   cos θ − sin θ 0 0 sin θ cos θ 0 0   B =  , θ ∈ [0 , 2 π ) , φ ∈ R ,   0 0 cosh φ sinh φ  0 0 sinh φ cosh φ – simple rotations (elliptic): lead to new solutions – simple rotations (hyperbolic): lead to new solutions – double rotations (elliptic-hyperbolic): lead to new solutions   1 0 0 0 0 1 − ξ ξ   C =  , ξ ∈ R .   1 − ξ 2 / 2 ξ 2 / 2 0 ξ  − ξ 2 / 2 1 + ξ 2 / 2 0 ξ – simple rotations (parabolic): lead to no solutions Florin Diacu The curved n -body problem

Relative equilibria (RE) in S 3 q = ( q 1 , q 2 , . . . , q n ) , q i = ( w i , x i , y i , z i ) , i = 1 , n,  w i ( t ) = r i cos( αt + a i )    x i ( t ) = r i sin( αt + a i )  [positive elliptic] : y i ( t ) = y i (constant)     z i ( t ) = z i (constant) , with w 2 i + x 2 i = r 2 i , r 2 i + y 2 i + z 2 i = 1 , i = 1 , n  w i ( t ) = r i cos( αt + a i )    x i ( t ) = r i sin( αt + a i )  [positive elliptic − elliptic] : y i ( t ) = ρ i cos( βt + b i )    z i ( t ) = ρ i sin( βt + b i ) ,  with w 2 i + x 2 i = r 2 i , y 2 i + z 2 i = ρ 2 i , r 2 i + ρ 2 i = 1 , i = 1 , n Florin Diacu The curved n -body problem

Relative equilibria (RE) in H 3  w i ( t ) = r i cos( αt + a i )    x i ( t ) = r i sin( αt + a i )  [negative elliptic] : y i ( t ) = y i (constant)    z i ( t ) = z i (constant) ,  with w 2 i + x 2 i = r 2 i , r 2 i + y 2 i − z 2 i = − 1 , i = 1 , n  w i ( t ) = w i (constant)    x i ( t ) = x i (constant)  [negative hyperbolic] : y i ( t ) = η i sinh( βt + b i )    z i ( t ) = η i cosh( βt + b i ) ,  with y 2 i − z 2 i = − η 2 i , w 2 i + x 2 i − η 2 i = − 1 , i = 1 , n  w i ( t ) = r i cos( αt + a i )    x i ( t ) = r i sin( αt + a i )  [negative elliptic − hyperbolic] : y i ( t ) = η i sinh( βt + b i )    z i ( t ) = η i cosh( βt + b i ) ,  with w 2 i + x 2 i = r 2 i , y 2 i − z 2 i = − η 2 i , so r 2 i − η 2 i = − 1 , i = 1 , n Florin Diacu The curved n -body problem