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May  2014, 34(5): 2091-2104. doi: 10.3934/dcds.2014.34.2091

On Hamiltonian flows whose orbits are straight lines

Hans Koch 1, and  Héctor E. Lomelí 2,

1. 

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712
2. 

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, United States

Received  February 2013 Revised  July 2013 Published  October 2013

We consider real analytic Hamiltonians on $\mathbb{R}^n \times \mathbb{R}^n$ whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q\in \mathbb{R}^n$. By a theorem of Moser [11], every polynomial Hamiltonian of degree $3$ reduces to such a $q$-independent Hamiltonian via a linear symplectic change of variables. We show that such a reduction is impossible, in general, for polynomials of degree $4$ or higher. But we give a condition that implies linear-symplectic conjugacy to another simple class of Hamiltonians. The condition is shown to hold for all nondegenerate Hamiltonians that are homogeneous of degree $4$.
Keywords: Hamiltonian, Iwasawa decomposition, symplectic map, polynomial map, jolt factorization., symplectic matrix.
Mathematics Subject Classification: Primary: 26C05; Secondary: 37J10, 37M15, 14E07, 58C2.
Citation: Hans Koch, Héctor E. Lomelí. On Hamiltonian flows whose orbits are straight lines. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2091-2104. doi: 10.3934/dcds.2014.34.2091
References:
[1]

A. Dragt and D. Abell, Jolt factorization of symplectic maps, Int. J. Mod. Phys. A (Proc. Suppl.) B, 2 (1993), 1019-1021. Google Scholar

[2]

M. de Bondt, Quasi-translations and counterexamples to the homogeneous dependence problem, Proc. Amer. Math. Soc., 134 (2006), 2849-2856. doi: 10.1090/S0002-9939-06-08335-3.  Google Scholar

[3]

M. d. Bondt and A. v. d. Essen, Singular Hessians, J. Algebra, 282 (2004), 195-204. doi: 10.1016/j.jalgebra.2004.08.026.  Google Scholar

[4]

A. J. Dragt and J. M. Finn, Lie series and invariant functions for analytic symplectic maps, J. Mathematical Phys., 17 (1976), 2215-2227. doi: 10.1063/1.522868.  Google Scholar

[5]

É. Forest, Geometric integration for particle accelerators, J. Phys. A, 39 (2006), 5321-5377. doi: 10.1088/0305-4470/39/19/S03.  Google Scholar

[6]

K. F. and D. Wang, Variations on a theme by Euler, J. Comput. Math., 16 (1998), 97-106.  Google Scholar

[7]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar

[8]

H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.  Google Scholar

[9]

H. E. Lomelí, Symplectic homogeneous diffeomorphisms, Cremona maps and the jolt representation, Nonlinearity, 18 (2005), 1065-1071. doi: 10.1088/0951-7715/18/3/008.  Google Scholar

[10]

R. I. McLachlan, H. Z. Munthe-Kaas, G. R. W. Quispel and A. Zanna, Explicit volume-preserving splitting methods for linear and quadratic divergence-free vector fields, Found. Comput. Math., 8 (2008), 335-355. doi: 10.1007/s10208-007-9009-6.  Google Scholar

[11]

J. Moser, On quadratic symplectic mappings, Mathematische Zeitschrift, 216 (1994), 417-430. doi: 10.1007/BF02572331.  Google Scholar

show all references

References:
[1]

A. Dragt and D. Abell, Jolt factorization of symplectic maps, Int. J. Mod. Phys. A (Proc. Suppl.) B, 2 (1993), 1019-1021. Google Scholar

[2]

M. de Bondt, Quasi-translations and counterexamples to the homogeneous dependence problem, Proc. Amer. Math. Soc., 134 (2006), 2849-2856. doi: 10.1090/S0002-9939-06-08335-3.  Google Scholar

[3]

M. d. Bondt and A. v. d. Essen, Singular Hessians, J. Algebra, 282 (2004), 195-204. doi: 10.1016/j.jalgebra.2004.08.026.  Google Scholar

[4]

A. J. Dragt and J. M. Finn, Lie series and invariant functions for analytic symplectic maps, J. Mathematical Phys., 17 (1976), 2215-2227. doi: 10.1063/1.522868.  Google Scholar

[5]

É. Forest, Geometric integration for particle accelerators, J. Phys. A, 39 (2006), 5321-5377. doi: 10.1088/0305-4470/39/19/S03.  Google Scholar

[6]

K. F. and D. Wang, Variations on a theme by Euler, J. Comput. Math., 16 (1998), 97-106.  Google Scholar

[7]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar

[8]

H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.  Google Scholar

[9]

H. E. Lomelí, Symplectic homogeneous diffeomorphisms, Cremona maps and the jolt representation, Nonlinearity, 18 (2005), 1065-1071. doi: 10.1088/0951-7715/18/3/008.  Google Scholar

[10]

R. I. McLachlan, H. Z. Munthe-Kaas, G. R. W. Quispel and A. Zanna, Explicit volume-preserving splitting methods for linear and quadratic divergence-free vector fields, Found. Comput. Math., 8 (2008), 335-355. doi: 10.1007/s10208-007-9009-6.  Google Scholar

[11]

J. Moser, On quadratic symplectic mappings, Mathematische Zeitschrift, 216 (1994), 417-430. doi: 10.1007/BF02572331.  Google Scholar

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