Lagrange’s equation of motion

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... 

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). 

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 ... 

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]. 

Engel, W. G. (1979). Lagrange-Hamilton s formalism and the chemical reaction, I. Lagrange s equation of motion for a single reaction. Hamiltonian treatment. The concept of chemical inertia. An. Acad, brasil. Cienc., 51, 195-201. 

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

Lagrange s equation of motion for the single degree of freedom is... 

One application of Lagrange s equations of motion is to a particle orbiting about a fixed point in a plane. We use plane polar coordinates p and f (with (f) measured in radians), and place the origin of coordinates at the fixed point. The component of the velocity parallel to the position vector is p, and the component perpendicular to this direction is pip, so that the Lagrangian is... 

Above we have assumed that each degree of freedom has an equal effective mass. We now relax this assumption, using Lagrange s equation of motion,... 

To illustrate this property of Lagrange s equations of motion we start with Eq. 3.15. Differentiating with respect to time yields, in concise vectorial form... 

Lagrange s equations of motion are invariant to the coordinate transformation and Eq. 3.24 is equivalent to... 

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