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Manuele Santoprete

Wilfrid Laurier University, Mathematics, Faculty Member

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Manuele Santoprete is a full professor of mathematics at the Wilfrid Laurier University. He is author of 40 articles in peer reviewed journals, and of a book. Prior to joining Laurier, he was a Visiting Assistant Professor of mathematics at University of California, Irvine.He received his B.S. in Physics from University of Pisa in 1999, and his PhD in Applied Mathematics under the direction of Florin Diacu in 2003, at University of Victoria. He was the recipient of the Governor's General Gold Medal in 2004 for achieving the highest academic standing at the graduate level. While a doctoral student at University of Victoria, he held a Howard E. Petch Research Scholarship (2001-2002) and a University of Victoria Fellowship (1999-2003). Specialities: Celestial Mechanics, Hamiltonian Systems, Dynamical Systems, Mathematical Sociology

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Papers by Manuele Santoprete

Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

by Manuele Santoprete and Giovanni Rastelli

Journal of Geometric Mechanics, 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Gravitational and harmonic oscillator potentials on surfaces of revolution

Journal of Mathematical Physics, 2008

In this paper, we consider the motion of a particle on a surface of revolution under the influenc... more In this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are closed and that there are exactly two on some surfaces with constant Gaussian curvature. The two potentials leading to closed orbits are suitable generalizations of the gravitational and harmonic oscillator potential. We also show that there could be surfaces admitting only one potential that leads to closed orbits. In this case, the potential is a generalized harmonic oscillator. In the special case of surfaces of revolution with constant Gaussian curvature, we prove a generalization of the well-known Bertrand theorem.

Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature

Journal of Differential Equations, 2009

We consider the Kepler problem on surfaces of revolution that are homeomorphic to S 2 and have co... more We consider the Kepler problem on surfaces of revolution that are homeomorphic to S 2 and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lentz vector. Then, using such first integrals, we determine the class of surfaces that lead to blockregularizable collision singularities. In particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature.

A Counterexample to a Generalized Saari's Conjecture with a Continuum of Central Configurations

Celestial Mechanics and Dynamical Astronomy, 2000

In this paper we show that in the n-body problem with harmonic potential one can find a continuum... more In this paper we show that in the n-body problem with harmonic potential one can find a continuum of central configurations for n = 3. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari's conjecture. This will help to refine our understanding and formulation of the Generalized Saari's conjecture, and in turn it might provide insight in how to solve the classical Saari's conjecture for n ≥ 4.

Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

Celestial Mechanics and Dynamical Astronomy, 2006

In this paper we study the linear stability of the relative equilibria for homogeneous and quasih... more In this paper we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. Firstly, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a > 2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.

Geometrical Obstruction to the Integrability of Geodesic Flows

Resorting to classical techniques of Riemannian geometry we develop a geometrical method suitable... more Resorting to classical techniques of Riemannian geometry we develop a geometrical method suitable to investigate the nonintegrability of geodesic flows and of natural Hamiltonian systems. Then we apply such method to the Anisotropic Kepler Problem (AKP) and we prove that it is not analytically integrable.

American Institute of Physics Conference Series

by Ernesto Pérez-chavela and Manuele Santoprete

In 1969, D. Saari conjectured that the only solutions of the Newtonian n—body problem that have c... more In 1969, D. Saari conjectured that the only solutions of the Newtonian n—body problem that have constant moment of inertia are relative equilibria. For n = 3, there is a computer assisted proof of this conjecture given by R. Moeckel in 2005, [10]. The collinear case was solved the same year by F. Diacu, E. Pérez‐Chavela, and M. Santoprete, [4], All the other cases are open. Denoting by Uthe potential energy, Saari’s homographic conjecture states that if along an orbit of the n—body problem IU2 is constant, then the orbit is a homographic solution, i.e. a solution whose initial configuration remains similar to itself. In this paper, we discuss both conjectures and survey the proof of the latter for a large set of initial data. This survey follows our previous paper on this subject, [5].

Central Configurations and Total Collisions for Quasihomogeneous n-Body Problems

by Ernesto Pérez-chavela and Manuele Santoprete

We consider $n$-body problems given by potentials of the form ${\alpha\over r^a}+{\beta\over r^b}... more We consider $n$-body problems given by potentials of the form ${\alpha\over r^a}+{\beta\over r^b}$ with $a,b,\alpha,\beta$ constants, $0\le a<b$. To analyze the dynamics of the problem, we first prove some properties related to central configurations, including a generalization of Moulton's theorem. Then we obtain several qualitative properties for collision and near-collision orbits in the Manev-type case $a=1$. At the end we

An approach to Mel'nikov theory in celestial mechanics

Using a completely analytic procedure - based on a suitable extension of a classical method - the... more Using a completely analytic procedure - based on a suitable extension of a classical method - the authors discuss an approach to the Poincaré-Mel'nikov theory, which can be conveniently applied also to the case of nonhyperbolic critical points, and even if the critical point is located at the infinity. In this paper, they concentrate their attention on the latter case,

On the Global Dynamics of the Anisotropic Manev Problem

by Florin Diacu and Manuele Santoprete

We study the global flow of the anisotropic Manev problem, which describes the planar motion of t... more We study the global flow of the anisotropic Manev problem, which describes the planar motion of two bodies under the influence of an anisotropic Newtonian potential with a relativistic correction term. We first find all the heteroclinic orbits between equilibrium solutions. Then we generalize the Poincare'-Melnikov method and use it to prove the existence of infinitely many transversal homoclinic orbits.

Mel'nikov method revisited

We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitabl... more We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be also applied to some ``degenerate'' situations, as to the case of nonhyperbolic unstable points, or of critical points located at

Symmetric Periodic Solutions of the Anisotropic Manev Problem

We consider the Manev potential in an anisotropic space, i.e., such that the force acts different... more We consider the Manev potential in an anisotropic space, i.e., such that the force acts differently in each direction. Using a generalization of the Poincare continuation method we study the existence of periodic solutions for weak anisotropy. In particular we find that the symmetric periodic orbits of the Manev system are perturbed to periodic orbits in the anisotropic problem.

Linear Transformations, Canonoid Transformations and BiHamiltonian Structures

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Symmetric Relative Equilibria in the Four-Vortex Problem with Three Equal Vorticities

by Manuele Santoprete and C. Tamayo

We examine in detail the relative equilibria of the 4-vortex problem when three vortices have equ... more We examine in detail the relative equilibria of the 4-vortex problem when three vortices have equal strength, that is, Γ 1 = Γ 2 = Γ 3 = 1, and Γ 4 is a real parameter. We give the exact number of relativa equilibria and bifurcation values. We also study the relative equilibria in the vortex rhombus problem.

Bifurcations of Central Configurations in the Four-Body Problem with some equal masses

We study the bifurcations of central configurations of the Newtonian four-body problem when some ... more We study the bifurcations of central configurations of the Newtonian four-body problem when some of the masses are equal. First, we continue numerically the solutions for the equal mass case, and we find values of the mass parameter at which the number of solutions changes. Then, using the Krawczyk method and some result of equivariant bifurcation theory, we rigorously prove the existence of such bifurcations and classify them.

Saari's Homographic Conjecture of the Three-Body Problem

by Ernesto Pérez-chavela and Manuele Santoprete

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari ... more Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets

On the topology of the double spherical pendulum

by Manuele Santoprete and Shengda Hu

Regular and Chaotic Dynamics, 2012

This paper studies the topology of the constant energy surfaces of the double spherical pendulum.

Nonhyperbolic homoclinic chaos

Physics Letters A, 1999

Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. ... more Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. We consider here, following a (suitably adjusted) classical analytic method, the case of non-hyperbolic points and show that, under a Melnikov-type condition plus an additional assumption, the negatively and positively asymptotic sets persist under periodic perturbations, together with their infinitely many intersections on the Poincaré section. We also examine, by means of essentially the same procedure, the case of (heteroclinic) orbits tending to the infinity; this case includes in particular the classical Sitnikov 3-body problem.

On the global dynamics of the anisotropic Manev problem

Physica D: Nonlinear Phenomena, 2004

We study the global flow of the anisotropic Manev problem, which describes the planar motion of t... more We study the global flow of the anisotropic Manev problem, which describes the planar motion of two bodies under the influence of an anisotropic Newtonian potential with a relativistic correction term. We first find all the heteroclinic orbits between equilibrium solutions. Then we generalize the Poincaré-Melnikov method and use it to prove the existence of infinitely many transversal homoclinic orbits. Invoking a variational principle and the symmetries of the system, we finally detect infinitely many classes of periodic solutions.

Escape dynamics in collinear atomic-like three mass point systems

by Manuele Santoprete and Daniel Paşca

Physica D: Nonlinear Phenomena, 2010

The present paper studies the escape mechanism in collinear three point mass systems with small-r... more The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise-interaction. Specifically, we focus on systems with non-negative total energy. We show that on the zero energy level set, most of the orbits lead to binary escape configurations and the set of initial conditions leading to escape configurations where all three separations infinitely increase as t → ∞ has zero Lebesque measure. We also give numerical evidence of the existence of a periodic orbit for the case when the two outer masses are equal.

Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

by Manuele Santoprete and Giovanni Rastelli

Journal of Geometric Mechanics, 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Gravitational and harmonic oscillator potentials on surfaces of revolution

Journal of Mathematical Physics, 2008

In this paper, we consider the motion of a particle on a surface of revolution under the influenc... more In this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are closed and that there are exactly two on some surfaces with constant Gaussian curvature. The two potentials leading to closed orbits are suitable generalizations of the gravitational and harmonic oscillator potential. We also show that there could be surfaces admitting only one potential that leads to closed orbits. In this case, the potential is a generalized harmonic oscillator. In the special case of surfaces of revolution with constant Gaussian curvature, we prove a generalization of the well-known Bertrand theorem.

Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature

Journal of Differential Equations, 2009

We consider the Kepler problem on surfaces of revolution that are homeomorphic to S 2 and have co... more We consider the Kepler problem on surfaces of revolution that are homeomorphic to S 2 and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lentz vector. Then, using such first integrals, we determine the class of surfaces that lead to blockregularizable collision singularities. In particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature.

A Counterexample to a Generalized Saari's Conjecture with a Continuum of Central Configurations

Celestial Mechanics and Dynamical Astronomy, 2000

In this paper we show that in the n-body problem with harmonic potential one can find a continuum... more In this paper we show that in the n-body problem with harmonic potential one can find a continuum of central configurations for n = 3. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari's conjecture. This will help to refine our understanding and formulation of the Generalized Saari's conjecture, and in turn it might provide insight in how to solve the classical Saari's conjecture for n ≥ 4.

Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

Celestial Mechanics and Dynamical Astronomy, 2006

In this paper we study the linear stability of the relative equilibria for homogeneous and quasih... more In this paper we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. Firstly, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a > 2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.

Geometrical Obstruction to the Integrability of Geodesic Flows

Resorting to classical techniques of Riemannian geometry we develop a geometrical method suitable... more Resorting to classical techniques of Riemannian geometry we develop a geometrical method suitable to investigate the nonintegrability of geodesic flows and of natural Hamiltonian systems. Then we apply such method to the Anisotropic Kepler Problem (AKP) and we prove that it is not analytically integrable.

American Institute of Physics Conference Series

by Ernesto Pérez-chavela and Manuele Santoprete

In 1969, D. Saari conjectured that the only solutions of the Newtonian n—body problem that have c... more In 1969, D. Saari conjectured that the only solutions of the Newtonian n—body problem that have constant moment of inertia are relative equilibria. For n = 3, there is a computer assisted proof of this conjecture given by R. Moeckel in 2005, [10]. The collinear case was solved the same year by F. Diacu, E. Pérez‐Chavela, and M. Santoprete, [4], All the other cases are open. Denoting by Uthe potential energy, Saari’s homographic conjecture states that if along an orbit of the n—body problem IU2 is constant, then the orbit is a homographic solution, i.e. a solution whose initial configuration remains similar to itself. In this paper, we discuss both conjectures and survey the proof of the latter for a large set of initial data. This survey follows our previous paper on this subject, [5].

Central Configurations and Total Collisions for Quasihomogeneous n-Body Problems

by Ernesto Pérez-chavela and Manuele Santoprete

We consider $n$-body problems given by potentials of the form ${\alpha\over r^a}+{\beta\over r^b}... more We consider $n$-body problems given by potentials of the form ${\alpha\over r^a}+{\beta\over r^b}$ with $a,b,\alpha,\beta$ constants, $0\le a<b$. To analyze the dynamics of the problem, we first prove some properties related to central configurations, including a generalization of Moulton's theorem. Then we obtain several qualitative properties for collision and near-collision orbits in the Manev-type case $a=1$. At the end we

An approach to Mel'nikov theory in celestial mechanics

Using a completely analytic procedure - based on a suitable extension of a classical method - the... more Using a completely analytic procedure - based on a suitable extension of a classical method - the authors discuss an approach to the Poincaré-Mel'nikov theory, which can be conveniently applied also to the case of nonhyperbolic critical points, and even if the critical point is located at the infinity. In this paper, they concentrate their attention on the latter case,

On the Global Dynamics of the Anisotropic Manev Problem

by Florin Diacu and Manuele Santoprete

We study the global flow of the anisotropic Manev problem, which describes the planar motion of t... more We study the global flow of the anisotropic Manev problem, which describes the planar motion of two bodies under the influence of an anisotropic Newtonian potential with a relativistic correction term. We first find all the heteroclinic orbits between equilibrium solutions. Then we generalize the Poincare'-Melnikov method and use it to prove the existence of infinitely many transversal homoclinic orbits.

Mel'nikov method revisited

We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitabl... more We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be also applied to some ``degenerate'' situations, as to the case of nonhyperbolic unstable points, or of critical points located at

Symmetric Periodic Solutions of the Anisotropic Manev Problem

We consider the Manev potential in an anisotropic space, i.e., such that the force acts different... more We consider the Manev potential in an anisotropic space, i.e., such that the force acts differently in each direction. Using a generalization of the Poincare continuation method we study the existence of periodic solutions for weak anisotropy. In particular we find that the symmetric periodic orbits of the Manev system are perturbed to periodic orbits in the anisotropic problem.

Linear Transformations, Canonoid Transformations and BiHamiltonian Structures

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Symmetric Relative Equilibria in the Four-Vortex Problem with Three Equal Vorticities

by Manuele Santoprete and C. Tamayo

We examine in detail the relative equilibria of the 4-vortex problem when three vortices have equ... more We examine in detail the relative equilibria of the 4-vortex problem when three vortices have equal strength, that is, Γ 1 = Γ 2 = Γ 3 = 1, and Γ 4 is a real parameter. We give the exact number of relativa equilibria and bifurcation values. We also study the relative equilibria in the vortex rhombus problem.

Bifurcations of Central Configurations in the Four-Body Problem with some equal masses

We study the bifurcations of central configurations of the Newtonian four-body problem when some ... more We study the bifurcations of central configurations of the Newtonian four-body problem when some of the masses are equal. First, we continue numerically the solutions for the equal mass case, and we find values of the mass parameter at which the number of solutions changes. Then, using the Krawczyk method and some result of equivariant bifurcation theory, we rigorously prove the existence of such bifurcations and classify them.

Saari's Homographic Conjecture of the Three-Body Problem

by Ernesto Pérez-chavela and Manuele Santoprete

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari ... more Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets

On the topology of the double spherical pendulum

by Manuele Santoprete and Shengda Hu

Regular and Chaotic Dynamics, 2012

This paper studies the topology of the constant energy surfaces of the double spherical pendulum.

Nonhyperbolic homoclinic chaos

Physics Letters A, 1999

Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. ... more Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. We consider here, following a (suitably adjusted) classical analytic method, the case of non-hyperbolic points and show that, under a Melnikov-type condition plus an additional assumption, the negatively and positively asymptotic sets persist under periodic perturbations, together with their infinitely many intersections on the Poincaré section. We also examine, by means of essentially the same procedure, the case of (heteroclinic) orbits tending to the infinity; this case includes in particular the classical Sitnikov 3-body problem.

On the global dynamics of the anisotropic Manev problem

Physica D: Nonlinear Phenomena, 2004

We study the global flow of the anisotropic Manev problem, which describes the planar motion of t... more We study the global flow of the anisotropic Manev problem, which describes the planar motion of two bodies under the influence of an anisotropic Newtonian potential with a relativistic correction term. We first find all the heteroclinic orbits between equilibrium solutions. Then we generalize the Poincaré-Melnikov method and use it to prove the existence of infinitely many transversal homoclinic orbits. Invoking a variational principle and the symmetries of the system, we finally detect infinitely many classes of periodic solutions.

Escape dynamics in collinear atomic-like three mass point systems

by Manuele Santoprete and Daniel Paşca

Physica D: Nonlinear Phenomena, 2010

The present paper studies the escape mechanism in collinear three point mass systems with small-r... more The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise-interaction. Specifically, we focus on systems with non-negative total energy. We show that on the zero energy level set, most of the orbits lead to binary escape configurations and the set of initial conditions leading to escape configurations where all three separations infinitely increase as t → ∞ has zero Lebesque measure. We also give numerical evidence of the existence of a periodic orbit for the case when the two outer masses are equal.