# Fooling around with Hamilton-Jacobi theory: can this be right?

• pellman
In summary, the conversation discusses the concept of Hamilton's principal function, which is a function that satisfies the equation H(x,\frac{\partial S}{\partial x})+\frac{\partial S}{\partial t}=0. The conversation also clarifies that the notation p=\frac{\partial S}{\partial x} is ambiguous and not generally applicable to p(x,t). It also explains that this notation only holds for specific solutions in phase space, where p(t)=\frac{\partial S}{\partial x}|_{x=x(t)}. The speaker also asks for confirmation that their understanding is correct.
pellman
A system with one particle in one dimension x, momentum p, and hamiltonin H(x,p). Hamilton's principal function S(x,t) is a function satisfying.

$$H(x,\frac{\partial S}{\partial x})+\frac{\partial S}{\partial t}=0$$

Now when we say that

$$p=\frac{\partial S}{\partial x}$$

this is somewhat nonsensical. The LHS, p, is a dynamical variable, independent of x, whereas the RHS is a function of x and t. It makes sense to refer to x(t) and p(t) separately, but not generally of p(x,t).

What $$p=\frac{\partial S}{\partial x}$$ must mean is that along any particular solution (x(t), p(t)) in the flow of solutions in phase space, it is the case that

$$p(t)=\frac{\partial S}{\partial x}|_{x=x(t)}$$

That is, we can speak of a function $$\hat{p}(x,t)\equiv\frac{\partial S}{\partial x}$$ and then it is true that if x(t),p(t) are solutions to the equations of motions then for all t

$$p(t)=\hat{p}(x(t),t)$$.

Let me pause here before proceeding to the question proper. Is what I say so far correct?

Last edited:
You are correct that partial derivative notation is ambiguous (cf. the additional notation which accompanies thermodynamic identities), although it should not normally cause confusion.

Thanks, confinement.

Actually, I don't need to post the actual question now. I figured out where I was going wrong. However, I would be interested in any comments about the OP.

## 1. What is Hamilton-Jacobi theory?

Hamilton-Jacobi theory is a mathematical theory that describes the motion of particles in a system. It is based on the principle of least action, which states that the path taken by a particle between two points is the one that minimizes the action, a quantity that describes the energy of the system.

## 2. How is Hamilton-Jacobi theory used in science?

Hamilton-Jacobi theory has many applications in various fields of science, such as classical mechanics, quantum mechanics, and optics. It is used to study the motion of particles, the behavior of waves, and the evolution of physical systems over time.

## 3. Can Hamilton-Jacobi theory be applied to all systems?

No, Hamilton-Jacobi theory is limited to systems that can be described by a Lagrangian function. This includes most physical systems, but there are cases where it may not be applicable, such as systems with non-conservative forces or non-deterministic behavior.

## 4. How does Hamilton-Jacobi theory relate to other theories in physics?

Hamilton-Jacobi theory is closely related to other theories in physics, such as the Hamiltonian formalism and the principle of least action. It also has connections to other areas of mathematics, such as differential equations and calculus of variations.

## 5. Is Hamilton-Jacobi theory still relevant in modern science?

Yes, Hamilton-Jacobi theory is still widely used in modern science, particularly in theoretical physics and engineering. It has also been extended and generalized to apply to more complex systems, making it a valuable tool in many areas of research and practical applications.

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