Abstract
According to Hamilton’s principle in classical mechanics, the trajectory of a particle between times t 1 and t 2 is such that*
where L is called the Lagrangian, the integration is over time, q j (j=,2,…) represent the generalized coordinates and dots represent differentiation with respect to time. Equation (1) is referred to as the Hamilton’s principle of least action. From {zyEq.(1)|33-1} it is possible to derive the Lagrange’s equations of motion [1]:
In this chapter we will write Fermat’s principle in the form of Eq.(1) and derive the ray equation using Cartesian coordinates. We will obtain explicit solutions of the ray equation. In the next chapter we will obtain the optical Lagrangian in cylindrical coordinates and derive ray equations valid for optical fibers which are characterized by cylindrically symmetric refractive index distribution.
Keywords
- Sound Propagation
- Refractive Index Profile
- Refractive Index Variation
- Sound Velocity Profile
- Launching Condition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass. (1960).
M. Born and E. Wolf, Principles of optics, Pergamon Press, Oxford (1975).
Ghatak and K. Thyagarajan, Introduction to Fiber Optics, Cambridge University Press (1998).
W.S. Burdic, Underwater Acoustic System Analysis, Prentice Hall, Englewood Cliffs, N.J. (1984).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media New York
About this chapter
Cite this chapter
Lakshminarayanan, V., Ghatak, A.K., Thyagarajan, K. (2002). The Optical Lagrangian and the Ray Equation. In: Lagrangian Optics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1711-5_3
Download citation
DOI: https://doi.org/10.1007/978-1-4615-1711-5_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-7923-7582-1
Online ISBN: 978-1-4615-1711-5
eBook Packages: Springer Book Archive