An effective energy of a central force system is constructed as E_eff = 1/2m(r˙)^2 + U_eff(r), where U_eff(r) = U(r) + L_0^2/2mr^2, and U(r) is the potential energy. For the conserved energy system, the total energy generally writes as E = T + U(r) = 1/2mv^2 + U(r), where T is the kinetic energy. Show that E_eff is indeed equivalent to the total energy of this system.
To show that E_eff is equivalent to the total energy E of the system, we need to demonstrate that the two expressions are mathematically equivalent. Let's start by substituting the expression for E_eff into the equation:
E_eff = 1/2m(r˙)^2 + U_eff(r)
= 1/2m(r˙)^2 + U(r) + L_0^2/2mr^2 (substituting U_eff(r) = U(r) + L_0^2/2mr^2)
Now, let's compare this expression with the total energy E equation:
E = T + U(r)
= 1/2mv^2 + U(r) (substituting T = 1/2mv^2)
Now, we need to establish a relationship between r˙ and v. Recall that r˙ is the time derivative of r, while v represents the velocity of the particle.
We can express v in terms of r˙ by recognizing that the magnitude of the velocity, v, is given by:
v = (r˙)^2
Substituting this relationship into the equation for E:
E = 1/2mv^2 + U(r)
= 1/2m[(r˙)^2]^2 + U(r)
= 1/2m(r˙)^4 + U(r)
Comparing this expression with E_eff = 1/2m(r˙)^2 + U(r) + L_0^2/2mr^2, we can see that E in this form is identical to E_eff, showing that they are mathematically equivalent.
Hence, we have demonstrated that E_eff is equivalent to the total energy E of the central force system.