## Abstract

We consider the normal mode problem of a vibrating string loaded with n identical beads of equal spacing, which involves an eigenvalue problem. Unlike the conventional approach to solving this problem by considering the difference equation for the components of the eigenvector, we modify the eigenvalue equation by introducing matrix-valued Lagrange undetermined multipliers, which regularize the secular equation and make the eigenvalue equation non-singular. Then, the eigenvector can be obtained from the regularized eigenvalue equation by multiplying the indeterminate eigenvalue equation by the inverse matrix. We find that the inverse matrix is nothing but the adjugate matrix of the original matrix in the secular determinant up to the determinant of the regularized matrix in the limit that the constraint equation vanishes. The components of the adjugate matrix can be represented in simple factorized forms. Finally, one can directly read off the eigenvector from the adjugate matrix. We expect this new method to be applicable to other eigenvalue problems involving more general forms of the tridiagonal matrices that appear in classical mechanics or quantum physics.

Original language | English |
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Pages (from-to) | 1079-1088 |

Number of pages | 10 |

Journal | Journal of the Korean Physical Society |

Volume | 79 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2021 Dec |

### Bibliographical note

Funding Information:As members of the Korea Pragmatist Organization for Physics Education (KPOP), the authors thank the remaining members of KPOP for useful discussions. The work is supported in part by the National Research Foundation of Korea (NRF) under the BK21 FOUR program at Korea University, Initiative for science frontiers on upcoming challenges. The work of JL is supported in part by grants funded by the Ministry of Science and ICT under Contract no. NRF-2020R1A2C3009918. The work of DWJ and CY is supported in part by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education 2018R1D1A1B07047812 (DWJ) and 2020R1I1A1A01073770 (CY), respectively. All authors contributed equally to this work.

Publisher Copyright:

© 2021, The Author(s).

## Keywords

- Classical mechanics
- Eigenvalue problem
- Lagrange multiplier
- Normal mode
- String vibration

## ASJC Scopus subject areas

- Physics and Astronomy(all)