March  2022, 14(1): 91-104. doi: 10.3934/jgm.2021019

Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations

†. 

Department of Mathematics, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium

‡. 

Department of Mathematics, University of Antwerp, Middelheimlaan 1, 2020 Antwerpen, Belgium

* Corresponding author: Tom Mestdag

Dedicated to Professor Tony Bloch on the occasion of his 65th birthday

Received  April 2021 Published  March 2022 Early access  August 2021

The so-called method of phase synchronization has been advocated in a number of papers as a way of decoupling a system of linear second-order differential equations by a linear transformation of coordinates and velocities. This is a rather unusual approach because velocity-dependent transformations in general do not preserve the second-order character of differential equations. Moreover, at least in the case of linear transformations, such a velocity-dependent one defines by itself a second-order system, which need not have anything to do, in principle, with the given system or its reformulation. This aspect, and the related questions of compatibility it raises, seem to have been overlooked in the existing literature. The purpose of this paper is to clarify this issue and to suggest topics for further research in conjunction with the general theory of decoupling in a differential geometric context.

Citation: Willy Sarlet, Tom Mestdag. Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations. Journal of Geometric Mechanics, 2022, 14 (1) : 91-104. doi: 10.3934/jgm.2021019
References:
[1]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and T. S. Ratiu, Dissipation induced instabilities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 37-90.  doi: 10.1016/S0294-1449(16)30196-2.

[2]

F. CantrijnW. SarletA. Vandecasteele and E. Martínez, Complete separability of time-dependent second-order ordinary differential equations, Acta Appl. Math., 42 (1996), 309-334.  doi: 10.1007/BF01064171.

[3]

H. Goldstein, Classical Mechanics, 2$^{nd}$ edition, Addison-Wesley Series in Physics. Addison-Wesley Publishing Co., Reading, Mass., 1980.

[4]

D. T. KawanoM. Morzfeld and F. Ma, The decoupling of defective linear dynamical systems in free motion, J. Sound Vib., 330 (2011), 5165-5183.  doi: 10.1016/j.jsv.2011.05.013.

[5]

F. MaA. Imam and M. Morzfeld, The decoupling of damped linear systems in oscillatory free vibration, J. Sound Vib., 324 (2009), 408-428.  doi: 10.1016/j.jsv.2009.02.005.

[6]

F. MaM. Morzfeld and A. Imam, The decoupling of damped linear systems in free or forced vibration, J. Sound Vib., 329 (2010), 3182-3202.  doi: 10.1016/j.jsv.2010.02.017.

[7]

E. MartínezJ. F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection, Differential Geom. Appl., 2 (1992), 17-43.  doi: 10.1016/0926-2245(92)90007-A.

[8]

E. MartínezJ. F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection. II, Differential Geom. Appl., 3 (1993), 1-29.  doi: 10.1016/0926-2245(93)90020-2.

[9]

E. MartínezJ. F. Cariñena and W. Sarlet, Geometric characterization of separable second-order differential equations, Math. Proc. Cambridge Philos. Soc., 113 (1993), 205-224.  doi: 10.1017/S0305004100075897.

[10]

M. Morzfeld and F. Ma, The decoupling of damped linear systems in configuration and state spaces, J. Sound Vib., 330 (2011), 155-161.  doi: 10.1016/j.jsv.2010.09.005.

[11]

M. MorzfeldF. Ma and B. N. Parlett, The transformation of second-order linear systems into independent equations, SIAM J. Appl. Math., 71 (2011), 1026-1043.  doi: 10.1137/100818637.

[12] R. M. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum Press, New York-London, 1977. 
[13]

R. G. Salsa Jr., D. T. Kawano, F. Ma and G. Leitmann, The inverse problem of linear Lagrangian dynamics, ASME J. Appl. Mech., 85 (2018), 031002.

[14]

W. Sarlet, Complete decoupling of systems of ordinary second-order differential equations, In: N. H. Ibragimov, F. M. Mahomed, D. P. Mason and D. Sherwell (Eds.), Proc. 4th Workshop on Differential Equations and Chaos, (1997), 237–264.

[15]

W. Sarlet, Different forms of separability of second-order equations, Nonlinear Anal., 47 (2001), 6135-6146.  doi: 10.1016/S0362-546X(01)00682-4.

[16]

W. Sarlet and G. Thompson, Complex second-order differential equations and separability, Appl. Algebra Engrg. Comm. Comput., 11 (2001), 333-357.  doi: 10.1007/s002000000049.

show all references

References:
[1]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and T. S. Ratiu, Dissipation induced instabilities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 37-90.  doi: 10.1016/S0294-1449(16)30196-2.

[2]

F. CantrijnW. SarletA. Vandecasteele and E. Martínez, Complete separability of time-dependent second-order ordinary differential equations, Acta Appl. Math., 42 (1996), 309-334.  doi: 10.1007/BF01064171.

[3]

H. Goldstein, Classical Mechanics, 2$^{nd}$ edition, Addison-Wesley Series in Physics. Addison-Wesley Publishing Co., Reading, Mass., 1980.

[4]

D. T. KawanoM. Morzfeld and F. Ma, The decoupling of defective linear dynamical systems in free motion, J. Sound Vib., 330 (2011), 5165-5183.  doi: 10.1016/j.jsv.2011.05.013.

[5]

F. MaA. Imam and M. Morzfeld, The decoupling of damped linear systems in oscillatory free vibration, J. Sound Vib., 324 (2009), 408-428.  doi: 10.1016/j.jsv.2009.02.005.

[6]

F. MaM. Morzfeld and A. Imam, The decoupling of damped linear systems in free or forced vibration, J. Sound Vib., 329 (2010), 3182-3202.  doi: 10.1016/j.jsv.2010.02.017.

[7]

E. MartínezJ. F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection, Differential Geom. Appl., 2 (1992), 17-43.  doi: 10.1016/0926-2245(92)90007-A.

[8]

E. MartínezJ. F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection. II, Differential Geom. Appl., 3 (1993), 1-29.  doi: 10.1016/0926-2245(93)90020-2.

[9]

E. MartínezJ. F. Cariñena and W. Sarlet, Geometric characterization of separable second-order differential equations, Math. Proc. Cambridge Philos. Soc., 113 (1993), 205-224.  doi: 10.1017/S0305004100075897.

[10]

M. Morzfeld and F. Ma, The decoupling of damped linear systems in configuration and state spaces, J. Sound Vib., 330 (2011), 155-161.  doi: 10.1016/j.jsv.2010.09.005.

[11]

M. MorzfeldF. Ma and B. N. Parlett, The transformation of second-order linear systems into independent equations, SIAM J. Appl. Math., 71 (2011), 1026-1043.  doi: 10.1137/100818637.

[12] R. M. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum Press, New York-London, 1977. 
[13]

R. G. Salsa Jr., D. T. Kawano, F. Ma and G. Leitmann, The inverse problem of linear Lagrangian dynamics, ASME J. Appl. Mech., 85 (2018), 031002.

[14]

W. Sarlet, Complete decoupling of systems of ordinary second-order differential equations, In: N. H. Ibragimov, F. M. Mahomed, D. P. Mason and D. Sherwell (Eds.), Proc. 4th Workshop on Differential Equations and Chaos, (1997), 237–264.

[15]

W. Sarlet, Different forms of separability of second-order equations, Nonlinear Anal., 47 (2001), 6135-6146.  doi: 10.1016/S0362-546X(01)00682-4.

[16]

W. Sarlet and G. Thompson, Complex second-order differential equations and separability, Appl. Algebra Engrg. Comm. Comput., 11 (2001), 333-357.  doi: 10.1007/s002000000049.

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