Relativistic angular momentum and cyclic coordinates

In summary, the conversation was about a relativistic Lagrangian for a particle in a central potential and the calculation of angular momentum. The question was whether the derivative of the Lagrangian with respect to the angular velocity should give the conserved angular momentum, but instead resulted in a different equation. However, upon further calculation, it was discovered that the correct answer was obtained.
  • #1
maverick_starstrider
1,119
6
I'm getting myself confused here. If my relativistic Lagrangian for a particle in a central potentai is

[tex]L = \frac{-m_0 c^2}{\gamma} - V(r) [/tex]

should

[tex] \frac{d L}{d \dot{\theta}} [/tex]

not give me the angular momentum (which is conserved)? Instead I get

[tex] \frac{d L}{d \dot{\theta}} = -4 m v r^2 \dot{\theta}\gamma [/tex]
 
Physics news on Phys.org
  • #2
Anyone?
 
  • #3
[tex]L = - {m_0}{c^2}\sqrt {1 - \frac{{{{\dot r}^2} + {r^2}{{\dot \theta }^2}}}{{{c^2}}}} - V\left( r \right)[/tex]

so

[tex]\frac{{\partial L}}{{\partial \dot \theta }} = \gamma {m_0}{r^2}\dot \theta [/tex]

What's the problem?
 
  • #4
Absolutely nothing apparently. I just did it again this morning and got the right answer. Sorry for the time waste.
 

1. What is relativistic angular momentum?

Relativistic angular momentum is a mathematical concept that describes the rotational motion of a particle in a relativistic system, taking into account the effects of special relativity. It is a conserved quantity and is related to the particle's mass, velocity, and distance from the axis of rotation.

2. How is angular momentum related to cyclic coordinates?

Cyclic coordinates are variables in a physical system that do not directly affect the equations of motion. The principle of least action states that the equations of motion can be expressed in terms of these cyclic coordinates and the corresponding generalized momenta, one of which is angular momentum. This allows for simpler and more elegant solutions to complex systems.

3. What is the difference between classical and relativistic angular momentum?

The main difference between classical and relativistic angular momentum is that classical angular momentum only considers the rotational motion of a particle, while relativistic angular momentum takes into account the particle's linear motion and its relativistic effects. Additionally, relativistic angular momentum is conserved in all inertial reference frames, while classical angular momentum is only conserved in an inertial frame of reference.

4. Can relativistic angular momentum be negative?

Yes, relativistic angular momentum can be negative. In a relativistic system, the sign of angular momentum is determined by the direction of the particle's motion relative to the axis of rotation. If the particle is moving in the opposite direction of the axis of rotation, its angular momentum will be negative.

5. How is relativistic angular momentum used in physics?

Relativistic angular momentum is a fundamental concept in physics and is used in various fields such as quantum mechanics, special relativity, and general relativity. It is used to describe and analyze the rotational motion of particles in relativistic systems, and is also a crucial component in understanding the behavior of objects in gravitational fields.

Similar threads

  • Special and General Relativity
Replies
8
Views
947
  • Special and General Relativity
Replies
4
Views
1K
Replies
5
Views
691
  • Special and General Relativity
Replies
3
Views
1K
  • Special and General Relativity
Replies
15
Views
1K
  • Special and General Relativity
2
Replies
36
Views
3K
  • Special and General Relativity
2
Replies
44
Views
1K
  • Special and General Relativity
Replies
13
Views
1K
  • Special and General Relativity
Replies
5
Views
260
  • Special and General Relativity
Replies
12
Views
1K
Back
Top