a “problem for the 21st century” Tianran Chen Department of Mathematics Michigan State University Nov. 5, 2012 T.R.Chen (MSU) N-body Nov. 5, 2012 1 / 28
(xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
(xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
(xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
(xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
(xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems Orbits can be chaotic for n 3 T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
(xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems Orbits can be chaotic for n 3 Long term simulation can be hard T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
(xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems Orbits can be chaotic for n 3 Long term simulation can be hard Orbits near singularity are hard to model T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass The combined force of all other bodies mimic a single central force T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass The combined force of all other bodies mimic a single central force F1 F2 F3 m1, x1(t) T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass The combined force of all other bodies mimic a single central force F1 F2 F3 m1, x1(t) c F1 + F2 + F3 m1, x1(t) T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) Orbits are uniform rotations around the center of mass xj(t) = R(t)xj(0) T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) Orbits are uniform rotations around the center of mass xj(t) = R(t)xj(0) Prediction can be made with arbitrary accuracy T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) Orbits are uniform rotations around the center of mass xj(t) = R(t)xj(0) Prediction can be made with arbitrary accuracy The system is “stable” (as long as no collision) T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
collision Analyze separation We see everywhere Generalizations: Einstein’s n-body problem Point charges problem T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
collision Analyze separation We see everywhere Generalizations: Einstein’s n-body problem Point charges problem Vortex problem T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
there only finite many of them? If finite, how many? What are they? How do they behave? Do they exist in nature? T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses Are there only finite many of them? T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses Are there only finite many of them? No one knows! T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses Are there only finite many of them? No one knows! Problem (Smale’s 6th problem) Is the number of equivalent classes for planar central configurations (relative equilibria) for the n-body problem finite? T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? Dark matter problem T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? Dark matter problem The vortex problem T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? Dark matter problem The vortex problem Many application in computer simulation T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
xj = n k=1,k j mk (xk − xj) r3 jk λj(c − xj) = n k=1,k j mk (xk − xj) r3 jk No derivative respect to t: we can fix t = 0 (initial configuration) Still need to capture solutions form equivalent classes Scaling Translation Permutation Rotation T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
xj = n k=1,k j mk (xk − xj) r3 jk λj(c − xj) = n k=1,k j mk (xk − xj) r3 jk No derivative respect to t: we can fix t = 0 (initial configuration) Still need to capture solutions form equivalent classes Scaling Translation Permutation Rotation can be done via change of variables T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
Algebraic equations I.e. get a polynomial system P(z1, . . . , zm) = P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) Equivalent classes of planar central configurations have a one-to-one correspondence to (positive) solutions of P = 0. Equivalent question Is the number of positive solutions of P = 0 finite? T.R.Chen (MSU) N-body Nov. 5, 2012 24 / 28
Algebraic equations I.e. get a polynomial system P(z1, . . . , zm) = P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) Equivalent classes of planar central configurations have a one-to-one correspondence to (positive) solutions of P = 0. Equivalent question Is the number of positive solutions of P = 0 finite? Stronger question Is the number of complex solutions of P = 0 finite? T.R.Chen (MSU) N-body Nov. 5, 2012 24 / 28
Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 Classical algebraic geometry: probably impossible for n > 5 T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 Classical algebraic geometry: probably impossible for n > 5 More sophisticated numerical algorithms are needed T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 Classical algebraic geometry: probably impossible for n > 5 More sophisticated numerical algorithms are needed There may be a connection to the homotopy continuation methods T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? Can we numerically find central configurations? (Open for n > 4) T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? Can we numerically find central configurations? (Open for n > 4) Can we use numerical methods to understand the orbits? T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? Can we numerically find central configurations? (Open for n > 4) Can we use numerical methods to understand the orbits? The applications: T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
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