Lagrange’s equations

From Lagrange s equations it therefore follows that... 

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) ... 

Generalized momenta are defined by pk = and generalized forces are defined by Qk = f - When applied using t as the independent variable, Euler s variational equation for the action integral / takes the form of Lagrange s equations of motion... 

Generalizing Newton s equations of motion, Lagrange s equations also set the time rate of change of momenta p equal to forces Q. Hamilton recognized that these generalized momenta could replace the time derivatives q as fundamental variables of the theory. This is most directly accomplished by a Legendre transformation, as described in the following subsection. 

Lagrange s equations follow by demanding that S is stationary under variations of the generalized coordinates q. Thus, for an arbitrary change in the field ... 

In terms of these parameters, the acceleration of each atom can be written in terms of the forces on it (or, the Lagrangian for the system can be constructed, and Lagrange s equation can be written for each component of each u ). These provide a set of simultaneous equations of the form... 

For a system with N degrees of freedom, q, i = 1 to N, this equation is obtained for each of the N coordinates qi. These are Lagrange s equations of motion, the equations of motion for a system obeying classical mechanics. Thus, the Lagrangian, which minimizes the value of the action integral along the true trajectory between the times tj and fj, is also the function which yields the equations of motion when inserted into the Euler equation (8.50). 

Applying Lagrange s equation to Eqs. (40) and (41) leads to an infinite number of second-order differential equations in r" ", with an infinite number of normal-mode frequencies. However, the assumption of normal modes for the helix is equivalent to assuming that all r" vary with the same frequency and with a phase factor that depends only on m, viz., that... 

In a similar manner the homogeneity in space leads to the law of conservation of linear momentum [52] [43]. In this case L does not depend explicitly on qi, i.e., the coordinate qi is said to be cyclic. It can then be seen exploring the Lagrange s equations (2.14) that the quantity dLjdqi is constant in time. By use of the Lagrangian definition (2.6), the relationship can be written in terms of more familiar quantities ... 

Given the Lagrange s equations of motion (2.14) and the Hamiltonian function (2.22), the next task is to derive the Hamiltonian equations of motion for the system. This can be achieved by taking the differential of H defined by (2.22). Each side of the differential of H produces a differential expressed as ... 

If E and H are constant helds (i.e., independent of the space and time coordinates), then they would come out of the integral signs in Eq. (9). With the rehection nonsymmetric Lagrangian density Lmt= j]i(Av. +Bfi), Lagrange s equation of motion then reduces to the following equation of motion of a test body with charge q ... 

We have so far rather limited the types of systems considered, but Lagrange s equations are much more general than we have indicated and by a proper choice of the function L nearly all dynamical problems can be treated with their use. These equations are therefore frequently chosen as the fundamental postulates of classical mechanics instead of Newton s laws. 

The form taken by Lagrange s equations (Eq. 1-29) when the definition of p is introduced is... 

If the potential energy given by Eq. 1.35 did not include any cross-products such as qtqp the problem could be solved directly by using Lagrange s equation ... 

Each of Lagrange s equations is a differential equation of the second order. In many cases, particularly for work of a general character, it is desirable to replace them by a system of twice as many differential equations of the first order. The simplest way of accomplishing this is to put qk=sk, and then to take these additional equations into account, treating the sk s, as well as the qks, as unknown quantities. A much more symmetrical form is obtained as follows ... 

This equation has the same form as Lagrange s equation... 

Action integrals, which are at the heart of the Feynman path-integral formalism, are employed as a formal tool in classical mechanics for generating Lagrange s equations of motion. A brief review of this postulate and its application is appropriate here and allows us to introduce some notation. Of course, the reader may consult any of a number of excellent texts for more detailed treatments of this subject [67-69]. 

The first approach of Lagrangian dynamics consists of transforming to a set of independent generalized coordinates and making use of Lagrange s equations of the first kind, which do not involve the forces of constraint. The equations of constraint are implicit in the transformation to independent gen-... 

The second approach, which uses the Lagrange multiplier technique, consists of retaining the set of constrained coordinates and making use instead of Lagrange s equations of the second kind, which involve the forces of constraints. The Lagrange equations of the second kind together with the equations of constraints are used to solve for both the coordinates and the forces of constraints. Use of this approach with Cartesian coordinates has come to be known as constraint dynamics. This chapter is concerned with the various methods of constraint dynamics. 

To caicuiate the equation of motion one must consider the dynamics of the two degrees of freedom of the problem the dynamic position of the surface, z = h(x, y t) (where t is the time) and the fluid velocity, v. One writes down the total surface, gravitational, and kinetic energy and uses Lagrange s equations to solve for the dynamics. 

The energy method approach uses Lagrange s equation (and/or Hamilton s principle, if appropriate) and differs from die newtonian approach by the dependence upon scalar quantities and velocities. This approach is particularly useful if the dynamic system has several degrees of freedom and the forces experienced by the system are derived from potential functions. In summary, the energy method approach often simplifies the derivation of the equations of motion for complex multibody systems involving several degrees of freedom as seen in human biodynamics. 

Within this section, several applications of the Lagrangian approach are presented and discussed. In particular, Lagrange s equation is used to derive the equations of motion for several dynamic systems that are mechanically analogous to the musculoskeletal system. A brief introduction of l grange s equation is provided, however, the derivation and details are left to other sources iden-... 

The application of Lagrange s equation to a model of a dynamic system can be conceptualized in six steps (Baruh, 1999) ... 

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