Is conservation of momentum and energy valid for non-inertial frames?

Question

Conservation laws of momentum and energy are said to be the most basic principles of physics. Are they also valid for non-inertial frames, and in what way?

Answer 1

Regarding total momentum conservation, the point is that in non-inertial reference frames inertial forces are present acting on every physical object. Momentum conservation is valid in the absence of external forces.

However, if these forces are directed along a fixed axis, say $e_x$, or are always linear combinations of a pair of orthogonal unit vectors, say $e_x,e_y$, (think of a frame of axes rotating with respect to an inertial frame around the fixed axis $e_z$ with a constant angular velocity), conservation of momentum still holds in the orthogonal direction, respectively. So, for instance, in a non-inertial rotating frame about $e_z$, conservation of momentum still holds referring to the $z$ component.

Mechanical energy conservation is a more delicate issue. A general statement is that, for a system of points interacting by means of internal conservative forces, a notion of conserved total mechanical energy can be given even in non-inertial reference frames provided a technical condition I go to illustrate is satisfied.

Let us indicate by $I$ an inertial reference frame and by $I'$ the used non-inertial frame. Assume that our physical system is made of a number of points interacting by means of conservative true forces depending on the differences of position vectors of the points, so that a potential energy is defined and it does not depend on the reference frame.

If the origin of $I'$ has constant acceleration with respect to $I$ and the same happens for the angular velocity $\omega$ of $I'$ referred to $I$ (it is constant in magnitude and direction), then only three types of inertial forces take place in $I'$ and all them are conservative but one which does not produce work (Coriolis' force). In this case, the sum of the kinetic energy in $I'$, the potential energy of the true forces acting among the points and the potential energy of the inertial forces appearing in $I'$ turns out to be conserved in time along the evolution of the physical system.

Answer 2

Balance of momentum and energy (and also angular momentum) is valid in any frame and any coordinate system. It's important here to make a distinction between a balance law and a conservation law.

For extensive quantities like momentum and energy, we can measure three things:

• how much is in a given volume at a given time;
• how much is being transferred through the volume's boundary, in the unit of time, at a given time; technically called flux;
• how much is being generated within the volume, in the unit of time, at a given time; technically called supply or source.

The balance law for momentum and energy say that the rate of change of the amount in the volume must equal the amount transferred through the volume's boundary (flux), plus the amount created within (source), in the unit of time: $$ \frac{\mathrm{d}\text{(content)}}{\mathrm{d}t} = \text{(flux through boundary)} + \text{(internal source)} $$ For momentum, the "flux through boundary" is what we call a contact force (it's exerted on, and scales with, an area), like friction; the "internal source" is what we call a body force (it's exerted on, and scales with, a volume), like the gravitational force. For energy, the "flux though boundary" is what we call heat and work or power done by a contact force; the "internal source" is what we call the work done by a body force (and sometimes volume heating, as it occurs in a microwave oven for instance).

Now, in the special case where there never is an internal source, that is, the second term on the right in the balance above is always zero, then we speak of a conservation law.

What happens with momentum and energy is that in some reference frames their "internal sources" are always zero. This is true in a freely-falling frame for instance.

In a frame at rest with respect to the Earth's surface (as it's typically chosen in physics problems, and called inertial), strictly speaking momentum is not conserved, but only balanced, because the gravitational force is a body force, not a contact one.

For energy in such a frame we have two choices:

1. Include gravitational potential energy as part of the "total energy". In this case this total energy is conserved in this frame.

2. Only count internal and kinetic energy as "total energy". In this case this (differently defined) total energy is balanced but not, strictly speaking, conserved, because the work done by the gravitational force constitutes an internal supply.

For momentum in a non-inertial frame, additional forces – what we call "inertial forces" – will appear as additional internal sources. Correspondingly we will have additional internal sources of energy: the work done by these additional forces.

Also, we must not forget that the numerical values of the volume content, flux, and source will generally be different in different frames. They will always satisfy a balance law, but with different values.

Unfortunately some texts use the terms "conservation"/"conserved" as synonyms of "balance"/"balanced", so the subtle distinction above is lost. If we use the term "conserved" in this more general sense, then we can say that momentum and energy are conserved in any frame.

The distinction becomes especially important in the (Newtonian) mechanics of continua and fluid mechanics. So if you read texts which focus on these more general disciplines, you get a clearer picture. Here are some example references:

• Lautrup: Physics of Continuous Matter

• Müller, Müller: Fundamentals of Thermodynamics and Applications

• Marsden, Hughes: Mathematical Foundations of Elasticity

• Samohýl, Pekař: The Thermodynamics of Linear Fluids and Fluid Mixtures

• Chung: General Continuum Mechanics

• Bird, Stewart, Lightfoot: Transport Phenomena

• Truesdell, Toupin: The Classical Field Theories

• Šilhavý: The Mechanics and Thermodynamics of Continuous Media