[1] R. Abraham and J.E. Marsden, “Foundations of Mechanics, 2nd ed.,” Addison- Wesley (1978).
[2] R. Abraham, J.E. Marsden and T. Ratiu, “Manifolds, Tensor Analysis, and Ap- plications, 2nd ed.,” App. Math. Sci., Vol. 75, Springer-Verlag, New York, 1988.
[3] V.I. Arnold, “Mathematical Methods of Classical Mechanics, 2nd ed., translated by K. Vogtmann and A. Weinstein,” SpringerーVerlag, 1989.
[4] H. Attouch, X. Goudou and P. Redont, “The Heavy Ball with Friction Method. I. The Continuous Dynamical System: Global Exploration of the Local Minima of a Real-Valued Function by Asymptotic Analysis of a Dissipative Dynamical System,” Commun. Contemp. Math., Vol. 2, pp. 1–34 (2000).
[5] B. Bateman, “On Dissipative Systems and Related Variational Principle,” Phys. Rev., Vol. 38, No. 4, 815–819 (1931).
[6] J. Bensa, S. Bilbao, R. Kronland-Martient and J.O. Smith, “The Simulation of Piano String Vibration: From Physical Models to Finite Difference Schemes and Digital Waveguides,” J. Acoust. Soc. Am., Vol. 114, pp. 1095–1107 (2003).
[7] J. Bensa, O. Gipouloux and R. Kronland–Martinet, “Parameter Fitting for Piano Sound Synthesis by Physical Modeling,” J. Acoust. Soc. Am., Vol. 118, pp. 495– 504 (2005).
[8] S. Bilbao, “Direct Simulation for Wind Instrument Synthesis,” in Proc. 11th Int. Conf. Digital Audio Effects, pp. 145–152 (2008).
[9] S. Bilbao, “Numerical Sound Synthesis: Finite Difference Schemes and Simula- tion in Musical Acoustics,” John Wiley and Sons, Chichester, 2009.
[10] G. Bogfjellmo, H. Marthinsen, “High-order Symplectic Partitioned Lie Group Methods,” Found. Comput. Math., Vol. 16, pp. 493–530 (2016).
[11] N.M. Bou-Rabee and J.E. Marsden, “Hamilton–Pontryagin Integrators on Lie Groups. I. Introduction and Structure-preserving Properties,” Found. Comput. Math., Vol. 9, pp. 197–219 (2009).
[12] L. Brugnano, F. Iavernaro and D. Trigiante, “Hamiltonian Boundary Value Meth- ods (Energy Preserving Discrete Line Integral Methods),” J. Numer. Anal. Ind. Appl. Math., Vol. 5, pp. 17–37 (2010).
[13] L. Brugnano, G. Frasca Caccia and F. Iavernaro, “Hamiltonian Boundary Value Methods (HBVMs) and Their Efficient Implementation,” Math. Eng. Sci. Aero., Vol. 5, pp. 343–411 (2014).
[14] L. Brugnano, F. Iavernaro and D. Trigiante, “Analysis of Hamiltonian Boundary Value Methods (HBVMs): A Class of Energy-preserving Runge-Kutta Meth- ods for the Numerical Solution of Polynomial Hamiltonian Systems,” Commun. Nonlinear Sci. Numer. Simul., Vol. 20, pp. 650–667 (2015).
[15] J.C. Butcher, “Coefficients for the Study of Runge–Kutta Integration Processes,” J. Austral. Math. Soc., Vol. 3, pp. 185–201 (1963).
[16] J.C. Butcher, “On Runge–Kutta Processes of High Order,” J. Austral. Math. Soc., Vol. 4, pp. 179–194 (1964).
[17] J.C. Butcher, “Implicit Runge–Kutta Processes,” Math. Comput., Vol. 18, pp. 50–64 (1964).
[18] J.C. Butcher, “A History of Runge–Kutta Methods,” Appl. Numer. Math., Vol. 20, pp. 247–260 (1996).
[19] J.C. Butcher, “Numerical Methods for Ordinal Differential Equations Second Edition,” John Wiley & Sons, 2008.
[20] O. Calin and D.C. Chang, “Geometric Mechanics on Riemannian Manifolds,” Birkh¨auser, Boston, 2004.
[21] E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D. O’Neale, B. Owren and G.R.W. Quispel, “Preserving Energy resp. Dissipation is Numerical PDEs Using the “Average Vector Field” method,” J. Comput. Phys., Vol. 231, pp. 6770–6789 (2012).
[22] E. Celledoni, R.I. McLachlan, B. Owren and G.R.W. Quispel, “Geometric Prop- erties of Kahan’s Method,” J. Phys. A, Vol. 46, 025201 (2013).
[23] E. Celledoni, R.I. McLachlan, B. Owren and G.R.W. Quispel, “Integrability Properties of Kahan’s Method,” J. Phys. A, Vol. 47, 365202 (2014).
[24] E. Celledoni, H. Marthinsen and B. Owren, “An Introduction to Lie Group Inte- grators – Basics, New Developments and Applications,” J. Comput. Phys., Vol. 257, pp. 1040–1061 (2014).
[25] E. Celledoni and B. Owren, “Preserving First Integrals with Symmetric Lie Group Methods,” Discrete Contin. Dyn. Syst., Vol. 34, pp. 977–990 (2014).
[26] E. Celledoni, R.I. McLachlan, B. Owren and G.R.W. Quispel, “Geometric and In- tegrability Properties of Kahan’s Method: the Preservation of Certain Quadratic Integrals,” J. Phys. A, Vol. 52, 065201 (2019).
[27] E. Celledoni, S. Eidnes, B. Owren and T. Ringholm, “Energy-preserving Methods on Riemannian Manifolds,” Math. Comput.
[28] J. Chabassier, A. Chaigne and P. Joly, “Time Domain Simulation of a Piano. Part 1: Model Description,” M2AN, Vol. 48, pp. 1241–1278 (2014).
[29] J. Chabassier, M. Durufl´e and P. Joly, “Time Domain Simulation of a Piano. Part 2: Numerical Aspects,” M2AN, Vol. 50, pp. 93–133 (2016).
[30] A. Chaigne and A. Askenfelt, “Numerical Simulations of Piano Strings. II. Com- parisons with Measurements and Systematic Exploration of Some Hammer-String Parameters,” J. Acoust, Soc. Am., Vol. 95, pp. 1631–1640, (1994).
[31] P. Chartier, E. Faou and A. Murua, “An Algebraic Approach to Invariant Pre- serving Integrators: The Case of Quadratic and Hamiltonian Invariants,” Numer. Math., Vol. 103, pp. 575–590 (2006).
[32] P.E. Crouch and R. Grossman, “Numerical Integration of Ordinary Differential Equations on Manifolds,” J. Nonlinear Sci., Vol. 3, pp. 1–33 (1993).
[33] M. Dahlby and B. Owren, “A General Framework for Deriving Integral Preserv- ing Numerical Methods for PDEs,” SIAM J. Sci. Comput., Vol. 33, pp. 2318–2340 (2011).
[34] L. Euler, “Institutionum Calculi Integralis,” Volumen Primum, Opera Omnia, 1768.
[35] K. Feng and M.Z. Qin, “Symplectic Geometric Algorithms for Hamiltonian Systems,” Zhejiang Science & Technology Publishing House, Hangzhou and Springer-Verlag, Berlin, 2010.
[36] D. Furihata, “Finite Difference Schemes for Equation ∂u = ( ∂ )α δG that Inherit Energy Conservation or Dissipation Property,” J. Comput. Phys., Vol. 156, pp. 181–205 (1999).
[37] D. Furihata, “Finite Difference Schemes for Nonlinear Wave Equation that In- herit Energy Conservation Property, J. Comput. Appl. Math., Vol. 134, pp. 35–57 (2001).
[38] D. Furihata and T. Matsuo, “Discrete Variational Derivative Method: A Structure-Preserving Method for Partial Differential Equations,” CRC Press, Boca Raton, 2011.
[39] M. Ghosh, “Experimental Study of the Duration of Contact of an Elastic Hammer Striking a Damped Pianoforte String,” Indian J. Phys., Vol. 7, pp. 365–382 (1932).
[40] O. Gonzalez, “Time Integration and Discrete Hamiltonian Systems,” J. Nonlinear Sci., Vol. 6, pp. 449–467 (1996).
[41] E. Hairer, S. P. Nørsett and G. Wanner, “Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd Revised Edition,” Springer, Berlin, 1993.
[42] E. Hairer, C. Lubich and G. Wanner, “Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, 2nd ed.,” Springer-Verlag, Berlin, 2006.
[43] E. Hairer, “Energy-preserving Variant of Collocation Methods,” J. Numer. Anal. Ind. Appl. Math., Vol. 5, pp.73–84 (2010).
[44] L. Herrera, L. Nu´n˜ez, A. Patin˜o, H. Rago, “A Variational Principle and the Classical and Quantum Mechanics of the Damped Harmonic Oscillator,” Amer. J. Phys., Vol. 54, pp. 273–277 (1986).
[45] D.D. Holm, T. Schmah and C. Stoica, “Geometric Mechanics and Symmetry,” Oxford University Press Inc., New York, 2009.
[46] N.H. Ibragimov, “Integrating Factors, Adjoint Equations and Lagrangians,” J. Math. Anal. Appl., Vol. 318, pp. 742–757 (2006).
[47] N.H. Ibragimov, “A New Conservation Theorem,” J. Math. Anal. Appl., Vol. 333, pp. 311–328 (2007).
[48] 井手 剛, 杉山 将, “異常検知と変化検知,” 講談社, 2015.
[49] A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett and A. Zanna, “Lie-group Methods,” Acta Numer, Vol. 9, pp. 215–365 (2000).
[50] A. Iserles and G.R.W. Quispel, “Why Geometric Numerical Integration?, ” arXiv: 1602.07755 (2016).
[51] A. Ishikawa and T. Yaguchi, “Invariance of Furihata’s Discrete Gradient Schemes for the Webster Equation with Different Riemannian Structures,” AIP Conf. Proc., Vol. 1648, 180003.
[52] A. Ishikawa and T. Yaguchi, “Geometric Investigation of the Discrete Gradient Method for the Webster Equation with a Weighted Inner Product,” JSIAM Lett., Vol. 7, pp. 17–20 (2015).
[53] 石川歩惟, 谷口隆晴, “Webster 方程式に対する離散勾配法とその力学的不変性について,” 数理解析研究所講究録, Vol. 1957, pp. 14–26 (2015).
[54] A. Ishikawa and T. Yaguchi, “Application of the Variational Principle to Deriving Energy-preserving Schemes for the Hamilton Equation,” JSIAM Lett., Vol. 8, pp. 53–56 (2016).
[55] 石川歩惟, 谷口隆晴, “ハミルトン方程式に対する離散勾配法の Riemann 構造不変性,” 日本応用数理学会論文誌, Vol.26, pp. 381–415 (2016).
[56] A. Ishikawa, D.L. Michels and T, Yaguchi, “Geometric-Integration Tools for the Simulation of Musical Sounds,” Japan J. Indust. Appl. Math., Vol. 35, pp. 511– 540 (2018).
[57] A. Ishikawa and T. Yaguchi, “Automatic Discrete Differentiation and Its Appli- cations,” arXiv:1905.08604.
[58] A. Ishikawa, “An Efficient Energy-Preserving Numerical Scheme for the Hamilton Equations on the Lie Group,” J. Comput. Appl. Math., submitted.
[59] T. Itoh and K. Abe, “Hamiltonian-conserving Discrete Canonical Equations Based on Variational Difference Quotients,” J. Comput. Phys., Vol. 76, pp. 161- 183 (1988).
[60] W.S. Koon, J.E. Marsden, S.D. Ross, M. Lo and D.J. Scheeres, “Geometric Mechanics and the Dynamics of Asteroid Pairs,” Ann. N. Y. Acad. Sci., Vol. 1017, pp. 11–38 (2004).
[61] M. Kraus and O. Maj, “Variational Integrators for Nonvariational Partial Differ- ential Equations,” Physica D., Vol. 310, pp. 37–71 (2015).
[62] M. Kraus, E. Tassi and D. Grasso, “Variational Integrators for Reduced Magne- tohydrodynamics,” J. Comput. Phys., Vol. 32, pp. 435–458 (2016).
[63] R.A. LaBudde and D. Greenspan, “Discrete Mechanics–A General Treatment,” J. Comput. Phys., Vol. 15, pp. 134–167(1974).
[64] S.K. Lele, “Compact Finite Difference Schemes with Spectral-like Resolution,” J. Comput. Phys., Vol. 103, pp. 16–42 (1992).
[65] T. Lee, M. Leok and N.H. McClamroch, “Lie Group Variational Integrators for the Full Body Problem in Orbital Mechanics,” Celestial Mech. Dyn. Astr., Vol. 98, pp. 121–144 (2007).
[66] T. Lee, “Computational Geometric Mechanics and Control of Rigid Bodies,” PhD thesis, University of Michigan, 2008.
[67] T. Lee, M. Leok, N.H. McClamroch, “Optimal atitude Control of a Rigid Body Using Geometrically Exact Computations on SO 3 ,” J. Dyn. Control Syst., Vol. 14, pp. 465–487 (2008).
[68] T. Lee, M. Leok and N.H. McClamroch, “Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds: A Geometric Approach to Modeling and Analysis,” Springer, Cham, 2018.
[69] M. Leok and J. Zhong, “Discrete Hamiltonian Variational Integrators,” IMA J. Numer. Anal., Vol. 31, pp. 1497–1532 (2011).
[70] A. Lew, J.E. Marsden, M. Ortiz and M. West, “Asynchronous Variational Inte- grators,” Arch. Rat. Mech. Anal., Vol. 167, pp.85–146 (2003).
[71] A. Lew, J.E. Marsden and M. West, “Variational Time Integrators,” Int. J. Numer. Math, Engng., Vol. 60, pp. 153–212 (2004).
[72] D. Lewis and J.C. Simo, “Conserving Algorithms for the Dynamics of Hamilto- nian Systems on Lie Group,” J. Nonlinear Sci., Vol. 4, pp. 25–299 (1994).
[73] L. Markus and K.R. Meyer, “Generic Hamiltonian Dynamical Systems are neither Integrable nor Ergodic,” American Mathematical Society Providence (1974).
[74] J.E. Marsden and T.S. Ratiu, “Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems,” Springer–Verlag (1999).
[75] J.E. Marsden and M. West, “Discrete Mechanics and Variational Integrators,” Acta Numer., Vol. 10, pp. 357–514 (2001).
[76] T. Matsuo, M. Sugihara, D. Furihata and M. Mori, “Linearly Implicit Finite Dif- ference Schemes Derived by the Discrete Variational Method,” RIMS Kokyuroku, Vol. 1145, pp. 121–129 (2000).
[77] T. Matsuo, “New Conservative Schemes with Discrete Variational Derivatives for Nonlinear Wave Equations,” J. Comput. Appl. Math., Vol. 203, pp. 32–56 (2007).
[78] T. Matsuo, “Dissipative/Conservative Galerkin Method Using Discrete Partial Derivatives for Nonlinear Evolution Equations,” J. Comput. Appl. Math., Vol. 218, pp. 506–521 (2008).
[79] T. Matsuo and Y. Miyatake, “微分方程式に対する構造保存型数値解法,” 日本応用数理学会論文誌, Vol. 22, No.3, pp.213–251 (2012).
[80] R.I. McLachlan, G.R.W. Quispel and N. Robidoux, “Geometric Integration Using Discrete Gradients,” Phil. Trans. Roy. Soc. A, Vol. 357, pp. 1021–1946 (1999).
[81] R.I. McLachlan and N. Robidoux, “Antisymmetry, Pseudo Spectral Methods, Weighted Residual Discretization, and Energy Conserving Partial Differential Equations,” arXiv:math/9907024.
[82] Y. Miyatake, “An Energy-preserving Exponentially-fitted Continuous Stage Runge–Kutta Method for Hamiltonian Systems,” BIT, Vol. 54, pp. 777–799 (2014).
[83] Y. Miyatake and J.C. Butcher, “A Characterization of Energy-preserving Meth- ods and the Construction of Parallel Integrators for Hamiltonian Systems,” SIAM J. Numer. Anal., Vol. 54, pp. 1993–2013 (2016).
[84] 宮武勇登, “長時間積分用のエネルギー保存解法,” 数理解析研究所講究録, Vol. 1995, pp.48–56 (2016).
[85] H. Munthe-Kaas, “Lie–Butcher Theory for Runge–Kutta Methods,” BIT, Vol. 35, pp. 572–587 (1995).
[86] H. Munthe-Kaas, “Runge–Kutta Methods on Lie Group,” BIT, Vol. 38, pp. 92– 111 (1998).
[87] H. Munthe-Kaas, “High Order Runge–Kutta Methods on Manifolds,” Appl. Nu- mer. Math., Vol. 29, pp. 115–127 (1999).
[88] 室田一雄, 杉原正顯, “東京大学工学課程 基礎系 数学 線形代数 I,” 丸善出版, 2015.
[89] 中野良平, “ニューラル情報処理の基礎数理,” 数理工学社, 2005.
[90] E. Noether, “Invariante Variation Probleme,” Kgl. Ges. Wiss. Nachr. G¨ottingen. Math. Physik, Vol. 2, pp. 235–257 (1918).
[91] S. Ober-Bl¨obaum, O. Junge and J.E. Marsden, “Discrete Mechanics and Optimal Control: an Analysis,” Control. Optimisation Calc. Var., Vol. 17, pp. 322–352 (2011).
[92] 岡谷貴之, “深層学習,” 講談社, 2015.
[93] G.R.W. Quispel and G.S. Turner, “Discrete Gradient Methods for Solving ODEs Numerically while Preserving a First Integral,” J. Phys. A, Vol. 29, pp. 341–349 (1999).
[94] G.R.W. Quispel and D.I. McLaren, “A New Class of Energy-preserving Numer-ical Integration for Hamiltonian Systems,” J. Phys. A, Vol. 41, 045206 (2008).
[95] L.R. Rabiner and R.W. Schafer, “Digital Processing of Speech Signals,” Prentice–Hall (1978).
[96] S. Reich, Backward Error Analysis for Numerical Integrators,” SIAM J. Numer. Anal., Vol. 36, pp. 1549–1570 (1999).
[97] M.L. Rosemblun, “Automatic Differentiation: Overview and Application to Sys- tems of Parameterized Nonlinear Equations,” Technical Report CRPC-TR92267, Center for Research on Parallel Comput., Rice Univ, Houston, TX, 1992.
[98] A.H. Salas and J.E. Castillo, “Exact Solution to Duffing Equation and the Pen- dulum Equation,” Appl. Math. Sci., Vol. 8, No. 176, pp. 8781–8789 (2014).
[99] J.M. Sanz-Serna, “Symplectic Integrators for Hamiltonian Problems: An Overview,” Acta Numer., Vol. 1, 243–286 (1992).
[100] J.M. Sanz-Serna and M.P. Calvo, “Numerical Hamiltonian Problems,” Chap- man and Hall, London, 1994.
[101] C.K.W. Tam and J.C. Webb, “Dispersion-relation-preserving Finite Difference Schemes for Computational Acoustics,” J. Comput. Phys., Vol. 229, pp. 4382– 4423 (2010).
[102] T. Yaguchi, T. Matsuo and M. Sugihara, “An Extension of the Discrete Varia- tional Method to Nonuniform Grids,” J. Comput. Phys., Vol. 229, pp. 4382–4423 (2010).
[103] T. Yaguchi, T. Matsuo and M. Sugihara, “The Discrete Variational Derivative Method Based on Discrete Differential Forms,” J. Comput. Phys., Vol. 231, pp. 3963–3986 (2012).
[104] 谷口隆晴, “Lagrange 力学に基づく局所エネルギー保存型数値解法導出法と線形波動方程式に対する無反射境界条件への応用,” 日本応用数理学会論文誌, Vol. 22, No.3, pp.143–169 (2012).
[105] T. Yaguchi, “Lagrangian Approach to Deriving Energy-Preserving Numerical Schemes for the Euler–Lagrange Partial Differential Equation,” M2AN, Vol. 47, pp. 1493–1513 (2013).
[106] K. Yoshida, “Functional Analysis,” Springer-Verlag (1995).
[107] G. Zhong and J.E. Marsden, “Lie–Poisson Hamilton–Jacobi Theory and Lie– Poisson Integrators,” Phys. Lett. A, Vol. 133, pp. 134–139 (1988).
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