When I took a course in classical mechanics, I was confronted with the two mystery equations of Hamiltonian Formalism. In this post, I want to derive the two equations abiding by the subtle dependencies between quantities and the partial derivative relations.
Denote the coordinates as \(q_i\) and momentum as \(p_i\). Occasionally, we will drop the subscript for simplicity.
For a conservative system with natural coordinates, the kinetic energy always has a quadratic dependence with respect to velocity, i.e. \[T(u) \ \ = \ A(q) \dot q^2\] (Refer to Taylor 7.8). This observation allows a direct correspondence between the velocity derivative of the Lagrangian and velocity. Also, recall that a conservative potential must exclusively depend on the position, but none of the derivatives.
\[\frac{\partial L}{\partial \dot q} \ = \ \frac{\partial}{\partial \dot q} \left(T(q, \dot q) + U(q)\right) \ = \ 2A(q) \dot q\]
The velocity derivative of the Lagrangian satisfies a constraining equation along with position and velocity. Thus, it is possible to rewrite velocity as a combination of position, and this new derivative which we baptize as general momentum, \(p = frac{\partial L}{\partial \dot x}\).
Take the partial derivative of the Hamiltonian defined as
\[H(p, q) \ = \ p \dot q(p, q) – L(q, \dot q(p, q))\]
with respect to the two arguments \(p, q\). Invoke the chain rule on the intermediate variable \(\dot q(p, q)\)
\[\frac{\partial H}{\partial q} \ = \ p \frac{\partial \dot q}{\partial q} – \frac{\partial L}{\partial q} – \frac{\partial L}{\partial \dot q} \frac{\partial \dot q}{\partial q} \]
Recognize that the first and the third term cancel out. Recall Lagrange’s equation which states that the spacial derivative of the Lagrangian is the time derivative of \(\frac{\partial L}{\partial \dot q} \). We derive the first Hamilton’s equation
\[\frac{\partial H}{\partial q} \ = \ -\frac{d}{dt}\frac{\partial L}{\partial \dot x} \ = \ -\dot p .\]
By a similar method, it is possible to derive the second equation
\[\frac{\partial H}{\partial p} \ = \ \dot q. \]
In most systems, the kinetic energy does not have a spacial dependence, i.e. \(A(q)\) is a constant such as half the mass. As a consequence, the spacial derivative of the Hamiltonian equals to that of the Potential energy. The first Hamilton’s equation thus implies
\[\frac{\partial H}{\partial q} \ = \ \frac{\partial U}{\partial q} \ = \ -\dot p \ = \ -F \]
Moreover, this relation implies
\[F = -\nabla U.\]
This conclusion is ubiquitously applied in many classical models. Given a potential energy function defined over any space, each particles are nudged in the direction that minimizes the potential energy. For example, it is possible to describe atomic nucleus of solids in higher dimensional space, where the computation of the potential depends on quantum mechanics but the behavior of each nuclei is approximated to be classical. A system of \(n\) atoms will be modeled by an abstract space of dimension \(3n\), and the evolution of the system is exactly described by Hamilton’s equations.
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