# Where does Hamilton's Principle come from?

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• PhilipsPhysics
In summary, Hamilton's Principle and the physical entity action are key concepts in modern physics. The Lagrangian is used to find a system's Hamiltonian, which is then used in Hamilton's Principle to determine the stationary action. This idea is rooted in the principle of relativity and the Poincare group, leading to the Hamiltonian dynamics of Wigner and Dirac.

#### PhilipsPhysics

TL;DR Summary
Hamiltons Principle
Hamiltons Principle and the physcial entity action are the terms in which modern physics is formulated. How do you know that you can always find a Lagrangian for a System which is then used for Hamitlons Principle and the formulation of Action? Why is the action stationary? What have you found so far on this question?

Thanks! Pretty much nothing, I know classical mechanics, quantum field theory, electromagnetism can formulated this way by I don't see why (I understand the math, I don't see where it comes from).

In my opinion, the deepest origin of the Hamiltonian formalism is the symmetry associated with the principle of relativity.

As all inertial observers are equivalent, there has to be a Lie group of transformations connecting different observers. In relativistic physics this is the Poincare group. According to Wigner, transformations from this group are represented by unitary operators in the Hilbert space of any isolated physical system. Generators of this representation form a 10-dimensional Lie algebra of Hermitian operators. The generator of time translations is the Hamiltonian.

This path will lead you to the relativistic Hamiltonian dynamics of Wigner and Dirac.

Eugene.

## 1. What is Hamilton's Principle?

Hamilton's Principle, also known as the Principle of Least Action, is a fundamental principle in classical mechanics that describes the motion of a system as the one that minimizes the action, which is the integral of the Lagrangian function over time.

## 2. Who developed Hamilton's Principle?

Hamilton's Principle was developed by Irish mathematician and physicist William Rowan Hamilton in the 1830s.

## 3. What is the significance of Hamilton's Principle?

Hamilton's Principle is significant because it provides a powerful and elegant way to describe the motion of a mechanical system, and it can be extended to other areas of physics such as electromagnetism and quantum mechanics.

## 4. How is Hamilton's Principle derived?

Hamilton's Principle is derived from the principle of stationary action, which states that the true path of a system is the one that makes the action stationary. The action is calculated by integrating the Lagrangian function over time and the stationary condition is achieved when the variation of the action with respect to the system's coordinates is equal to zero.

## 5. Can Hamilton's Principle be applied to all physical systems?

While Hamilton's Principle is most commonly used in classical mechanics, it can also be applied to other physical systems such as fluid mechanics, quantum field theory, and general relativity. However, its applicability may be limited in certain cases, and other principles may need to be used instead.