Fytb14 Extra Notes: Motion in a Central Force Field

Fytb14 extra notes: Motion in a Central Force Field

A central force F = F (r)er results from a spherically symmetric potential function V (r), as F (r)= −V ′(r). The motion is restricted to a plane, and in plane-polar coordinates (r, ϕ) the Lagrangian is m m L = r˙2 + r2ϕ˙ 2 − V (r) (1) 2 2 which gives the equations of motion (note the cyclicity of ϕ)

′ mr¨ = mrϕ˙ 2 − V (r) (2) d mr2ϕ˙ = 0 (3) dt L ⇒ ϕ˙ = (4) mr2 where L is the value of the conserved angular momentum. Inserting the expression (4) forϕ ˙ into the r equation yields 2 L ′ mr¨ = − V (r) (5) mr3 Note that the radial force as described by (5) can be seen as formally resulting from an eﬀective 2 2 2 2 potential, Veﬀ(r), where a repulsive centrifugal term, L /(2mr ) ≡ mr ϕ˙ /2 representing the angular part of the kinetic energy, is added to V (r), yielding

L2 Veﬀ(r)= V (r)+ (6) 2mr2 A ﬁrst integration of eq. (5) is easily done, yielding

m L2 r˙2 + + V (r)= const. = E (7) 2 2mr2 which shows the conservation of energy. This implies

2 r˙ = ± (E − Veff(r)) (8) rm with the interpretation that in a bound orbit, the radius will oscillate over an interval with Veff(r) < E, bounded by a pair of turning points r1, r2, characterized by Veff = E.

Shape of the Orbit

To derive an equation for the trajectory shape, r(ϕ), we can use the expression (4) forϕ ˙ to convert time derivatives to ϕ derivatives, d d L d =ϕ ˙ = (9) dt dϕ mr2 dϕ Using this on the r equation, and denoting ϕ derivatives with a prime, we get

2 ′ 2 L d −r L ′ − − + V (r)=0 (10) mr2 dϕ r2 mr3

1 where V ′(r)= dV (r)/dr. In terms of u(ϕ) = 1/r(ϕ), this simpliﬁes to the generic Binet’s equation

2 L ′′ ′ (u + u)+ W (u) = 0. (11) m Here, W (u) = V (1/u) = V (r) is the potential energy function expressed as a function of u, and W ′(u) ≡ dW (u)/du. We have also used d d = −u2 (12) dr du A ﬁrst integration of Binet’s equation is easily done. It yields 2 L ′ u 2 + u2 + W (u)= const. = E, (13) 2m expressing the conservation of energy, in accordance, term-by-term, with eq. (7).

A circular orbit corresponds to a stationary value of r, i.e. a constant u(ϕ); according to Binet’s eq. (11), this reqires a local minimum (or maximum) of the eﬀective potential Weﬀ(u) ≡ W (u)+ L2u2/(2m), and an exactly matching energy.

Special case: the Gravitational Force

For the special case of the gravitational force, V (r)= −GMm/r, i.e.

W (u)= −GMmu (14)

Binet’s equation simpliﬁes to 2 L ′′ (u + u) − GMm = 0 (15) m ′′ 2 2 2 2 or u + u − GMm /L = 0, which is easily solved to yield u(ϕ)= GMm /L + C cos(ϕ − ϕ0), i.e.

2 L −1 r(ϕ)= (1 + e cos(ϕ − ϕ0)) , (16) GMm2 describing a planetary trajectory of excentricity e, and with perihelium at ϕ0.

The energy associated with such a trajectory is obtained from eq. (13), and becomes

G2M 2m3 E = e2 − 1 (17) 2L2 This gives a relation between the energy E, the angular momentum L, and the excentricity e. In particular, we see that a negative E is necessary for a closed orbit (bounded r).

A circular orbit means a stationary (constant) r = R, which requires e = 0. This implies R = L2 2 3 2 GMm2 . Using that L = mR ϕ˙, this means R ϕ˙ = GM. For a circular path,ϕ ˙ = 2π/T , where T is the period of the orbit, and we get Kepler’s relation

4π2R3 = GMT 2

between the radius R and the period T of a circular planetary orbit. In reality, the solar mass M should be replaced by M + m.