Two classical examples of integrable systems Florian Beck 37th day in quarantine String Math Seminar (Summer 2020) Recap Equations of motion in Darboux coordinates (pi ; qj ) @H @H q_j = ; p_i = − : @pj @qi The Hamiltonian system ((M;!); H) is called Liouville/completely integrable if there are n independent, Poisson commuting conserved quantities F1;:::; Fn, fH; Fj g = 0. Liouville theorem: let Mf be a level set of n F := (F1;:::; Fn): M ! R : In Darboux coordinates around m 2 Mf define the action Z q X S(F ; q) := pi (F ; q)dqi ; pi = pi (F ; q): q0 i Recap (I) Phase space of a mechanical / Hamiltonian system is a symplectic 2n manifold (M ;!) with Hamiltonian H : M ! R. 1 @H @H q_j = ; p_i = − : @pj @qi The Hamiltonian system ((M;!); H) is called Liouville/completely integrable if there are n independent, Poisson commuting conserved quantities F1;:::; Fn, fH; Fj g = 0. Liouville theorem: let Mf be a level set of n F := (F1;:::; Fn): M ! R : In Darboux coordinates around m 2 Mf define the action Z q X S(F ; q) := pi (F ; q)dqi ; pi = pi (F ; q): q0 i Recap (I) Phase space of a mechanical / Hamiltonian system is a symplectic 2n manifold (M ;!) with Hamiltonian H : M ! R. Equations of motion in Darboux coordinates (pi ; qj ) 1 The Hamiltonian system ((M;!); H) is called Liouville/completely integrable if there are n independent, Poisson commuting conserved quantities F1;:::; Fn, fH; Fj g = 0. Liouville theorem: let Mf be a level set of n F := (F1;:::; Fn): M ! R : In Darboux coordinates around m 2 Mf define the action Z q X S(F ; q) := pi (F ; q)dqi ; pi = pi (F ; q): q0 i Recap (I) Phase space of a mechanical / Hamiltonian system is a symplectic 2n manifold (M ;!) with Hamiltonian H : M ! R. Equations of motion in Darboux coordinates (pi ; qj ) @H @H q_j = ; p_i = − : @pj @qi 1 Liouville theorem: let Mf be a level set of n F := (F1;:::; Fn): M ! R : In Darboux coordinates around m 2 Mf define the action Z q X S(F ; q) := pi (F ; q)dqi ; pi = pi (F ; q): q0 i Recap (I) Phase space of a mechanical / Hamiltonian system is a symplectic 2n manifold (M ;!) with Hamiltonian H : M ! R. Equations of motion in Darboux coordinates (pi ; qj ) @H @H q_j = ; p_i = − : @pj @qi The Hamiltonian system ((M;!); H) is called Liouville/completely integrable if there are n independent, Poisson commuting conserved quantities F1;:::; Fn, fH; Fj g = 0. 1 In Darboux coordinates around m 2 Mf define the action Z q X S(F ; q) := pi (F ; q)dqi ; pi = pi (F ; q): q0 i Recap (I) Phase space of a mechanical / Hamiltonian system is a symplectic 2n manifold (M ;!) with Hamiltonian H : M ! R. Equations of motion in Darboux coordinates (pi ; qj ) @H @H q_j = ; p_i = − : @pj @qi The Hamiltonian system ((M;!); H) is called Liouville/completely integrable if there are n independent, Poisson commuting conserved quantities F1;:::; Fn, fH; Fj g = 0. Liouville theorem: let Mf be a level set of n F := (F1;:::; Fn): M ! R : 1 Recap (I) Phase space of a mechanical / Hamiltonian system is a symplectic 2n manifold (M ;!) with Hamiltonian H : M ! R. Equations of motion in Darboux coordinates (pi ; qj ) @H @H q_j = ; p_i = − : @pj @qi The Hamiltonian system ((M;!); H) is called Liouville/completely integrable if there are n independent, Poisson commuting conserved quantities F1;:::; Fn, fH; Fj g = 0. Liouville theorem: let Mf be a level set of n F := (F1;:::; Fn): M ! R : In Darboux coordinates around m 2 Mf define the action Z q X S(F ; q) := pi (F ; q)dqi ; pi = pi (F ; q): q0 i 1 F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. • Example of Hamiltonian reduction. • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates 2 Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. • Example of Hamiltonian reduction. • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . 2 Hence completely integrable systems are solvable `by quadrature'. Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. • Example of Hamiltonian reduction. • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): 2 Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. • Example of Hamiltonian reduction. • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. 2 • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. • Example of Hamiltonian reduction. • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. Plan for today: 2 • Separation of variables. • Example of Hamiltonian reduction. • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. 2 • Example of Hamiltonian reduction. • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. 2 • Relation to algebraic curves. Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. • Example of Hamiltonian reduction. 2 Recap (II) Then the following are conjugate coordinates F1;:::; Fn; 1 := @S=@F1; : : : ; n := @S=@Fn in a neighborhood of m 2 Mf . Equations of motion in these coordinates: F_i = fH; Fi g = 0; _j = fH; j g =: Ωj ; Ωj = Ωj (F ): Hence completely integrable systems are solvable `by quadrature'. Plan for today: • Two classical examples (Kepler and Neumann problem) with direct application of Liouville theorem. • Separation of variables. • Example of Hamiltonian reduction. • Relation to algebraic curves. 2 Kepler problem Historically, the first (higher-dimensional) integrable system introduced by Kepler (∼ 1600) and first solved by Newton (1687). Hamiltonian system: 3 6 X • (M := R ;!) where ! = dxi ^ dyi i=1 (x1; x2; x3) 2 M center of mass coordinates. • Hamiltionian: 3 1 X 2 H = 2 yi + V(r) ; r = k(x1; x2; x3)k: i=1 In two-body problem: V (r) = C=r. For us: any potential dependent only on r. Statement (I) 3 Motion of two (point) particles in R only interacting with each other. 3 Hamiltonian system: 3 6 X • (M := R ;!) where ! = dxi ^ dyi i=1 (x1; x2; x3) 2 M center of mass coordinates. • Hamiltionian: 3 1 X 2 H = 2 yi + V(r) ; r = k(x1; x2; x3)k: i=1 In two-body problem: V (r) = C=r. For us: any potential dependent only on r. Statement (I) 3 Motion of two (point) particles in R only interacting with each other. Historically, the first (higher-dimensional) integrable system introduced by Kepler (∼ 1600) and first solved by Newton (1687). 3 3 6 X • (M := R ;!) where ! = dxi ^ dyi i=1 (x1; x2; x3) 2 M center of mass coordinates. • Hamiltionian: 3 1 X 2 H = 2 yi + V(r) ; r = k(x1; x2; x3)k: i=1 In two-body problem: V (r) = C=r. For us: any potential dependent only on r. Statement (I) 3 Motion of two (point) particles in R only interacting with each other. Historically, the first (higher-dimensional) integrable system introduced by Kepler (∼ 1600) and first solved by Newton (1687). Hamiltonian system: 3 (x1; x2; x3) 2 M center of mass coordinates.