Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

Błaszak, Maciej; Ma, Wen‐Xiu

doi:10.1016/s0393-0440(02)00173-0

Cited by 13 publications

(27 citation statements)

References 29 publications

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“…and ϕ 1 (θ 1 ),...,ϕ N (θ N ) are arbitrary functions of one variable. More details can be found in the papers [7,8,4].…”

Section: Integrable Quasilinear Systems In Riemann Invariantsmentioning

confidence: 99%

Integrable multi-phase thermodynamic systems and Tsallis' composition rule

Nittis¹,

Lorenzoni²,

Moro³

2014

J. Phys.: Conf. Ser.

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We derive a class of equations of state for a multi-phase thermodynamic system associated with a finite set of order parameters that satisfy an integrable system of hydrodynamic type. As particular examples, we discuss one-phase systems such as the van der Waals gas and the effective molecular field model. The case of N −phase systems is also discussed in detail in connection with entropies depending on the order parameter according to Tsallis' composition rule.

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“…and ϕ 1 (θ 1 ),...,ϕ N (θ N ) are arbitrary functions of one variable. More details can be found in the papers [7,8,4].…”

Section: Integrable Quasilinear Systems In Riemann Invariantsmentioning

confidence: 99%

Integrable multi-phase thermodynamic systems and Tsallis' composition rule

Nittis¹,

Lorenzoni²,

Moro³

2014

J. Phys.: Conf. Ser.

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“…Here the local part is defined by the metricg ij given by (56),Γ is the Levi-Civita connection ofg, and the nonlocal terms λ (62) and (4). In particular, the curvature tensor ofg ij is…”

Section: Similarly If We Interchange the Independent Variables (Thatmentioning

confidence: 99%

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems

Ferapontov

Pavlov

2003

Journal of Mathematical Physics

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Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our results to linearly degenerate semi-Hamiltonian systems in Riemann invariants, a typical example beingSince all such systems are linearizable by appropriate (generalized) reciprocal transformations, our formulae provide an infinity of mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by n arbitrary functions of one variable. MSC: 37K18, 37K25, 37K35

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“…Proposition 7 The bi-cofactor system (19) has on T * Q the following bi-quasihamiltonian representation:…”

Section: Propositionmentioning

confidence: 99%

“…This proposition can be proved either by direct calculation or by observing that the underlying bi-cofactor system (19) has in the variables (q, t) the potential-cofactor form (22) and in the variables ( q, t) the cofactorpotential form (23) and using arguments similar to those used in the proof of Proposition 5. Proof.…”

Section: Propositionmentioning

confidence: 99%

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Non-Hamiltonian systems separable by Hamilton–Jacobi method

Marciniak

Błaszak

2008

Journal of Geometry and Physics

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We show that with every separable calssical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.

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Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

Cited by 13 publications

References 29 publications

Integrable multi-phase thermodynamic systems and Tsallis' composition rule

Integrable multi-phase thermodynamic systems and Tsallis' composition rule

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems

Non-Hamiltonian systems separable by Hamilton–Jacobi method

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