# What is Hamiltonian action?

• Identity
In summary, the Hamiltonian function of action is a function used in Hamilton's principle of least action, which states that a mechanical system will behave in a way that minimizes the action integral. The action is a weighted sum of the kinetic and potential energies of the system. It is a fundamental concept in classical mechanics and is also important in quantum mechanics, where it represents the phase of the quantum mechanical amplitude of a process. The Euler-Lagrange equations can be derived from the action, allowing for the derivation of equations of motion without dealing with forces. While it may not have a clear intuitive meaning, it is a powerful tool in understanding the behavior of physical systems.

#### Identity

$$W = \int_{t_0}^t \mathcal{L}\,dt$$

Apparently Schrodinger used it along with the Hamilton-Jacobi equation to derive the Schrodinger equation so it's a pretty important part of quantum physics history.

According to wikipedia it's a function which takes the trajectory of a system and returns a real number. But this seems quite vague to me, what is the significance of the scalar it outputs?

I've also looked in several books, but the mathematics of analytical mechanics is mind-boggling, I don't really have the time right now to fully digest the whole of the subject. The only definitions I've gotten from various book seem to be very vague and mathematical.

Thanks :)

it is energy weighted with time

Hamilton's principle of least action states that a mechanical system will behave in such a way as to minimize this action integral. The Lagrangian L that is integrated is T-V, the kinetic energy of the system minus the potential energy of the system.
Starting from this, all of classical mechanics can be derived very elegantly (as an alternative to starting with Newton's laws), so it's not just important in quantum mechanics but also in classical mechanics.

In practice, you don't really need to work with the action but using the Euler-Lagrange equations $$\frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L}{\partial \dot{x}} = 0$$ (with the dot representing the derivative with respect to time) you can derive the equations of motion for the system. The Euler-Lagrange equations are a consequence of the least action principle, which you will learn when you study the calculus of variations.

You don't really need an intuitive idea for what this action is, all that is important is that it automagically assumes a minimum, and that once you know the Lagrangian (kinetic and potential energy of your system), you can derive the equations of motion without messing around with forces how you are used to from Newtonian mechanics.

For example a particle moving vertically near the Earth's surface has the kinetic energy $$T = \frac{m}{2} \dot{z}^2$$, a potential energy $$V = mgz$$ and therefore a Lagrangian $$L = \frac{m}{2} \dot{z}^2 - mgz$$. In this case, $$\frac{\partial L}{\partial z} = -mg, \frac{\partial L}{\partial \dot{z}} = m \dot{z}$$, therefore $$mg + m\ddot{z} = 0$$ or $$\ddot{z} = -g$$, which of course you knew already, but the point is that this works for much more complicated systems too.

Hope this helps, if not I guess you'll have to wait until you take a class on classical mechanics again.

Thanks niklaus, that helped a lot

it still not says anything what the action "is"...

also niklaus post can be read at wikipedia basically

Well, if the hamiltonian action only really has an abstract meaning then I guess I'll have to settle with that

A deeper understanding of the action is provided by quantum mechanics: the action for a process is the phase of the quantum mechanical amplitude of that process.

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## 1. What is the definition of Hamiltonian action?

Hamiltonian action is a mathematical concept used in the field of physics, specifically in the study of classical mechanics. It refers to the equations of motion that describe the evolution of a system over time based on its energy, known as the Hamiltonian. It is a fundamental principle in understanding the behavior of physical systems.

## 2. How is Hamiltonian action related to the Hamiltonian function?

The Hamiltonian function is an important component of Hamiltonian action. It is a mathematical function that represents the total energy of a system and is used to calculate the equations of motion for that system. The Hamiltonian function is derived from the Hamiltonian action and is a key aspect of its definition.

## 3. What is the difference between Hamiltonian action and Lagrangian action?

Both Hamiltonian action and Lagrangian action are mathematical concepts used in classical mechanics to describe the behavior of physical systems. The main difference is that Hamiltonian action is based on the Hamiltonian function, while Lagrangian action is based on the Lagrangian function. Hamiltonian action is often used to describe systems with a large number of particles, while Lagrangian action is more suitable for systems with a smaller number of particles.

## 4. How is Hamiltonian action used in quantum mechanics?

In quantum mechanics, Hamiltonian action is used to describe the dynamics of a system over time. However, there are some key differences from its use in classical mechanics. In quantum mechanics, the Hamiltonian function represents the system's total energy, but it also includes the effects of quantum uncertainty. This allows for a more accurate description of the system's behavior on a microscopic level.

## 5. What are some real-world applications of Hamiltonian action?

Hamiltonian action has numerous applications in physics, engineering, and other fields. It is used to study the behavior of physical systems such as planets, satellites, and atoms. It is also used in the design of control systems, such as in robotics and aerospace engineering. Additionally, Hamiltonian action is used in the development of new technologies, such as quantum computing and quantum cryptography.