From hamilton's principle obtain the lagrange equation of motion
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Both Hamilton's principle and Lagrange principle are based on energy principle for deriving the equation of motion of a system. As energy is a scalar quantity, the derivation of equation of motion is more straight forward unlike the derivation based on Newton's 2nd Law or d' Alembert's principle which are vector based approach. In the Newton's or d'Alembert's approach, with increase in degrees of freedom of the system it is very difficult and time consuming to draw the free body diagrams to find the equation of motion using force or moment equilibrium. Hence it is advantageous to go for energy based approach. While in Hamilton's principle one uses a integral based approach, in Lagrange principle a differential approach is followed. Hence, use of Lagrange principle is easier than the Hamilton's principle. Though all these methods in principle can be applied to any system, however it is better to use Newton's 2nd Law or d'Alembert's principle for single or two degree of freedom systems, Lagrange principle for multi degree of freedom and extended Hamilton's principle for continuous systems.
In Lagrange principle, generally the equations of motion are derived using generalized coordinates. Let us consider a system with N physical coordinates and n generalized coordinates. The kinetic energy T for a system of particles can be given by
................................................................................ (2.5.1)
Where are the position and velocity vector of a typical particles of mass mi ( i =1,2,.., N ). Considering as the displacement and velocity in terms of kth generalized coordinates, one may write,
................................................................................................................(2.5.2)
So using generalized coordinate one may write,
..................................................................................(2.5.3)
Hence, ..........................................................................(2.5.4)
The virtual work ( ) performed by the applied force can be written in terms of generalized forces and virtual displacement or
............................................................................................................ (2.5.5)
where, . ............................................................................(2.5.6)
The over bar in shows that the work done is a path function. Substituting (2.5.4) and (2.5.5) into the extended Hamilton's Principle,
In Lagrange principle, generally the equations of motion are derived using generalized coordinates. Let us consider a system with N physical coordinates and n generalized coordinates. The kinetic energy T for a system of particles can be given by
................................................................................ (2.5.1)
Where are the position and velocity vector of a typical particles of mass mi ( i =1,2,.., N ). Considering as the displacement and velocity in terms of kth generalized coordinates, one may write,
................................................................................................................(2.5.2)
So using generalized coordinate one may write,
..................................................................................(2.5.3)
Hence, ..........................................................................(2.5.4)
The virtual work ( ) performed by the applied force can be written in terms of generalized forces and virtual displacement or
............................................................................................................ (2.5.5)
where, . ............................................................................(2.5.6)
The over bar in shows that the work done is a path function. Substituting (2.5.4) and (2.5.5) into the extended Hamilton's Principle,
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