Existence and uniqueness of classical paths under quadratic potentials
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Calculus of Variations
Existence and uniqueness of classical paths under quadratic potentials Kazuki Narita1 · Tohru Ozawa2 Received: 28 November 2019 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Minimization problem on the action given by the Lagrangean is studied on the tangent bundle over a real Hilbert space. The existence of minimizers is proved by a compactness argument. The uniqueness of classical paths is proved under quadratic potentials. Mathematics Subject Classification Primary 49S05; Secondary 34B60 · 70H03
1 Introduction The existence and uniqueness of classical paths is basic to Lagrangean Mechanics and is also the major promise of the Feynman path integrals (see [2,3,5–9,11,15] and references therein for instance). In this paper, we revisit the fundamental problem on the classical path. The purpose of this paper is to formulate Lagrangean Mechanics in the tangent bundle over a real Hilbert space as a minimization problem on the action given by the prescribed Lagrangean with two point boundary condition, to prove the existence of classical paths as the corresponding minimizers, and to give a characterization on the optimal length of time intervals that ensures the uniqueness of classical paths. As regards the length of time intervals paths, we extend Fujiwara’s √ensures the existence and uniqueness of classical √ √ that bound 8/ M (limit excluded) [9] to a new bound π/ M (limit excluded) and prove its optimality by an explicit example of periodic paths, where M is a bound of the Hessian of quadratic potentials to be specified below. The method of the proof of the existence of classical paths as minimizers depends on a compactness argument on function spaces with values in a Hilbert space [12,16]. The method of the proof of the uniqueness of classical paths depends on the standard Poincaré–Wirtinger type inequality (reformulated on functions with values in a Hilbert space) [1,4,10].
Communicated by Y.Giga.
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Tohru Ozawa [email protected]
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Department of Pure and Applied Physics, Graduate School of Advanced Science and Engineering, Waseda University, Tokyo 169-8555, Japan
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Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan 0123456789().: V,-vol
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K. Narita, T. Ozawa
This paper is organized as follows. In Sect. 2, Lagrangean Mechanics is reformulated in the tangent bundle over a real Hilbert space. In Sect. 3, basic inequalities in this paper are collected with proofs. In Sect. 4, we characterize classical paths as minimizers of the action of the prescribed Lagrangean. In Sect. 5, we examine the uniqueness of classical paths with maximal length of the time interval from the initial state to the final state.
2 Minimization problem on the action Let H be a real Hilbert space with scalar product (·|·). We regard H as the configuration space and with any q ∈ H we associate the tangent space at q as Tq H = {q} × H and regard Tq H as the momentum space. We call Lagrangean a C 1 -function on the tangent b
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