# Without Lagrangian, show that angular momentum is conserved

• jack476
In summary, the conversation discusses the possibility of showing that rotational invariance about an axis implies the conservation of angular momentum about that axis without using the Lagrangian formalism or Noether's theorem. The proposed solution involves considering the interactions of a system with the outside world in terms of a potential energy function and using vector geometry to show that the force and torque experienced by the system are both 0, leading to the conservation of angular momentum. The person asking for help is unsure if this solution is valid and is seeking confirmation.
jack476

## Homework Statement

I'd like to show, if possible, that rotational invariance about some axis implies that angular momentum about that axis is conserved without using the Lagrangian formalism or Noether's theorem. The only proofs I have been able to find use a Lagrangian approach and I'm wondering if it's possible to do this using only geometry and vector mechanics.

## Homework Equations

Definition of vector angular momentum and torque, basic vector geometry in coordinate-free form.

## The Attempt at a Solution

Suppose that the interactions of a system S with the outside world can be described in terms of a potential energy function U, and that we measure U at some point P with position vector r with respect to some coordinate system embedded in S. Suppose that S is rotated by an arbitrarily small angle Δθ about one of the axes (Δθ points in the direction of the axis). Then the position vector of point P in this new coordinate system is r+Δθr. Suppose that this has not changed the measured value of U so that U(r) - U( r+Δθr) =0. This is true for an arbitrarily small rotation and therefore $$\lim_{\mid \Delta \vec{\theta} \times \vec{r} \mid \to 0} \frac{U(\vec{r} +\Delta {\vec{\theta}} \times \vec{r}) - U(\vec{r})}{\mid \Delta \vec{\theta} \times \vec{r} \mid} = 0$$

But this is simply the directional derivative of U in the direction of Δθr, which means that the force at point P in that direction is 0. By Newton's third law, this means that the torque about the axis of rotation experienced by S is also 0. Therefore angular momentum about that axis is constant in any motion of this system.

There's no specific thing here that I'm not sure of, I just have this nagging feeling that it shouldn't be this simple.

Bump?

## 1. How is angular momentum defined?

Angular momentum is a measure of the amount of rotational motion an object possesses. It is defined as the product of an object's moment of inertia and its angular velocity.

## 2. What is the Lagrangian method?

The Lagrangian method is a mathematical approach used in classical mechanics to describe the motion of a system of particles. It uses a single function, called the Lagrangian, to represent the total energy of the system.

## 3. Why is the Lagrangian method important in understanding conservation of angular momentum?

The Lagrangian method allows us to analyze the forces and motion of a system without explicitly considering the individual forces acting on each particle. This allows us to easily show that angular momentum is conserved, as the Lagrangian itself is conserved.

## 4. How does the Lagrangian method prove conservation of angular momentum?

The Lagrangian method shows that the Lagrangian is constant over time for a system with no external forces, which means that the total energy of the system is conserved. Since angular momentum is a form of energy, it must also be conserved.

## 5. Can angular momentum still be conserved without using the Lagrangian method?

Yes, angular momentum can still be conserved without using the Lagrangian method. It can be shown using other mathematical methods, such as using Newton's laws of motion and the principle of conservation of energy. However, the Lagrangian method provides a more elegant and concise way of proving conservation of angular momentum.

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