Hamilton’s variational principle

Since dt cannot be singled out for special treatment, the covariant generalization of Hamilton s variational principle for a single particle requires an invariant action integral... 

Any multicomponent system whose dynamical behavior is governed by coupled linear equations can be modelled by an effective Lagrangian, quadratic in the system variables. Hamilton s variational principle is postulated to determine the time behavior of the system. A dynamical model of some system of interest is valid if it satisfies the same system of coupled equations. This makes it possible, for example,... 

Extending Hamilton s variational principle to piezoelectric media gives an equivalent description of the above boundary value problem (BVP) ... 

It has been shown [22] that the time-dependent non- linear Schrodinger Eq. (3.1) can be obtained from the Hamilton principle when a suitable QM Lagrangian density is defined (see Appendix A.l), and that this Lagrangian density implies an extension of the time-dependent Frenkel s variational principle [23]. 

Following Hamilton s principle in classical mechanics, the required time dependence can be derived from a variational principle based on a seemingly artificial Lagrangian density, integrated over both space and time to define the functional... 

Hamiltonian mechanics refers to a mathematical formalism in classical mechanics invented by the Irish mathematician William Rowan Hamilton (1805-1865) during the early 1830 s arising from Lagrangian mechanics which was introduced about 50 years earlier by Joseph-Louis Lagrange (1736-1813). The Hamiltonian equations can however be formulated on the basis of a variational principle without recourse to Lagrangian mechanics [95] [2j. 

In many cases the system of equations (1) is equivalent to a variation principle, known as Hamilton s Principle, viz. ... 

Here L ig a certain function of the co-ordinates and velocities of all the particles, and, in certain circumstances, also an explicit function of the time, and equation (2) as an expression of Hamilton s Principle is to be interpreted as follows the configuration (co-ordinates) of the system of particles is given at the times l1 and t2 and the motion is sought (i.e. the co-ordinates as function of the time) which will take the system from the first configuration to the second in such a way that the integral will have a stationary value.2 The chief advantage of such a variation principle is its independence of the system of coordinates. 

By using the calculus of variations, the integral form of Lagrange s method can be obtained. This is known as Hamilton s principle (Goldstein et al. 2002). It can be stated as follows ... 

Hamilton s principle states that the natural motion of the system described by the Lagrangian L is a stationary point of the classical action which implies that the 0 s) term above should vanish for any smooth variation q t) with q a) = q(P) = 0, i.e.. 

In cases where equations of motion are desired for deformable bodies, methods such as the extended Hamilton s principle may be employed. The energy is written for the system and, in addition to the terms used in Lagrange s equation, strain energy would be included. Application of Hamilton s principle will yield a set of equations of motion in the form of partial differential equations as well as the corresponding boundary conditions. Derivations and examples can be found in other sources (Baruh, 1999 Benaroya, 1998). Hamilton s principle employs the calculus of variations, and there are many texts that will be of benefit (Lanczos, 1970). 

Using a variational formulation, it can be shown that Hamilton s principle of least action leads to the following Lagrange s equations of motion ... 

Generalized Hamilton s principle with dissipation. The first variational of (3.2) is... 

The independent variation of ip and rj vanish at arbitrary temporal values t and in Hamilton s principle (3.3) yielding the following scaled boundary value problem ... 

Such a variation principle may be looked upon as the extension from one to three dimensions of Hamilton s Principle of Stationary Action in dynamics. We will show below that the analogous extension from one to three dimensions of Hamilton s less known Principle of Variable Action " (which regards the integral I in its dependence on both the boundary B of D and on the boundary values of ip) throws significant light on the structure of colloid theory. 

The principle of virtual work is suitable for solving a wide range of problems. There are tasks however where different but related formulations might be more useful. Thus, two prominent variational principles will be extended here to take into account materials with electromechanical couplings. This novel approach to Dirichlet s principle of minimum potential energy will be employed later in Section 6.3.2. In comparison to the principle of virtual work, the extended general Hamilton s principle is considered to be equivalent and even more versatile, but only its derivation will be demonstrated here. 

In the first place, the laws formulated as variational principles are themselves important, irrespective of their mathematical equivalence to those expressed in a set of differential equations, as seen from the importance of Hamilton s and Maupertuis principles in theoretical mechanics. Physical insight into the behavior of rather complicated phenomena can be acquired much more easily from laws in this form than from those in the form of a complicated set of differential equations, and often more intuitive conceptions of the phenomena can be obtained. 

The equations of motion of classical mechanics can be derived from variational principles such as Hamilton s principle of least action [74,75]. This principle states that a physical path connecting a given initial configuration with a given final configuration in time T makes the action... 

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