2 Body Radial Equation, Effective Potential derivation

In summary, the derivation for the effective potential U_eff in the textbook involves substituting the angular momentum after taking the derivatives, resulting in an effective potential of V(r) + l^2/2mu r^2. However, when the angular momentum is substituted first, the effective potential becomes V(r) - l^2/2mu r^2, which is the wrong result due to the action principle being affected by the implicit connection between r and phi(dot).
  • #1
vancouver_water
77
10
According to my textbook, in the derivation for the effective potential [itex]U_{eff}[/itex], starting with the Lagrangian [itex]L = \frac{1}{2}\mu(\dot r^2 +r^2\dot\phi^2) -V(r)[/itex], substituting into Lagrange's equation gives [itex]\mu\ddot r = -\frac{\partial V}{\partial r} + \frac{l^2}{\mu r^3} = -\frac{\partial}{\partial r}(V(r) + \frac{l^2}{2\mu r^2})[/itex], where the substitution [itex]\dot\phi = \frac{l}{r^2\mu}[/itex] is made after substituting into Lagrange's equation and the effective potential is [itex]V(r) + \frac{l^2}{2\mu r^2}[/itex] and [itex]l[/itex] is the angular momentum.

However, when I do the derivation by first substituting the angular momentum, it goes like this: Starting with [itex]L = \frac{1}{2}\mu(\dot r^2 +\frac{l^2}{\mu^2 r^2}) -V(r)[/itex], then substituting into lagranges equation gives [itex]\mu\ddot r = -\frac{\partial V}{\partial r} - \frac{l^2}{\mu r^3} = -\frac{\partial}{\partial r}(V(r) - \frac{l^2}{2\mu r^2})[/itex], where the effective potential is [itex]V(r) - \frac{l^2}{2\mu r^2}[/itex].

This is the wrong result, but I can't see why substituting angular momentum before or after taking the derivatives should make any difference.

Thanks for any help!
 
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  • #2
vancouver_water said:
This is the wrong result, but I can't see why substituting angular momentum before or after taking the derivatives should make any difference.

actually we use r,theta description.
here theta is replaced by phi...
no matter but r, dr/dt, phi, dphi/dt are the generalised coordinates and velocities ;
naturally one has two Lagranges equations - and if the result of second Lagranges equation is put in the definition of L itself to describe r-ewuation ; you have problem - the two are independent equations in r and phi.
so the action principle must be getting affected.
r^2. phi(dot) is a constant -where r and phi(dot) both can vary such that the product is a constant.
the error lies in the r-equation as this condition which implicitly connects r and phi(dot) is no longer an independent equation of motion in r.
 

1. What is the 2 Body Radial Equation?

The 2 Body Radial Equation is a mathematical expression used in physics to describe the motion of two particles interacting with each other through a central force. It takes into account the masses of the particles, the distance between them, and the strength of the force.

2. What is the significance of the Effective Potential in the derivation?

The Effective Potential is a key component in the derivation of the 2 Body Radial Equation. It represents the potential energy experienced by one of the particles in the system, taking into account the influence of the other particle. This allows us to simplify the equations and better understand the dynamics of the system.

3. How is the Effective Potential derived?

The Effective Potential is derived by considering the total energy of the system, which includes the kinetic energy of both particles and the potential energy between them. By manipulating the equations and taking into account the symmetry of the system, we can arrive at an expression for the Effective Potential.

4. What assumptions are made in the derivation of the 2 Body Radial Equation?

The main assumptions made in the derivation are that the particles are point masses, the force between them is central, and the potential energy is conservative. These assumptions allow for a simplified mathematical model that is still accurate enough to describe the motion of real particles.

5. How is the 2 Body Radial Equation used in real-world applications?

The 2 Body Radial Equation has many applications in physics, such as in celestial mechanics to describe the motion of planets and satellites, and in atomic and molecular physics to study the interactions between particles. It is also used in fields such as aerospace engineering and astrophysics to model and analyze the motion of objects in space.

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