Hamilton-Jacobi Equation

Prerequisites: Hamiltonian Formalism. Full path (13 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Newton's or Hamilton's equations tell us how one particle moves step by step, starting from a given position and velocity. The Hamilton-Jacobi equation takes a different point of view. Instead of following a single trajectory, it describes a whole family of trajectories at once through one function, the action, considered as a function of the endpoint position and time.

A useful picture comes from optics. Light can be described by rays or by wavefronts. The rays are the lines perpendicular to the wavefronts, and if we know how the wavefronts move, we know all the rays. The Hamilton-Jacobi equation does the same for mechanics. The surfaces on which the action has a constant value play the role of wavefronts, and the particle trajectories run perpendicular to them, with the momentum given by how steeply the action changes from one surface to the next. The equation itself is the condition that these wavefronts move in a way that is consistent with the energy of the particle.

This point of view is useful for two reasons. In practice, if we can find a suitable action, for example by separating variables, the equations of motion are solved almost automatically. Conceptually, it shows that classical mechanics already looks like the limit of a wave theory, in the same way that ray optics is the short-wavelength limit of wave optics. Quantum mechanics is the corresponding wave theory, which is why the Hamilton-Jacobi equation is closely related to the Schrödinger equation.

Concrete

For a very nice explanation of the origin of the Hamilton-Jacobi equation and its meaning see chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan

The Hamilton-Jacobi equation is essentially a dispersion relation for a complex wave. This is easy to see in the context of non-relativistic quantum mechanics. If a quantum amplitude is expressed in the form ψ = Rexp(iS/h¯), then the Hamilton-Jacobi equation relates p = ∂S/∂q to E = −∂S/∂t by the condition p2(q) = 2m(E −V). This is a relation between the wave vector k = p/h¯ and the frequency ω = E/h¯ of the “matter wave” associated with the particle.chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan

Take note that while the Hamilton equation for time-evolution is closely related to the Heisenberg equation in quantum mechanics, the Hamilton-Jacobi equation is closely connected to the Schrödinger equation. See, for example, Section 4.8.1 in Tong's lecture notes.

• No-Nonsense Classical Mechanics by Jakob Schwichtenberg, section "Hamilton-Jacobi Mechanics"

Abstract

We consider a manifold $Q$ and a Lagrangian $L \colon TQ\rightarrow\mathbb{R}$. Now, we define Hamilton's principal function $$W \colon Q\times\mathbb{R}\times Q\times\mathbb{R}\longrightarrow\mathbb{R}$$ by $$W(q_0,t_0; q_1, t_1) = \inf_{q\in\Upsilon} S(q)$$ where $$\Upsilon=\bigl\{ q \colon [t_0,t_1]\rightarrow Q,\, q(t_0)=q_0,\text{ \& }q(t_1)=q_1 \bigr\}$$ and $$S(q) = \int_{t_0}^{t_1} L\Bigl(q(t),\dot{q}(t)\Bigr)\,dt .$$ Here $W$ is the least action for some path from $(q_0,t_0)$ to $(q_1,t_1)$. The infimum is attained by the classical path only when $t_1-t_0$ is small enough; in general $W$ is the action evaluated on the classical path, a stationary point of $S$. We have $$\frac{\partial}{\partial q_1^i}W(q_0,q_1) = (p_1)_i,$$

where $p_1$ is the momentum of the particle going from $q_0$ to $q_1$, at time $t_1$, and $$\frac{\partial W}{\partial q_0^i}= -(p_0)_i ,\qquad\text{(-momentum at time $t_0$)}$$ $$\frac{\partial W}{\partial t_1}= -H_1,\qquad\text{(-energy at time $t_1$)}$$ $$\frac{\partial W}{\partial t_0}= H_0,\qquad\text{(+energy at time $t_0$)}$$

Take note that $H_1=H_0$ since energy is conserved for a time-independent Lagrangian. Combining the first and third of these equations gives the Hamilton-Jacobi equation $\frac{\partial W}{\partial t_1} + H\left(q_1, \frac{\partial W}{\partial q_1}\right) = 0$, and likewise at $(q_0,t_0)$.

Path to this page

1. Calculus
2. Linear Algebra
3. Functional
4. Legendre Transformation
5. Newton's Second Law
6. Vector Calculus
7. Newtonian Mechanics
8. Variational Calculus
9. Configuration Space
10. Euler-Lagrange Equations
11. Lagrangian Formalism
12. Phase Space
13. Hamiltonian Formalism