Consider again the motion of a simple pendulum. Since it is one dimensional, use arc length as a coordinate. Since radius is ﬁxed, use the angular displacement, θ, as a generalized coordinate. The equation of motion involves as it should, although the coordinate is dimensionless.
Problem 5: Simple pendulum Choose θ as the generalized coordinate for a simple pendulum.
What is an appropriate generalized momentum, so that its time derivative is equal to the force? What is the engineering dimension of the generalized momentum. Draw phase space trajectories for the pendulum: periodic motion corresponds to closed trajectories.
What is the dimension of the area enclosed by such a trajectory?
What is the physical interpretation of this area?
Following the motion of N particles requires keeping track of N vectors, x1 , x2 , · · · , xN . The conﬁguration space has 3N dimensions; the phase space has 6N dimensions. We say that there are 3N degrees of freedom. Phase space volume has engineering dimension of (energy × time)3N .
The equations of motion are
If these are subject to some non-holonomic constraints, then there is no reduction in the number of degrees of freedom. If there are M scalar equations expressing holonomic constraints, then the number of degrees of freedom reduces to D = 3N − M . There is a consequent change in the dimension of phase space and the engineering dimension of phase space volume.
If there are M constraints of the form fα (x1 , x2 , · · · , xN ) = 0 with 1 ≤ α ≤ M , then all the coordinates of the N particles are given in terms of generalized coordinates qi where 1 ≤ i ≤ D = 3N − M .
In other words, one has N vector-valued functions xj = xj (q1 , q2 , · · · qD , t ). If the generalized coordinates are to provide a complete description of the dynamics then knowledge of all the qk should be equivalent to specifying all the xj . A counting of the number of scalar equations shows that this is possible.
Clearly, the velocities are
As a result one has the important identity
The equations of motion are equivalent to the principle that if one makes an instantaneous virtual displacement of a mechanical system, then the work done by the forces goes into a change of the total kinetic energy. In other words
Now one can use the generalized coordinates to rewrite the work done by the forces
where one has deﬁned the generalized forces
The change in kinetic energy
One can write
Equation of motion
Since the virtual displacements of the generalized coordinates are all independent, one can set each coeﬃcient independently to zero.
Then we have
If the particles move in a ﬁeld of conservative forces then
Then the equations of motion can be written in terms of the Lagrangian function L = T − V ,
Particle in an electromagnetic ﬁeld
The Lorentz force on a particle in an electromagnetic ﬁeld is
where q = charge, c = speed of light, v the velocity, E and B the electric and magnetic ﬁelds, and φ and A the scalar and vector potentials.
The Lagrangian formalism continues to be useful if one can write down a velocity dependent potential such that
Now using the identity
The problem of dissipative forces lies a little away from the developments made till now. However, models of frictional forces show that they are proportional to the velocity. Hence, for the dissipative forces on a body one may write the relation
This introduces the Rayleigh term, , which is usually chosen to be quadratic in The equations of motion are then written as
In order to describe the motion of a body in a dissipative environment both the Lagrangian L and the Rayleigh term need to be speciﬁed.