The Lagrangian is a natural function of coordinate and velocity.
Reverting once again to our example of a single particle
moving in one dimension, this means that we have 
.
Another useful variable
is the particle momentum. Recall the naive definition of momentum is simply
The more general definition of a momentum comes from the Lagrangian. The momentum conjugate to a given coordinate x is
Thus, for 
, the momentum is simply 
.
This definition holds even in generalized coordinates. The momenutm 
conjugate to a
given generalized coordinate 
is
An independent formulation of classical mechanics can be obtained in
terms of a natural function of coordinates and their conjugate momenta. This is the so called Hamiltonian formulation. The Hamiltonian is defined in terms of the Lagrangian as follows:
where 
indicates that the velocities must be expressed as functions of the momenta.
The above relation between the Hamiltonian and the Lagrangian is an example of what is called
a Legendre Transform, which plays an important role in thermodynamics.
For the single particle moving in one dimension, the definition is
Now, if 
, then
Thus, the Hamiltonian is
Note that the Hamiltonian is just the total energy expressed as a function of coordinates and their conjugate momenta.
Following the prescription, we can work out the Hamiltonian in generalized coordinates and momenta. First, compute the velocities as functions of momenta:
Now, substituting into the definition, we find
The equations of motion of the system are given by Hamilton's equations:
These are precisely equivalent to the Lagrangian formulation and, therefore, to Newton's laws of motion. For example, for a single particle on one dimension, we have
which is just Newton's second law.
Home: Top
Mark Tuckerman