Duduli

A double pendulum consists of two masses connected by an inextensible light rod. The system of two masses and the rod is suspended by another inextensible and weightless rod fastened to one of the masses as shown in the figure In another way a double pendulum may be described as follows: As the name indicates a double pendulum may be supposed to be formed of two pendular oscillating in the same plane; one has a Bob of mass m1 suspended from a flat hinge at O by an inextensible and weightless rod of length l1, while the other has a rod of mass m2 suspended from another hinge in mass m1 by a similar rod of length l2.

A frame attached with the centre of mass of an isolated system of particles is called at the centre of mass coordinate system or C-system.

It is to be noted that the Hamiltonian equations are specially used to solve the problems involving cyclic coordinates. A coordinates qk is cyclic if it does not appear in the lagrangian. Cyclic coordinate is the coordinate for which corresponding momentum remains conserved.

If the work done by a force is independent of path, the force is said to be conservative. If the forces acting on the particles are conservative: it's mechanical energy(kinetic energy+potential energy) remains constant.This is called conservation of energy.

According to Hamilton's principle,“The path actually traversed in a conservative, holonomic dynamical system from t1 & t2 is one over which the integral of the lagrangian between limits t1 & t2 is stationary (i.e the time integral of the lagrangian is extremum.) Analytically it can be represented as    ∫L dt = J = extremum Where J is the extremum value of the time integral of the lagrangian and is known as Hamilton's principle function for the path. Or  d[ ∫L dt ]=0 Where 'd' is the variation symbol. This principle helps us to distinguish the actual path from the neighbouring paths.

The constraints which are dependent on time are known as rheonomic constraints. e.g a simple pendulum suspended with a moving support i.e pendulum having variable length is an example of rheonomic constraint.

The constraints which are independent of time are known as scleronomic constraints. e.g a simple pendulum suspended with fixed support i.e length of the pendulum remains constant is an example of scleronomic constraints.

Constraints are said to be holonomic if constraints imposed on the motion of the system can be expressed in the form of mathematical equation f(r1,r2,.....,rn,t)=0.  where r1,r2,.....,rn are position vector of system of particles & 't' is time. Example of holonomic constraints  ★The constraints involved in the motion of rigid bodies in which the distance between two particular points is always fixed. ★The constraints involved when a particle is restricted to move along a curve of surface are holonomic.

D' alembert's principle states that the virtual work done by effective force(external force+reversed effective force) on a dynamical system is zero.