Lagrangians and Hamiltonians
Lagrangians and Hamiltonians: elements of analytic dynamics
In order to proceed to an advanced theory of mechanics, you have to know the mathematical technique developed by Euler, Lagrange, Hamilton, etc., which is called Analytical Dynamics. Let's start from the simplest equation of motion (second law):
Now, let's recall the notion of the total energy E of a system; it is the sum of kinetic energy T and potential energy V, and it is conserved. Using these concepts, you can define
(1) L = T - V
and this is called the Lagrangian (of the system). This function has x (position) and dx/dt (velocity) as its components, and we may treat these two as if they are independent variables (although, in fact, they are not). The point of using this function is that we can rewrite Newton's equation in an interesting way; that is, in this reformulation, it becomes quite easier to see the connection of Newton's equation and the problem of obtaining the extremal value of a certain function (quantity). (By following this line, we arrive at Euler-Lagrange Equations, but we will skip this.)
In a nutshell, the connection appears as follows: Take (1) as a function of two independent variables. This function satisfies Newton's equation only in a special case, but not generally. The question is, exactly in what case does it satisfy Newton's equation? Answer is, only when a certain quantity is minimized. Thus appears what is called the "principle of least action", or more generally, the variational principle. Mechanical systems are such that they follow such trajectories as minimize a certain function. Newton's equation exactly matches such a minimization! To be more specific, the sure indication that a certain function takes an extremal value at a certain point is that its differential is zero at that point; i.e., the tangent of the curve of this function becomes horizontal.
Then, apply the same idea to the equation expressed in terms of x (position) and p (momentum). The total energy E expressed in terms of these are called the Hamiltonian.
(2) H = (1/2m)pp + V
Again, we regard the position and the momentum as two independent variables; the two may vary quite wildly, and then Newton's equation is not satisfied at all. However, if and only if we require the conditions as in the following figure, Newton's equation holds; and this corresponds to the minimization of a certain quantity.
Lagrangians and Hamiltonians can be extended to a system with n particles; since each particle has two variables and there are three dimensions of space, there are 6n variables in total. However, by considering an abstract space with 6n dimensions (called Phase Space), the state of the whole system at any given instant can be expressed by a single point in this space. Thus, in terms of Lagrangians and Hamiltonians in this phase space, a very powerful technique can be developed for treating a dynamical system. For more on this, see Takahashi (1978).
Reference
高橋康（1978）『量子力学を学ぶための解析力学入門』講談社。
Last modified March 30, 2003. (c) Soshichi Uchii
