Section 13–3 Summation of energy

(Time derivative of energy / Energy between the pairs / Sum of the works)

In this section, Feynman discusses the time derivative of energy, gravitational potential energy between any pairs of particles, and the sum of works done that brought in the particles.

1. Time derivative of energy:

“… we see that Eqs. (13.16) and (13.15) are precisely the same, but of opposite sign, so that the time derivative of the kinetic plus potential energy is indeed zero (Feynman et al., 1963, section 13–3 Summation of energy).”

Feynman moves on to a more general consideration of what happens to the total energy of a large number of particles. He suggests that we can prove the constancy of summation of energies by differentiating the kinetic energies of all particles and gravitational potential energies of all pairs of particles in a system. This is not a conventional way to show the total energy is a constant. Alternatively, one may explain that the total energy of a closed (isolated) system is constant. On the other hand, the total energy of an open system is not constant because it allows energy transfer. The constancy of summation of energies is dependent on how we define a system of particles.

In chapter 4, Feynman states that the law of conservation of energy is closely related to the symmetry of time in quantum mechanics. In his Ph.D. thesis titled A New Approach to Quantum Theory, he writes that “in the theory of action at a distance, the kinetic energy of the particles is not conserved. To find a conserved quantity, one must add a term corresponding to the ‘energy in the field’ (Feynman, 1942, p. 11).” He proceeds to show the constancy of total energy of particles (or Noether’s theorem) using a transformation of time-displacement. Based on an interview with Feynman, Mehra (1994) writes that “Feynman proved a thing called Noether’s theorem, not knowing that it was known (p. 132).”

2. Energy between the pairs:

“…the potential energy is also simple, it being also just a sum of contributions, the energies between all the pairs (Feynman et al., 1963, section 13–3 Summation of energy).”

Feynman explains that the involvement of potential energy is simple because it is only about the gravitational potential energy between all pairs of particles. In addition, the symbol S means that given values of i and j occur only once and we can keep track of this if we let i range over all values 1, 2, 3, …, and for each i, we can restrict the range of j to be greater than i. Simply phrased, one may specify the condition i < j (or state it as a subscript) to prevent a possibility of ‘double counting’ in listing the terms. It could be clearer if Feynman also writes down all the terms, for example, -Gm₁m₂/r₁₂ -Gm₂m₃/r₂₃ -Gm₁m₃/r₁₃, for a system of three particles.

Physics teachers could explain that the gravitational potential energy is a property of a system of particles instead of an individual particle. In other words, we cannot divide the total gravitational potential energy by specifying the quantity of energy that belongs to every particle. Some textbook authors elaborate that “if M >> m, as is true for Earth (mass M) and a baseball (mass m), we often speak of ‘the potential energy of the baseball.’ We can get away with this because, when a baseball moves in the vicinity of Earth, changes in the potential energy of the baseball – Earth system appear almost entirely as changes in the kinetic energy of the baseball, since changes in the kinetic energy of Earth are too small to be measured (Halliday, Resnick, & Walker, 2005).” In essence, it is inappropriate to specify the gravitational potential energy of individual object if the objects in a system are comparable in mass.

3. Sum of the works:

“Therefore the work done is the sum of the works done by each because if F₃ can be resolved into the sum of two forces, F₃ = F₁₃ + F₂₃… (Feynman et al., 1963, section 13–3 Summation of energy).”

We can understand the sum of works done (or change in potential energy of particles) as follows: Firstly, when we bring in the first particle, there is no work done because no other particles are present to exert a force on it. Secondly, when we bring in the second particle, there is only work done between the two particles, W₁₂ = −Gm₁m₂/r₁₂. Thirdly, when we bring in the third particle, the work done is the sum of the works done against the first two particles because F₃ can be resolved as a sum of two forces, F₃ = F₁₃ + F₂₃. Mathematically, the net work done can be written as ½(−Gm₁m₂/r₁₂ −Gm₂m₃/r₂₃ −Gm₁m₃/r₁₃) and it is independent of the order of the particles that are brought to their final positions. However, one may explain that the potential energy of particles is stored in the fields instead of sharing among the particles.

In the case of electrostatic energy, it can be expressed as “U = ½[q₁ϕ(1)+q₂ϕ(2)]… That is why we need the factor ½ (Feynman et al., 1964, Chapter 8).” Using Feynman’s notations, the gravitational potential energy stored in three objects can also be written as U = ½[m₁Y(1) + m₂Y(2) + m₃Y(3)]. Initially, Feynman explains that the idea of locating the energy in the field is inconsistent with the assumption of the existence of point charges. To resolve this inconsistency, he suggests that “[o]ne way out of the difficulty would be to say that elementary charges, such as an electron, are not points but are really small distributions of charge… (Feynman et al., 1964, Chapter 8).” One may use the principle of mass-energy equivalence to argue that the gravitational field has energy, and thus it has mass.

Questions for discussion:

1. What is/are the implication(s) if the time derivative of the kinetic plus potential energy is zero?

2. How would you explain the gravitational potential energy is a property of a system of particles?

3. How would you explain the work done is the sum of the works done by each force?

The moral of the lesson: we can write the potential energy as a sum over each pair of particles because gravitational forces obey the principle of superposition of forces.

References:

1. Feynman, R. P. (1942/2005). Feynman’s thesis: A New Approach to Quantum Theory. Singapore: World Scientific.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

4. Halliday, D., Resnick, R., & Walker, J. (2005). Fundamentals of Physics (7th ed.). New York: Wiley.

5. Mehra, J. (1994). The Beat of a Different Drum: The life and science of Richard Feynman. Oxford: Oxford University Press.