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A Tour Around Central Configurations

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A Tour Around Central Configurations A tour around Central Configurations Josep M. Cors Departament de Matem`atiques Universitat Polit`ecnicade Catalunya Campus Manresa GSDUAB Online Seminar December 14, 2020 J.M. Cors Central Configurations A tour with... Martha Alvarez Esther Barrab´es Montserrat Corbera Jos´eLino Cornelio Antonio Fernandes Dick Hall Jaume LLibre Rick Moeckel Merc`eOll´e Ernesto Perez-Chavela Gareth Roberts Claudio Vidal J.M. Cors Central Configurations Central configurations A central configuration is a special arrangement of point masses interacting by Newton's law of gravitation with the following property: the gravitational acceleration vector produced on each mass by all the others should point toward the center of mass and be proportional to the distance to the center of mass. Figure: A central configuration of five equal masses with gravitational acceleration vectors. Figure by Rick Moeckel (2014), Scholarpedia, 9(4):10667. J.M. Cors Central Configurations The role of Central Configurations Central configurations (or CC's) play an important role in the study of the Newtonian n-body problem. they lead to the only explicit solutions of the equations of motion, they govern the behavior of solutions near collisions, they influence the topology of the integral manifolds. J.M. Cors Central Configurations Explicit solutions of the n-body problem Released from rest, a central configuration maintains the same shape as it heads toward total collision (homothetic motion). lagrangehomothetic Given the correct initial velocities, a central configuration will rigidly rotate about its center of mass. Such a solution is called a relative equilibrium eulerRE Any Kepler orbit (elliptic, hyperbolic, ejection-collision) can be attached to a central configuration to obtain a solution to the full n-body problem. Above is an example of an asymmetric 8-body c.c. with elliptic homographic motion (eccentricity 0.8). homographic Simulations by Rick Moeckel (2014), Scholarpedia, 9(4):10667. J.M. Cors Central Configurations Approaches to Studying CC's Existence: Fix n. Find all possible c.c.'s and investigate how they depend on the masses. Existence for special cases: For a particular choice of masses (or set of masses), what are the c.c.'s and are there any interesting bifurcations? An Inverse Problem: Given a fixed set of positions, what (if any) are the possible masses that make the configuration central? (no restriction on the center of mass) J.M. Cors Central Configurations Mass Mapping k +n M : D ⊂ R 7! R (suitably normalized) Notice: central configurations are not isolated. It is standard practice to fix a scaling and center of mass c, and then identify solutions that are equivalent under a rotation. J.M. Cors Central Configurations Four-body problem: Quadrilaterals Legend Quadrilaterals A, B, C, D are angles a, b, c, d are side lengths B All sides in one plane Not all sides in one plane b Skew a C Planar c A D No crossed sides d Two crossed Note that all properties are sides Simple inherited along the arrows All angles < 180° Complex One angle > 180° Arrow Antiparallelogram (Self-intersecting) Concave (aka Dart/Chevron) Convex Two pairs of Two pairs of opposite equal adjacent equal sides sides Opposite angles sum to 180° Opposite side One pair Equal One pair lengths have Perpendicular of parallel diagonals of opposite equal sums diagonals sides equal sides Tangential Watt Cyclic Ex-tangential Orthodiagonal A + C = B + D = 180° (Inscriptible) a + c = b + d 2 2 2 2 Trapezium Equidiagonal a + c = b + d quadrilateral a + c = b + d A + B = C + D = 180° One pair of Opposite side lengths parallel ides Non-equal Two pairs of equal have equal sums opposite adjacent sides sides parallel Two One pair of Non-parallel Two pairs J.M. CorsA second adjacentCentral Configurations parallel sides Opposite of equal sides equal (if circle r=∞) pair of parallel right sides Opposite adjacent sides angles equal sum angles sides equal sum Right-angled Isosceles Kite Tangential Parallelogram Bicentric (‘Rhomboid’ if adjacent trapezium trapezium sides are unequal) trapezium Two Four sides Four Four sides Four right Four Four right One pair of opposite equal sides equal angles right angles opposite right equal angles equal sides angles Opposite side Third lengths have equal sums side equal Four right angles Right kite Rhombus Quadric Rectangle 3-sides-equal Isosceles (‘Oblong’ if adjacent trapezium tangential sides are unequal) Four right angles Four Four equal equal sides Four right angles sides Four equal sides Square Equilic 60° Four-body problem: Quadrilaterals Legend Quadrilaterals A, B, C, D are angles a, b, c, d are side lengths B All sides in one plane Not all sides in one plane b Skew a C Planar c A D No crossed sides d Two crossed Note that all properties are sides Simple inherited along the arrows All angles < 180° Complex One angle > 180° Arrow Antiparallelogram (Self-intersecting) Concave (aka Dart/Chevron) Convex Two pairs of Two pairs of opposite equal adjacent equal sides sides Opposite angles sum to 180° Opposite side One pair Equal One pair lengths have Perpendicular of parallel diagonals of opposite equal sums diagonals sides equal sides Tangential Watt Cyclic Ex-tangential Orthodiagonal A + C = B + D = 180° (Inscriptible) a + c = b + d 2 2 2 2 Trapezium Equidiagonal a + c = b + d quadrilateral a + c = b + d A + B = C + D = 180° One pair of Opposite side lengths parallel ides Non-equal Two pairs of equal have equal sums opposite adjacent sides sides parallel Two One pair of Non-parallel Two pairs J.M. CorsA second adjacentCentral Configurations parallel sides Opposite of equal sides equal (if circle r=∞) pair of parallel right sides Opposite adjacent sides angles equal sum angles sides equal sum Right-angled Isosceles Kite Tangential Parallelogram Bicentric (‘Rhomboid’ if adjacent trapezium trapezium sides are unequal) trapezium Two Four sides Four Four sides Four right Four Four right One pair of opposite equal sides equal angles right angles opposite right equal angles equal sides angles Opposite side Third lengths have equal sums side equal Four right angles Right kite Rhombus Quadric Rectangle 3-sides-equal Isosceles (‘Oblong’ if adjacent trapezium tangential sides are unequal) Four right angles Four Four equal equal sides Four right angles sides Four equal sides Square Equilic 60° Classifying Convex Quadrilaterals CC Goal: Classify the full set of convex central configurations in the Newtonian planar four-body problem Answer The set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain D that is the +3 union of elementary regions in R J.M. Cors Central Configurations Four-body central configurations J.M. Cors Central Configurations Convex four-body central configurations J.M. Cors Central Configurations The set of convex central configuartions Recall Dziobek's equation: −3 0 −3 0 −3 0 −3 0 −3 0 −3 0 (r12 − λ )(r34 − λ ) = (r13 − λ )(r24 − λ ) = (r14 − λ )(r23 − λ ) Eliminating λ0 from the above, define F to be the function 3 3 3 3 3 3 3 3 3 3 3 3 F (a; b; c; θ) = (r24−r14)(r13−r12)(r23−r34)−(r12−r14)(r24−r34)(r13−r23) Up to an isometry, relabeling, and rescaling, the set of all four-body convex central configurations with positive masses is given by n +3 o E = s = (a; b; c; θ) 2 R × (0; π): F (s) = 0; and r13; r24 > r12 ≥ r14; r23 ≥ r34 J.M. Cors Central Configurations Good coordinates Three radial variables a; b; c > 0 and an angular variable θ 2 (0; π) J.M. Cors Central Configurations Defining the domain D Let D = D1 [ D2 denote the three-dimensional region, where Main result D is the domain of the function θ = f (a; b; c) and the projection of E into abc-space. J.M. Cors Central Configurations Plot of the boundary of D J.M. Cors Central Configurations Configurations on the boundary D these points correspond to configurations where two or more of mutual distances inequalities r13; r24 > r12 ≥ r14; r23 ≥ r34 become equalities. Moreover, only points for which is true lie on the boundary of D. J.M. Cors Central Configurations Back to special cases of central configurations J.M. Cors Central Configurations What's next +3 Study the mass map from D into R and show that is injective Given a particular ordering of the bodies, this would prove that there is a unique convex central configurations for any choice of four positive masses. J.M. Cors Central Configurations The 1 + n{body problem Of course this is the planetary case, which is very studied, but the investigation of the relative equilibria starts with an unpublished paper of G.R. Hall from 1988. Hall's Definition A relative equilibria of the planar 1 + n{body problem is a n{body configuration 2 q¯ = (¯q0; q¯1; :::; q¯n)q ¯i 2 R which is a limit of relative equilibria q" for the n{body problem with masses m0 = 1; mi = εµi ; i = 1; :::; n for some sequence " ! 0. The parameters µi determine the mass ratios of the small bodies. J.M. Cors Central Configurations Approach to the limit As " ! 0 Hall showed that Proposition All the relative equilibria of the planar 1 + n{body problem lie on a circle centered at q0 = 0. Convergence with clustering (collinear) Convergence with clustering J.M. Cors Central Configurations Approach to distinct limit positions We are interested in the case where the limit positions of the small bodies are distinct. The positions can be described by n angles 0 ≤ θ1 < θ2 < ··· < θn < 2π Convergence without clustering (convex) Convergence without clustering (convave) J.M. Cors Central Configurations Antisymmetric matrix For given angles θ = (θ1; : : : ; θn), equations of relative equilibria can be viewed as an n × n system of linear equations A(θ)µ = 0; for the mass vector µ = (µ1; : : : ; µn).
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