# Hamilton's principle

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## Hamilton's principle

A variational principle from which can be derived the equations of motion of a classical dynamical system in which friction or other forms of dissipation of energy do not occur. In the original formulation of Newton's laws of motion, the position of each particle of the system of interest is specified by the cartesian coordinates of that particle. In many cases, these coordinates are not all independent of each other or do not reflect the structure of the system in a convenient way. It is then advantageous to introduce a system of generalized coordinates which are independent of each other and do reflect any special features of the system such as its symmetry about some center. The number of degrees of freedom of the system, f, is the number of such coordinates required to specify the configuration of the system at any time. See Degree of freedom (mechanics)

The problem of determining how a system moves may be formulated in the following way: If the configuration of the system at time t1 is specified by the generalized coordinates q1(t1), …, qf(t1) and at the time t2 by q1(t2), …, qf(t2), then it is required to find the trajectory along which the system travels from the initial to the final configuration. Hamilton's principle addresses this problem similarly to the way that a geometer addresses the problem of finding the shortest path lying in a curved surface between two given points on the surface. The geometer specifies the distance ds between any two close-lying points in terms of the coordinates qi of the two points and their differences, the coordinate differentials dqi, as in Eq. (1).

(1) (2) The path length D between the two specified points, given by the integral in Eq. (2), is then required to be a minimum. Hamilton defined a characteristic function &PHgr;, analogous to D, by Eq. (3),
(3) using the lagrangian function L(q, …, t) of the system in a way analogous to the geometer's g. Hamilton's principle states that the system follows the trajectory that makes the integral in Eq. (3) have a minimum value, provided the time interval between times t1 and t2 is not too great. It can be shown that this principle implies Lagrange's equations of motion for the system, and that it follows from Lagrange's equations. See Lagrange's equations, Lagrangian function, Least-action principle, Minimal principles, Variational methods (physics)

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.

## Hamilton's principle

[′ham·əl·tənz ¦prin·sə·pəl]
(mechanics)
A variational principle which states that the path of a conservative system in configuration space between two configurations is such that the integral of the Lagrangian function over time is a minimum or maximum relative to nearby paths between the same end points and taking the same time.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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References in periodicals archive ?
In Section 2, based on Hamilton's principle and Reddy's third-order shear deformation theory, the formulas for the honeycomb sandwich plate subjected to in-plane and transverse excitations are derived.
Substituting (2) and (3) into the potential and kinetic functions, the equations of motion for the honeycomb sandwich plate are derived by using Hamilton's principle:
The governing equations are derived by using Hamilton's principle and Reddy's third-order shear deformation theory.
 deduced governing differential equations and general elastic boundary conditions of a nonuniform pretwist Timoshenko beam by using Hamilton's principle. Yao et al.
According to large deformation geometric relationship, the piston theory and the quasi-static thermal stress theory, Yuan and Qiu  established the aerodynamic model of a composite stiffened panel and used Hamilton's principle to derive the equations of motion for the system.
Recently, we have analyzed the nonlinear dynamic response of the cross string, the simplest net structure of strings such as the space antennas, using Hamilton's principle and the perturbation method .
Xiong and Hutton used Hamilton's principle to obtain the governing equation and the boundary conditions of a multi-guided rotating string , and they proved that Hamilton's principle could be used for the derivation of nonlinear vibration problems as we presented in a recent paper .
Even so, the overall motion obeys the Hamilton's principle which is founded on trajectories.
Other approaches include the use of virtual work, Lagrange formalism, Hamilton's principle, and Newton-Euler equations.
331) Of course, we know that Jefferson's version of republicanism perversely rationalized slavery and condemned commercialization, financialization, and urbanization as inherently corrupt, which is counter to Holloway's odd claim that he opposed Washington and Hamilton's principles and policies because he was "too committed to the individual rights doctrine informing the founding" (pp.
As for Ross Perot's brand of economic nationalism, it owes more to southwestern populism than to Hamilton's principles.

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