Hamiltonian Mechanics: Energy-Based Reformulation of Classical Physics

Table of Contents

1. Introduction
2. From Lagrangian to Hamiltonian
3. The Hamiltonian Function
4. Phase Space and Generalized Momentum
5. Deriving Hamilton’s Equations
6. Simple Example: Harmonic Oscillator
7. Hamiltonian vs Lagrangian Mechanics
8. Symmetries and Conservation Laws
9. Poisson Brackets
10. Canonical Transformations
11. Hamiltonian Formalism in Quantum Mechanics
12. Applications in Modern Physics
13. Conclusion

1. Introduction

While Lagrangian mechanics reformulated Newtonian mechanics in terms of energy differences and generalized coordinates, Hamiltonian mechanics takes this abstraction further. Instead of tracking position and velocity, it describes systems using coordinates and momenta, turning the problem into a study of energy evolution in phase space.

This approach is not just elegant—it’s foundational in quantum mechanics, statistical physics, and symplectic geometry. It brings powerful tools like Poisson brackets and canonical transformations, which streamline solving physical systems and analyzing conserved quantities.

2. From Lagrangian to Hamiltonian

Let’s recall the Lagrangian:

\[
L(q_i, \dot{q}_i, t) = T – U
\]

To build the Hamiltonian, we start by defining generalized momentum:

\[
p_i = \frac{\partial L}{\partial \dot{q}_i}
\]

Then, perform a Legendre transform to define the Hamiltonian:

\[
H(q_i, p_i, t) = \sum_i p_i \dot{q}_i – L
\]

Here, \( H \) is a function of positions \( q_i \), momenta \( p_i \), and time \( t \). This replaces the dependence on velocities \( \dot{q}_i \).

3. The Hamiltonian Function

In many systems, the Hamiltonian corresponds to the total energy:

\[
H = T + U
\]

However, this equivalence holds only under specific conditions—typically when the kinetic energy is quadratic in velocities.

For example, in Cartesian coordinates:

\[
T = \frac{1}{2}mv^2 = \frac{p^2}{2m}, \quad U = U(x)
\]

Then:

\[
H = \frac{p^2}{2m} + U(x)
\]

4. Phase Space and Generalized Momentum

Phase space is the combined space of positions and momenta ( \(q_i, p_i\) ). Each point represents a complete state of the system.

Unlike configuration space, which tracks position only, phase space tracks how fast and in which direction the system evolves. This makes it ideal for visualizing trajectories and understanding conserved quantities.

5. Deriving Hamilton’s Equations

From the Hamiltonian, we derive the Hamilton’s equations of motion:

\[
\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}
\]

These equations describe how coordinates and momenta evolve with time and together replace Newton’s second law or the Euler-Lagrange equations.

Think of \(\dot{q}_i\)​ as “how position changes” and (\dot{p}_i\)​ as “how force influences momentum”.

6. Simple Example: Harmonic Oscillator

Let’s apply the Hamiltonian formalism to a simple mass-spring system.

Lagrangian:

\[
L = \frac{1}{2}m\dot{x}^2 – \frac{1}{2}kx^2
\]

Generalized momentum:

\[
p = \frac{\partial L}{\partial \dot{x}} = m\dot{x}
\]

Hamiltonian:

\[
H = p\dot{x} – L = \frac{p^2}{2m} + \frac{1}{2}kx^2
\]

Hamilton’s equations:

\[
\dot{x} = \frac{\partial H}{\partial p} = \frac{p}{m}, \quad \dot{p} = -\frac{\partial H}{\partial x} = -kx
\]

7. Hamiltonian vs Lagrangian Mechanics

Lagrangian mechanics focuses on minimizing action using velocity-based equations, while Hamiltonian mechanics is energy-centric and evolution-oriented.

8. Symmetries and Conservation Laws

Hamiltonian mechanics naturally exposes symmetries via Noether’s theorem:

• If \( H \) does not depend on \( q_i \): \( p_i \) is conserved.
• If \( H \) does not depend on time: energy is conserved.

Symmetries are easier to identify in phase space, and this forms the backbone of modern theoretical physics, especially in quantum field theory.

9. Poisson Brackets

The Poisson bracket of two functions ( f ) and ( g ) is:

\[
{f, g} = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} – \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)
\]

The evolution of any observable ( f ) is:

\[
\frac{df}{dt} = {f, H} + \frac{\partial f}{\partial t}
\]

Example:

\[
{q_i, p_j} = \delta_{ij}
\]

10. Canonical Transformations

Hamiltonian mechanics supports canonical transformations, which preserve the structure of Hamilton’s equations.

A transformation from ( \(q, p)\ ) to ( \(Q, P\) ) is canonical if it preserves the Poisson bracket structure:

\[
{Q_i, P_j} = \delta_{ij}
\]

Canonical transformations simplify problems and are essential in advanced topics like action-angle variables, perturbation theory, and integrable systems.

These transformations simplify problems in celestial mechanics and quantum field theory.

11. Hamiltonian Formalism in Quantum Mechanics

The Hamiltonian becomes the energy operator in quantum mechanics:

\[
\hat{H} \psi = i\hbar \frac{\partial \psi}{\partial t}
\]

Poisson brackets become commutators:

\[
{f, g} \rightarrow \frac{1}{i\hbar}[\hat{f}, \hat{g}]
\]

Time evolution in quantum systems is driven by the Hamiltonian operator.

Thus, Hamiltonian mechanics is the natural precursor to the Schrödinger equation and the entire framework of quantum theory.

12. Applications in Modern Physics

• Quantum mechanics: Schrödinger and Heisenberg formulations
• Statistical mechanics: Hamiltonian governs microstate energies
• Celestial mechanics: Long-term stability of planetary orbits
• Field theory: Lagrangian → Hamiltonian density
• Chaos theory: Phase space analysis via Hamiltonian systems

13. Conclusion

Hamiltonian mechanics offers a flexible and powerful view of classical systems. By focusing on energy and phase space, it provides tools for exploring conservation laws, symmetries, and deeper connections to quantum theory.

For learners, it serves not only as an alternative to Newtonian thinking but also as a gateway into modern theoretical physics.