On the variational principle and applications for a class of damped vibration systems with a small forcing term

Shaomin Wang; Cunji Yang; Guozhi Cha; Shaomin Wang; Cunji Yang; Guozhi Cha

doi:10.3934/math.20231129

AIMS Mathematics

2023, Volume 8, Issue 9: 22162-22177. doi: 10.3934/math.20231129

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Research article

On the variational principle and applications for a class of damped vibration systems with a small forcing term

1.
School of Mathematics and Computer, Dali University, Dali, Yunnan, 671003, China
2.
School of Engineering, Dali University, Dali, Yunnan, 671003, China
3.
School of Aerospace Engineering, Xiamen University, Xiamen, Fujian, 361102, China

Received: 04 May 2023 Revised: 16 June 2023 Accepted: 20 June 2023 Published: 12 July 2023
MSC : 34C25, 58E30, 58E50

This paper is dedicated to studying the existence of periodic solutions to a new class of forced damped vibration systems by the variational method. The advantage of this kind of system is that the coefficient of its second order term is a symmetric $N \times N$ matrix valued function rather than the identity matrix previously studied. The variational principle of this problem is obtained by using two methods: the direct method of the calculus of variations and the semi-inverse method. New existence conditions of periodic solutions are created through several auxiliary functions so that two existence theorems of periodic solutions of the forced damped vibration systems are obtained by using the least action principle and the saddle point theorem in the critical point theory. Our results improve and extend many previously known results.
- forced damped vibration system,
- the variational principle,
- periodic solutions,
- the least action principle,
- the saddle point theorem
Citation: Shaomin Wang, Cunji Yang, Guozhi Cha. On the variational principle and applications for a class of damped vibration systems with a small forcing term[J]. AIMS Mathematics, 2023, 8(9): 22162-22177. doi: 10.3934/math.20231129

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Abstract

This paper is dedicated to studying the existence of periodic solutions to a new class of forced damped vibration systems by the variational method. The advantage of this kind of system is that the coefficient of its second order term is a symmetric $N \times N$ matrix valued function rather than the identity matrix previously studied. The variational principle of this problem is obtained by using two methods: the direct method of the calculus of variations and the semi-inverse method. New existence conditions of periodic solutions are created through several auxiliary functions so that two existence theorems of periodic solutions of the forced damped vibration systems are obtained by using the least action principle and the saddle point theorem in the critical point theory. Our results improve and extend many previously known results.

References

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