Hamilton's principle

(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.

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. Inthe 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 suchas 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 also Degree of freedom (mechanics).

The problem of determining how a system moves may be formulated in the following way: If theconfiguration of the system at time t₁ is specified by the generalized coordinates q₁(t₁), …, q_f(t₁) and at the time t₂ by q₁(t₂), …, q_f(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 onthe surface. The geometer specifies the distance ds between any two close-lying pointsin terms of the coordinates q_i of the two points and their differences, thecoordinate differentials dq_i, as in Eq. (1).
1.

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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 Φ, analogous to D, by Eq. (3),
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using the lagrangian function L(q, &qdot;, 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 theintegral 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 also Differential geometry; Lagrange's equations; Lagrangian function; Least-action principle; Minimal principles; Variational methods (physics).