On Hamiltonian flows whose orbits are straight lines

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	É. Forest, Geometric integration for particle accelerators, J. Phys. A, 39 (2006), 5321-5377. doi: 10.1088/0305-4470/39/19/S03.
[6]	K. F. and D. Wang, Variations on a theme by Euler, J. Comput. Math., 16 (1998), 97-106.
[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.
[8]	H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.
[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.
[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.
[11]	J. Moser, On quadratic symplectic mappings, Mathematische Zeitschrift, 216 (1994), 417-430. doi: 10.1007/BF02572331.

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.
[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.
[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.
[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.
[5]	É. Forest, Geometric integration for particle accelerators, J. Phys. A, 39 (2006), 5321-5377. doi: 10.1088/0305-4470/39/19/S03.
[6]	K. F. and D. Wang, Variations on a theme by Euler, J. Comput. Math., 16 (1998), 97-106.
[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.
[8]	H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.
[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.
[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.
[11]	J. Moser, On quadratic symplectic mappings, Mathematische Zeitschrift, 216 (1994), 417-430. doi: 10.1007/BF02572331.

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