Hamilton s principle

Hamilton s principle is equivalent to the statement that for the actual motion the average kinetic energy approaches the average potential energy as closely as possible. If the potential energy is a function of position only the equivalent mathematical statement is... 

The principal function S is the same quantity as the one defined by Hamilton s principle as in equation (2). It has the advantage of dealing with the scalars T and V only, whereas transformations of force components using Newton s laws deal with more complicated vector quantities. 

The Hamiltonian for a charged particle in an electromagnetic field can be obtained from Hamilton s principle and Lagrange s equations of motion (Section 3.3) ... 

By analogy with Hamilton s principle of least action, the simplest proposition that could solve the thermodynamic problem is that equilibrium also depends on an extremum principle. In other words, the extensive parameters in the equilibrium state either maximize or minimize some function. 

Following Atkins [68], the propagation of particles follows a path dictated by Newton s laws, equivalent to Hamilton s principle, that particles select paths between two points such that the action associated with the path is a minimum. Therefore, Fermat s principle for light propagation is Hamilton s principle for particles. The formal definition of action is an integral identical in structure with the phase length in physical optics. Therefore, particles are associated with wave motion, the wave-particle dualism. Hamilton s principle of least... 

Hamilton s principle exploits the power of generalized coordinates in problems with static or dynamical constraints. Going beyond the principle of least action, it can also treat dissipative forces, not being restricted to conservative systems. If energy loss... 

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... 

Of fundamental significance in the development of this theory is Hamilton s principle of least action. It states that the action integral... 

Special techniques are required to describe the symmetry of fields. Since fields are defined in terms of continuous variables it is desirable to formulate suitable transformations of dynamic variables pertaining to fields, in terms of continuous parameters. This is done by using Hamilton s principle and defining quantities such as momentum densities for any field. The most useful parameter to quantify the symmetry of a field is the Lagrangian density (T 3.3.1). 

The classical equations of motion for any generalized set of coordinates can be obtained from Hamilton s principle which states that the motion of a system from time to time is such that the line integral... 

The classical equations of motion in Hamilton s form can be obtained from a modification of Hamilton s principle as outlined above. The Lagrangian in the action integral is expressed in terms of the Hamiltonian using eqn (8.52) to yield the integral... 

In the first chapter no attempt will be made to give any parts of classical dynamics but those which are useful in the treatment of atomic and molecular problems. With this restriction, we have felt justified in omitting discussion of the dynamics of rigid bodies, non-conservative systems, non-holonomic systems, systems involving impact, etc. Moreover, no use is made of Hamilton s principle or of the Hamilton-Jacobi partial differential equation. By thus limiting the subjects to be discussed, it is possible to give in a short chapter a thorough treatment of Newtonian systems of point particles. 

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. 

Hamilton s Principle is also valid when the particles are constrained in a manner defined by equations... 

As already mentioned, the chief advantage of Hamilton s Principle is that it represents the laws of motion in a manner independent of any special choice of co-ordinates. If a number of equations of con-... 

The canonical transformations are characterised by the fact that they leave invariant the form of the equations of motion, or the stationary character of the integral [(6) of 5] expressing Hamilton s principle. This raises the question whether there are still other invariants in the case of canonical transformations. This is in fact the case, and we shall give here a series of integral invariants introduced by PoincarA1 We can show that the integral... 

Nearly two hundred years ago Maupertius tried to show that the principle of least action was one which best exhibited the wisdom of the Creator, and ever since that time the fact that a great many natural processes exhibit maximum or minimum qualities has attracted the attention of natural philosophers. In dealing with the available energy of chemical and physical phenomena, for example, the chemist seeks to find those conditions which make the entropy a maximum, or the free energy a minimum, while if the problems are treated by the methods of energetics, Hamilton s principle ... 

Hamilton s principle [68] establishes a procedure for generating the classical path. One establishes the Lagrangian function =S (x, x) ... 

Path-integral statistical mechanics is obtained in strict analogy to path-integral quantum dynamics [71]. The path-integral version of quantum dynamics focuses on the fact that a quantum mechanical system is allowed to violate Hamilton s principle. For example, tunneling (a nonclassical motion) is possible within quantum mechanics. Seizing on a comment by Dirac [72] on the relationship between S and the quantum propagator... 

In the EEM method. Gauss s principle of least constraint is invoked to derive the equations of motion of the system of particles with holonomic constraints. However, it is well known that when holonomic constraints are involved, the equations of motion can be derived from either D Alembert s principle, Hamilton s principle, or by means of a third approach. Application of Gauss s principle in this case offers no advantage over these other approaches. Gauss s principle is also exploited in the EEM method to enforce a nonholonomic temperature constraint in constant-temperature MD simulations. Again, the same equations of motion can be obtained by alternative means. ... 

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 of least action provides a mechanism for deriving equations of motion from a Lagrangian. Recall from Chap. 1 that the Lagrangian for the N-body system is defined by... 

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.. 

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