# Laplace's equation vs Principle of Least Action?

• Newton-reborn
In summary, there is a possible similarity between the solutions of Laplace's equation and the principle of least action. This can be seen in the case of a one-dimensional Laplace equation where the solution is a straight line and the curve that minimizes the action is also a straight line. It is unclear if one was derived from the other. A link to a lecture with a beautiful derivation is provided. In addition, there is a question about whether the principle of minimum entropy generation also applies in a quantum-mechanical description, but this remains unknown.
Newton-reborn
There seems to be a similarity between the solutions of laplace's equation and the principle of least action. e.g. the solution of a one dimensional laplace equation is a straight and the curve that minimizes the action is also a straight line. Was one derived from the other? Newbie here. Id appreciate any answer.

Dale
mitochan said:
Hi. You see the truth. You can see beautiful derivation in
http://www.feynmanlectures.caltech.edu/II_19.html

uhm, in the note after the lecture, talking about currents and velocity distributions Feynman says "The question is: Does the same principle of minimum entropy generation also hold when the situation is described quantum-mechanically? I haven’t found out yet. "

Does this have a known answer?

## 1. What is Laplace's equation?

Laplace's equation is a partial differential equation that describes the behavior of a scalar field in a given space. It is used to solve problems related to electrostatics, fluid dynamics, and heat transfer.

## 2. What is the Principle of Least Action?

The Principle of Least Action is a fundamental principle in physics that states that the path taken by a system between two points is the one that minimizes the action, which is the integral of the Lagrangian over time.

## 3. What is the relationship between Laplace's equation and the Principle of Least Action?

Laplace's equation and the Principle of Least Action are both used to describe physical systems, but they are different concepts. Laplace's equation is a mathematical tool used to solve problems, while the Principle of Least Action is a fundamental principle that describes the behavior of physical systems.

## 4. How are Laplace's equation and the Principle of Least Action applied in science?

Laplace's equation is used in many fields of science, including electromagnetism, fluid dynamics, and heat transfer, to solve problems related to these systems. The Principle of Least Action is used in theoretical physics to describe the behavior of physical systems and predict their future states.

## 5. What are the limitations of Laplace's equation and the Principle of Least Action?

Laplace's equation can only be applied to systems that are linear and have a steady state. The Principle of Least Action is limited to systems that can be described by a Lagrangian, and it does not take into account quantum effects. Additionally, both concepts have limitations in their applicability to complex systems.

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