Associate Professor

Mathematics

D.B.S. (P.G.) College

 Dehradun, Uttarakhand, India 

Abstract

Hamilton’s principle is one of the great achievements of analytics mechanics. It offers a methodical manner of deriving equations of motion for many systems. Hamilton’s principle can be used as solution the basis for approximate solution. Classical mechanics dictates that are always composed of the same particles. It allows insight into the manner that the system is modeled, as any modelling assumptions are clear and the effects of changing basic system properties become apparent and are accounted for in a consistent manner. Many researchers also corelates the insights and accuracy of the modelled equations. Simplifications may also be made and Hamilton’s principle can be used as the basis for an approximate solution. Classical mechanics dictates that Hamilton’s principle could only be used for systems that are always composed of the same particles. This has been to include systems whose constituent particles change with time, including open systems of changing mass. In this book chapter, I have included the principle and its extended version of equations and showed through application to give examples how it could be led to insightful observations about the system when it be modelled. 

Keywords: Variational principle, dynamical system and configuration.

Introduction

Hamilton’s principle was formulated in 1834 by the Irish mathematician William Hamilton. We have used the techniques of variational principles of Calculus of Variation to find the stationary path between two points. Hamilton’s principle is one of the variational principles in mechanics. All the laws of mechanics can be derived by using Hamilton’s principle. Hence it is one of the most fundamental and important principle of mechanics.

Hamilton’s Principle: Hamilton’s principle states that “the behaviour of any physical system will minimize the time interval of the Lagrangian.”  In other words, a physical system that involves over time from t₁ to t₂ will follow a path q = q(t) such that the integralis minimized. The condition for this integral to be minimized is given by the Eulers-Lagrange equations of the calculus of variations. By replacing q’s for y’s and t for x and Changning the order of the terms, the Eulers-Lagrange equation

becomes

which we recognize as Lagrange’s equation.

The integralis called the action. Using that terminology, Hamilton’s principle is:

The time development of a dynamical system will minimize the action.

Hamilton’s principle is some times referred to as, “the fundamental principle of mechanics.”

To make this a bit more explicit, consider a simple physical process, such as the motion of a falling rock. This physical system can be considered to evolve from some initial configuration at time t₁ to a different final configuration at time t₂. Initially the rock is at height h and has zero velocity. Therefore, the initial configuration is(q,q),(h=0) . The final configuration (just before it hits the ground) is given by q=0  and . Hamilton’s principle states that the behaviour of the rock (its position and velocity at any instant of time) is such as to minimize the quantity. This leads to the variant of the Euler-Lagrange equation called the Lagrange equation. Finally, the Lagrange equation generates the equation of motion, which for the falling rock, is simply .

Newton’s second law can be derived from the Lagrange equation. As an illustration, consider the one-dimensional motion of a particle of mass m. Assume the particle is acted upon by a conservative force F. In one dimension, if F is conservative, it can be obtained from the potential energy V=V(x) by.

as expected.Thus, Newton’s second law is a consequence of Lagrange’s equations and Lagrange’s equations are a consequence of Hamilton’s principle. I suppose you might conclude that Hamilton’s principle is more fundamental than Newton’s laws. However, I don’t think many physicists would agree with you. Newton’s laws are the basic relations from which all of physics has been derived. Hamilton’s principle is a beautiful, sophisticated, and elegant way of describing the way nature behaves, but it is not required for understanding physical processes. On the other hand, Newton’s laws are necessary and sufficient for our understanding of nature.

Conclusion

Hamilton’s principle has been derived, in its classical form. It is very powerful principle in that it will yield the correct equations of motion with proper use of initial velocities and accelerations is ensured.

References

1. D. B. Mclver. Hamilton’s principle for systems of changing mass. Journal of Engineering Mechanics, 7(3):249-261, 1972.

2. A. Wang, Z. Zhang, and F. Zhao. Stability analysis of viscoelastic curved pipes conveying fluid. Applied Mathematics and Mechanics, 26(6):807-813,2005.

3. Y. C. Fung. Foundations of Solid Mechanics. Prentice- Hall, Inc, Englewood Cliffs, New Jersey, 1965.

4. H. N. Chu and G. Herrmann. Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. Journal of Applied Mechanics, 23:532-540, 1956.

5. H, H. E. Leipholz. On an extension of Hamilton’s variational principle to nonconservative systems which are conservative in a higher sense. Inglenieur Archiv, 47:257-266, 1978.