PBC Lecture Notes Series in Mechanics: by Dr. Abhijit Kar Gupta, e-mail: kg.abhi@gmail.com

1

Classical Mechanics: Lecture-2 (generalized coordinates)

The motion of a system can be described by the n -independent coordinates. These coordinates can be length, angle or anything in general. These coordinates are called generalized coordinates as the system can be described without the specific reference to a coordinate system. We go over from ),,( zyx or ),,( φθr to ),,( 321 qqq , for example. This approach is called Lagrangian formulation. This has several advantages over Newtonian description. No reference of basis vectors is required. The position vector of the i -th particle can be written as the function of all the independent generalized coordinates:

),,......,( 21 tqqqrr nii =

The velocity, iv = =•

r ..........1

1

+∂∂

dtdq

qri + +

∂∂

dtdq

qr n

n

i

tri

∂∂

= tr

qqr i

l

n

l l

i

∂∂

+∂∂ •

=∑

1,

•

lq → The generalized velocity For no explicit dependence of time (scleoronomic constraint):

=•

r•

=∑ ∂

∂l

n

l l

i qqr

1

Differentiating again, we get the acceleration, =••

r = +

∂∂ •

=∑ l

n

l l

i qqr

dtd

1

••

∂∂

ll

i qqr

Now,

∂∂

=

∂∂

dtrd

qqr

dtd i

ll

i =

∂∂

∂∂ ∑

•

mm

m

i

l

qqr

q =

•

∑ ∂∂

mm lm

i qqqr2

∴••

r = +∂∂ ••

∑ lmml lm

i qqqqr

,

2 ••

∑ ∂∂

ll l

i qqr

To calculate work done, we consider displacement,

=ird l

n

l l

i dqqr∑

= ∂∂

1

∴Work done on the system, i

N

ii rdFdW ⋅= ∑

=1

= l

N

i

n

l l

ii dq

qrF∑ ∑

= =

∂∂

⋅1 1

PBC Lecture Notes Series in Mechanics: by Dr. Abhijit Kar Gupta, e-mail: kg.abhi@gmail.com

2

= l

n

l

N

i l

ii dq

qrF∑ ∑

= =

∂∂

⋅1 1

= ll

l dqQ∑ ,

where l

iN

iil q

rFQ

∂∂

⋅= ∑=1

, the generalized force.

Note: The particle index i goes from 1 to N whereas the generalized coordinates l run from 1 to n . Since the generalized coordinates may have any dimension (not necessarily the length), the generalized force also can have any dimension (not necessarily the dimension of force). Again, let us consider that the work done W is a function of all the generalized coordinates.

∴ We can write, l

n

l l

dqqWdW ∑

= ∂∂

=1

= ll

l dqQ∑ .

∴ Therefore, the components of generalized force can be obtained by differentiating the work with respect to the corresponding generalize coordinate. Note: For a particle moving in one dimension, the work done dxFdW x ⋅= , wherefrom we

write dx

dWFx = , the force along the x-direction.

ll q

WQ∂∂

=