Langrangian and Hamiltonian Dynamics, Generalized coordinates, Degrees of Freedom

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Newton’s Mechanics is not always valid. 

Because,

1. It deals only with an inertial frame of reference.

2. The particles’ motion having restrictions called constraints cannot be evaluated using Newton’s laws.

3. Sometimes, it’s easier to interpret particle’s motion using energies.

Difficulties in introducing the forces of constraints

Here are the following difficulties,

1. Equations of motion are not all independent.

2. Forces of constraints are not known.

So the algebraic equations become complex. Here, we use the Hamiltonian or Lagrangian approach where energy is related instead of forces. 

So we introduce Lagrangian and Hamiltonian here.

Lagrangian = T — V (K.E — P.E)

Hamiltonian = T + V (K.E + P.E)

Generalized Coordinates

Any convenient set of parameters that can be used to specify the configuration or state of the system is called generalized coordinates. 

Thus generalized coordinates can be any quantities that can be observed to change as practice moves and they are not necessarily geometrical quantities for example length or angles. In suitable circumstances, they can be electric currents.

We have Generalized coordinates, not a Generalized coordinate system.

For example,  

Generalized Momentum, Generalized force etc. 

Sometimes we take polar coordinates as Generalized coordinates if it makes problem evaluation easy. We can use cylindrical coordinates if it makes the problem easier.

Degrees of Freedom

It is the smallest number of coordinates required to specify completely the configuration or state of the system. 

Or 

It is the number of independent coordinates that can describe the state of a system.

For a free particle, the degree of freedom is 3.

For a system of N particles free from constraints, degree of freedom is 3N — so 3N independent coordinates needed.

If the system or N particles is subjected to k constraints, No. of independent coordinates for the system reduce to 

3N-k 

and so the degrees of freedom are 3N-k.

Proper Generalized coordinates

  

A set of n independent Generalized coordinates whose number equals the no. of degrees of freedom and that are not restricted by the constraints is called a proper set of Generalized coordinates. 

 

For the systems subject to conservative forces only, since these forces are derived from potential functions. So condition 1 is satisfied.