Hamilton's Canonical Equations

The Geometric Shift

Newtonian mechanics focuses on forces causing acceleration. It asks, "What pushes the particle?" Hamiltonian mechanics asks, "How does the state evolve?" In the Newtonian view, position and velocity are linked; you cannot specify one without the other in the equation of motion. In the Hamiltonian view, we treat position \( q \) and momentum \( p \) as independent variables defining a point in a new space called phase space. The evolution of the system is then described by how this point moves through that space. The key insight is that the time derivative of position is determined by the slope of the Hamiltonian with respect to momentum, while the time derivative of momentum is determined by the negative slope of the Hamiltonian with respect to position. This creates a perfect duality between the two variables.

Deriving the Canonical Equations

We begin with the Hamiltonian defined via the Legendre transform: \( H(q, p, t) = p \dot{q} - L(q, \dot{q}, t) \). To find the equations of motion, we examine the total differential of \( H \). Treating \( q \), \( p \), and \( t \) as independent variables, we have:

However, we can also compute \( dH \) by differentiating the definition \( H = p \dot{q} - L \). Using the product rule and the chain rule for \( L \):

Recall the definition of conjugate momentum \( p = \frac{\partial L}{\partial \dot{q}} \). The terms involving \( d\dot{q} \) cancel out exactly (\( p \, d\dot{q} - p \, d\dot{q} = 0 \)). We are left with:

By comparing the coefficients of \( dq \), \( dp \), and \( dt \) in the two expressions for \( dH \), we identify the partial derivatives of the Hamiltonian. This yields the fundamental relations:

From the Euler-Lagrange equation, we know that \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) = \frac{\partial L}{\partial q} \). Since \( p = \frac{\partial L}{\partial \dot{q}} \), this implies \( \dot{p} = \frac{\partial L}{\partial q} \). Substituting this into the first relation above gives \( \dot{p} = - \frac{\partial H}{\partial q} \). Combining this with the second relation, we arrive at Hamilton's canonical equations.

The Symmetric Pair

The result is a set of two first-order differential equations that are structurally identical but with a sign difference. They are:

This symmetry is profound. It treats \( q \) and \( p \) as partners. If you swap \( q \) and \( p \), and change the sign of time, the equations remain invariant. (Time reversal proper is \( t \to -t \) with \( p \to -p \), which leaves an ordinary Hamiltonian unchanged.) Furthermore, these equations are first-order, meaning that if you know the initial position \( q_0 \) and initial momentum \( p_0 \), you can integrate forward in time uniquely. In Newtonian mechanics, you needed initial position and velocity. Here, you need position and momentum. Since momentum is mass times velocity, the information content is identical, but the mathematical structure is more robust for complex systems.

Conservation and Energy

A critical feature of Hamilton's equations is their relationship to energy conservation. If the Hamiltonian does not depend explicitly on time, i.e., \( \frac{\partial H}{\partial t} = 0 \), then the total time derivative of \( H \) is:

Substituting the canonical equations into this expression:

Thus, \( H \) is a constant of motion. For standard mechanical systems where the potential energy depends only on position and the kinetic energy is quadratic in velocity, the Hamiltonian \( H \) is exactly equal to the total mechanical energy \( E = T + V \). Therefore, Hamilton's equations automatically encode the conservation of energy when the system is time-translation invariant. This is a direct consequence of the variational principle, not an additional assumption.