Hamiltonian Mechanics - As A Reformulation of Lagrangian Mechanics

As A Reformulation of Lagrangian Mechanics

Starting with Lagrangian mechanics, the equations of motion are based on generalized coordinates

and matching generalized velocities

We write the Lagrangian as

with the subscripted variables understood to represent all N variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as conjugate momenta. By doing so, it is possible to handle certain systems, such as aspects of quantum mechanics, that would otherwise be even more complicated.

For each generalized velocity, there is one corresponding conjugate momentum, defined as:

In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum. For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.

One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinate patches on the same symplectic manifold (see Mathematical formalism, below).

The Hamiltonian is the Legendre transform of the Lagrangian:

If the transformation equations defining the generalized coordinates are independent of t, and the Lagrangian is a sum of products of functions (in the generalized coordinates) which are homogeneous of order 0, 1 or 2, then it can be shown that H is equal to the total energy E = T + V.

Each side in the definition of produces a differential:

\mathrm{d}\mathcal{H} &= \sum_i \left + \left({\partial \mathcal{H} \over \partial t}\right) \mathrm{d}t\qquad\qquad\quad\quad \\ \\ &= \sum_i \left - \left({\partial \mathcal{L} \over \partial t}\right) \mathrm{d}t.

Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton:

\frac{\partial \mathcal{H}}{\partial q_j} = - \dot{p}_j, \qquad
\frac{\partial \mathcal{H}}{\partial p_j} = \dot{q}_j, \qquad
\frac{\partial \mathcal{H}}{\partial t } = - {\partial \mathcal{L} \over \partial t}.

Hamilton's equations consist of 2n first-order differential equations, while Lagrange's equations consist of n second-order equations. However, Hamilton's equations usually don't reduce the difficulty of finding explicit solutions. They still offer some advantages, since important theoretical results can be derived because coordinates and momenta are independent variables with nearly symmetric roles.

Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, such that a coordinate does not occur in the Hamiltonian, the corresponding momentum is conserved, and that coordinate can be ignored in the other equations of the set. Effectively, this reduces the problem from n coordinates to (n-1) coordinates. In the Lagrangian framework, of course the result that the corresponding momentum is conserved still follows immediately, but all the generalized velocities still occur in the Lagrangian - we still have to solve a system of equations in n coordinates.

The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in the theory of classical mechanics, and for formulations of quantum mechanics.

Read more about this topic:  Hamiltonian Mechanics

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