On global representations of the solutions of linear differential equations as a product of exponentials

Wei, James Cheng‐Chung; Norman, Edward

doi:10.1090/s0002-9939-1964-0160009-0

Cited by 326 publications

(117 citation statements)

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“…The results obtained here extend and unify those given in [8,10]; see also [20]. As the φ identify group orbits for the group G generated by the Lie algebra, we interpreṫ φ = z(φ) as an equation in orbit space, and the equation for (x, w) as an equation on the Lie group G. Methods for the solution of the latter are discussed in [21], see also [4]. ⊙ Remark 5.…”

Section: Embedding Systems With Invariance Relations In Quasilinear Ssupporting

confidence: 78%

Dimension Increase and Splitting for Poincaré-Dulac Normal Forms

Gaeta¹,

Walcher²

2005

JNMP

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Integration of nonlinear dynamical systems is usually seen as associated to a symmetry reduction, e.g. via momentum map. In Lax integrable systems, as pointed out by Kazhdan, Kostant and Sternberg in discussing the Calogero system, one proceeds in the opposite way, enlarging the nonlinear system to a system of greater dimension. We discuss how this approach is also fruitful in studying non integrable systems, focusing on systems in normal form.

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Section: Embedding Systems With Invariance Relations In Quasilinear Ssupporting

confidence: 78%

Dimension Increase and Splitting for Poincaré-Dulac Normal Forms

Gaeta¹,

Walcher²

2005

JNMP

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“…"Dynamical invariants" are quantities that remain constant, but are explicit functions of time-dependent parameters and of (implicitly time-dependent) expectation values; for the harmonic oscillator, the first such invariant was noted by Lewis [20]. Korsch and Koshual used dynamical algebras to derive the dynamical invariants which lie within the algebra [21,22] (i.e., invariants which are linear in the expectation values of the algebra's operators), and Sarris and Proto demonstrated that for our state (Equation 2) it is possible to generalize further and derive dynamical invariants that are outside the algebra [23], consisting of higher powers of the the algebra's expectation values (they actually consider maximum entropy states,ρ = exp n λ n (t)L n , but a product form and exponential sum are interchangeable (see [24], corollary to theorem 2)). One such invariant in our case was found to be [25] …”

Section: Internal Frictionmentioning

confidence: 91%

Reflections on Friction in Quantum Mechanics

Rezek

2010

Entropy

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Distinctly quantum friction effects of three types are surveyed: internal friction, measurement-induced friction, and quantum-fluctuation-induced friction. We demonstrate that external driving will lead to quantum internal friction, and critique the measurement-based interpretation of friction. We conclude that in general systems will experience internal and external quantum friction over and beyond the classical frictional contributions. Keywords: quantum friction; dissipation; open systemClassification: PACS 03. 05.70.Ln Friction is the price for moving too fast. An attempt to induce a rapid change in the state of a system would be accompanied by additional entropy generation and will be encumbered by energy costs. As an example of friction we can consider a body moving rapidly against a stationary background. Its kinetic energy is dissipated, generating heat and entropy in the environment. The amount of dissipation is proportional to the velocity. Another archtypical case of friction is a driven gas compression process. For a system that is thermally decoupled from its environment, rapid changes in the piston position that compresses the gas will result in internal heating of the gas and entropy generation. To restore the system to its slow quasi-static compression equivalent, heat has to be removed from the gas. This additional heat is equivalent to extra work against friction.Friction is typically modeled by a phenomenological theory within classical mechanics. How does it extend to the quantum domain? Can a first principles model of friction be developed? There are ample examples of quantum frictional phenomena. These include friction observed in micro-mechanical systems at low temperatures [1], in superfluid theory [2], and even in quantum cosmology [3] and more. Are such quantum examples of friction essentially classical phenomena that endure into the quantum

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“…If instead, the Lie algebra of G is semi-simple, then the integrability by quadratures is not assured [6,8,12,23,24].…”

Section: The Wei and Norman Methodmentioning

confidence: 99%

“…Thus, after a brief account of a generalization of the method proposed by Wei and Norman [6,8,12,23,24], to be used later, we will study the particular case where the Lie systems of interest are Hamiltonian systems as well, both in the classical and quantum frameworks. The theory is illustrated through the particularly interesting example of generic classical and quantum quadratic time-dependent Hamiltonians.…”

Section: Introduction: Lie Systemsmentioning

confidence: 99%

Applications of Lie systems in quantum mechanics and control theory

Cariñena

Ramos

2003

Classical and Quantum Integrability

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Abstract. Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems.We will show, in particular, that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, may be up to the replacement of the Lie group involved by a central extension of it.The geometric techniques developed for dealing with Lie systems are also used in problems of control theory. Specifically, we will study some examples of control systems on Lie groups and homogeneous spaces.

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On global representations of the solutions of linear differential equations as a product of exponentials

Cited by 326 publications

References 1 publication

Dimension Increase and Splitting for Poincaré-Dulac Normal Forms

Dimension Increase and Splitting for Poincaré-Dulac Normal Forms

Reflections on Friction in Quantum Mechanics

Applications of Lie systems in quantum mechanics and control theory

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