Mechanics Lagrangian

A more general formulation of the mechanics of particle systems is based on Hamilton s principle, or the principle of least action. This principle states that the action S defined as 

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 

1 See Landau and Lifshitz, 1980 in the Ibirther Reading section at the end of this chapter. 

1 Holonomic constraints are those that do not depend on the generalized velocities. 

For a system placed in a conservative force field an alternative form of the equations of motion is obtained by introducing a Lagrangian function defined as the difference between the kinetic energy, T(q, q), and the potential energy, Ep (, t)  

An example of the alternative differential method is the principle of Jean Le Rond d Alemert (1717-1783). Perspicuous descriptions of the d Alemert principle and the derivation of the Lagrangian equations are, for example, given by Greiner [37] and Panat [73]. 

The phrase degrees of freedom is not interpreted identically when it is used in the different branches of science. In physics and chemistry, each independent mode in which a particle or system may move or be oriented is one degree of freedom. In mechanical engineering, degrees of freedom describes flexibility of motion. In statistics, the degrees of freedom are the number of parameter values in probability distributions that are free to be varied. In statistical mechanics the number of degrees of freedom a given system has is equal to the minimum number of independent parameters necessary to uniquely determine the location and orientation of the system in physical space. 

It is emphasized that Lagrangian mechanics is the description of a mechanical system in terms of generalized coordinates q and generalized velocities q. 

The Hamiltonian variational principle states that the motion of the system between two fixed points, denoted by q,t)x and ( 7, )2, renders the action integrah  

Employing this concept to obtain a differential equation for L(q, q, t), which is defined by (2.6), the first variation of the action 5 is expressed by  

By expanding the first integral to first order, the variation can be expressed as  

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The equation of motion is derived from the Lagrangian mechanics ( L = T - U ). The rotational motion is derived from the rotational equation of motion for a rigid rod5. 

At the outset it is important to recognize that several mathematical frameworks for the description of dynamic systems are in common use. In this context classical mechanics can be divided into three disciplines denoted by Newtonian mechanics, Lagrangian mechanics and Hamiltonian mechanics reflecting three conceptually different mathematical apparatus of model formulation [35, 52, 2, 61, 38, 95, 60, 4],... 

Elementary Concepts in Classical Machanics 197 2.2.2 Lagrangian Mechanics... 

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. 

L(q, q, t) Lagrangian function in Lagrangian mechanics Lc integral scale of scalar segregation (m)... 

A simple illustration of Noether s theorem has been presented by Baez [9], For a single particle, its position should be represented by a generalized coordinate q. The generalized velocity of the particle is q. In terms of Lagrangian mechanics, the generalized momentum p and the generalized force F are as follows ... 

In another recent effort [2S], Li uses the theory of Lagrangian mechanics to formulate the dynamic equations of a manipulator. Similar to Lee and Lee [24] above, this formulation includes an algorithm for computing the elements of the Joint space inertia matrix. In this approach, Li is able to further reduce the required computations fw the inertia matrix, making this algorithm the most efficient serial algorithm prior to the present wwk. The computational complexity is 0 N ) and the equations are applied to revolute and/or prismatic Joint configurations only. 

Lagrangian mechanics, based on the concept of minimization of energy in a systan. 

The problem is that the two approaches are independent Lagrangian mechanics works only with energy varieties possessing the two subvarieties (mechanics and electrodynamics) and needs the time factor (Longair 2003), whereas thermodynamics treats only the capacitive subvariety and works in static or equilibrium conditions (in its usual acceptance, although a branch called thermodynamics of irreversible processes deals with out-of-equilibrium processes and time). 

The theory for the application of rigid body equations in a system based on Lagrangian mechanics was presented by Lane et al. (2007a). The only points that are included in... 

Repeat Example 1.2 using lagrangian mechanics. Note that the kinetic energy of the dipole is... 

L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics, 3rd ed. Vol.l, Pergamons, New York, 1988. [(An excellent treatment of Lagrangian mechanics is given in this text.) Other excellent theoretical mechanics texts include H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, MA, 1980. E.A. Desloge, Classical Mechanics, Vols. 1 and 11, Krieger Publishing, FL, 1989.]... 

From classical mechanics (e.g., Goldstein 1950), we can show that the presence of a vector potential requires that the Hamiltonian function must be constructed using the kinetic momentum (or mechanical momentum), which is the momentum that is given in nonrelativistic theory by m. We must express this momentum in terms of the canonical momentum of Lagrangian mechanics, because it is the canonical momentum to which the quantization rule p —ihV applies. Here (and hereafter) we will use p for the canonical momentum and n for the kinetic momentum. The relation between the two is... 

Lagrangian mechanics is a way of writing the classical mechanics of Newton in a way that has the same form in any coordinate system. It is convenient for problems in which Cartesian coordinates cannot conveniently be used. We specify the positions of the particles in a system by the coordinates , 3,..., qn, where n is the number of... 

Generalized potential energy of a given system in Lagrangian mechanics... 

Action integral in classical or Lagrangian mechanics Imaginary surface in phase space enclosing J7, used in classical mechanics Sensitivity (projection) matrix Specific entropy of mixture (kJ/kgK)... 

Generalized kinetic energy in Lagrangian mechanics Temperature scale in turbulent boundary layer theory (—) Dimensionless temperature in turbulent boundary layer theory (—)... 

A set of generalized momenta in Hamiltonian mechanics A set of generalized coordinates in Hamiltonian mechanics Generalized velocities in Lagrangian Mechanics Vector function in Enskog expansion Vector function in Enskog expansion Acceleration of a single particle (m/s )... 

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