Hamilton Lagrange Motion Equation

According to the generalized d'Alembert principle the virtual work performed by the effective forces through infinitesimal virtual displacements compatible with the system constraints is zero.
                 

                                                             
Consider the case where the displacements r_i (t) are independent, so that δ r_i are entirely arbitrary.

The virtual work done by the applied force can be given by
                                                           
Considering constant mass m_i one may write -

 
Here, T_i represents the kinetic energy of mass m_i

Hence, for the entire system -

where T is the kinetic energy of the entire system of particles.

Substituting (4) and (2) in (1)

One may represent the motion of the particle m_i using a moving coordinate system (X_i, Y_i, Z_i) as


Hence for a system of n particles one may require a 3n dimensional space to represent the motion of all the particles. This 3n dimensional space is known as the Configuration Space.

As time unfolds, the representative point P traces a curve in the configuration space called the true path, or the Newtonian path, or the dynamical path. At the same time let us think of a different representative point P’ resulting from imagining the system in a slightly different position defined by the virtual displacement δ r_i ( i = 1,2… n ). As time changes the point P’ traces a curve in the configuration space known as the Varied Path.  

Now multiplying dt in equation (5) and integrating between the time limits t₁ and t₂ of all the possible varied path one may obtains



Now considering only those paths for which the true path and varied path coincide at the two instants t₁ and t₂ as shown in figure 1, equation (6) reduces to

Equation (7) is known as the Extended Hamilton's Equation.

Equation (7) is written in terms of physical coordinates. But in many cases it is desirable to work with generalized coordinates. As δT and are independent of coordinates so one can write -
    


It may be noted that -
    
(i) The extended Hamilton’s principle is very general and can be used for a large variety of systems.
    
(ii) The only limitation is that the virtual displacement must be reversible which implies that the constraint forces must perform no work.
    
(iii) This principle cannot be used for systems with friction forces.

In general the total work done can be considered as the work done by the conservative and nonconservative forces. Hence,                                                   


 
But as the work done by the conservative forces is equal to the negative of the change in potential energy
 
(δW_c = 1 – V) ,

Now introducing Lagrangian L = T – V, equation (8) can be written as -
   


For a conservative system
 
Hence equation (9) for a conservative system becomes –

which is commonly known as the Hamilton's Principle..

Lagrange's Equation of Motion: -

It represents equations of motion in terms of generalized coordinates and can be obtained solely from two scalar expressions i.e. kinetic and potential energy a feature shared with Hamilton's Principle.

The kinetic energy T for a system of particles can be represented in terms of the displacement and velocity vector of a typical particle of mass m_i as –



Considering q_k and the displacement and velocity in terms of generalized coordinates.



i = 1,2,…. n , m = no of generalized coordinates

Hence in terms of generalized coordinates the kinetic energy can be written as





The virtual work performed by the applied force can be written in terms of generalized forces and virtual displacement as

Here Qk is the generalized force, which can be given by
        
Substituting (14) and (15) into extended Hamilton's Principle (7), one obtains –

Now,
 

Substituting (18) in (17) one obtains

Considering the arbitrariness of the virtual displacement, equation (19) will be satisfied for all values of δqk provided

Equation (20) is known as Lagrange's equation of motion.

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