Lagrange Equations for Non-Inertial Frame of Reference

Question

I am trying to expand my limited knowledge of Lagrange's equations for evaluating motion. Regarding the Lagrangian in a rotating coordinate system, the text Mechanics by Symon states "...we use the kinetic energy with respect to a coordinate system at rest, expressed in terms of the rotating coordinates, and not the kinetic energy as it would appear in the rotating system if we ignored the motion of the coordinate system." Perhaps this has something to do with expressing the generalized forces in an inertial frame (not including fictitious forces)? Can the community elaborate on this?

Answer 1

I will start of this answer by a "simpler" example. Lets consider a free particle. The Lagrangian for a non-relativistic free particle $$ \mathcal L = \frac{m}{2}\mathbf v^2 $$ where $m \in \mathbb R_{>0}$ is the particles mass and $\mathbf v \in \mathbb R^3$ is the particles velocity with respect to a chosen inertial frame. Lets now consider an translational accelerated frame of reference (with respect to the previously chosen inertial frame). We therefore use the transformation $$ \mathbf v \to \mathbf v' + \mathbf v_{\text{frame}}(t). $$ Let us consider for example a frame which is accelerated by a constant acceleration $\mathbf a_{\text{frame}}$. So the transformation becomes $$ \mathbf v \to \mathbf v' + \mathbf a_{\text{frame}}\, t . $$ So the lagrangian for the accelerated system is $$ \mathcal L = \frac{m}{2}(\mathbf v' + \mathbf a_{\text{frame}}\, t)^2 =\frac m 2 (\mathbf v' ^2 + 2\,\mathbf v'\,\mathbf a_{\text{frame}}\,t + (\mathbf a_{\text{frame}}\, t)^2) $$ which results in the equations of motion $$ \frac{d}{dt}(m\mathbf v'+m\,\mathbf a_{\text{frame}}\,t) = 0\\ \implies m\mathbf a' = -m\mathbf a_{\text{frame}} $$ which describes exactly the fictitious force we expected. So for rotating reference frames it is essentially the same. You start with the lagrangian in a inertial frame and transform your system to the accelerated frame. The transformation is then given by $$ \mathbf v \to \mathbf v' + \mathbf\omega\times\mathbf r' , $$ where $\mathbf \omega$ is the angular frequency of the rotating system. A good reference would be "Landau, Lifschitz - Mechanics".In my edition the treatment of accelerated frames of reference is discussed on p.155-159.

Answer 2

@AlmostCluesless gave a nice calculation, I will just add a bit of philosophy, while we're at it, since you were interested in the relevant concepts.

In classical mechanics, you're stuck with Newton's idea of absolute space and time. When those are given, you can decide for each coordinate frame of reference whether it is at rest or it follows linear or accelerated motion. Physics in a linearly moving coordinate system is equivalent to physics in the rest system, all these equivalent refrence frames are the inertial frames of reference. If you leave those to get into an accelerated frame of reference, you get fictitious forces. That's Newton's picture.

Lagrangian mechanics is a formulation of classical mechanics that, in its mathematical essence, can be given before choosing any specific frame of reference. The exact mathematical theory is often suppressed in introdutory courses to analytical mechanics. That makes sense, it requires establishing mathematical structures that are a bit lofty for everyday calculations. So, one ditches the full mathematical machinery and rather works with a certain set of generalized coordinates. One can, then, develop the relevant concepts hand in hand with intuition, just by stating that the Lagrangian density is $L = T - U$.

This is all fine, but one has to keep in mind that the kinetic energy, $T$, in this formulation has to be given with respect to an inertial frame of reference with respect to absolute space. Otherwise it won't work, because the extra terms that would arise don't fit the prescription $L = T - U$ in a straightforward way.

It's like talking about the length of a vector in $\mathbb{R}^2$. The concept can be explained without choosing a specific set of basis vectors for $\mathbb{R}^2$ (the relevant mathematical object is the so-called metric tensor, you may have already encountered it, otherwise ignore this). However, if you were to say that the length $L$ always follows the prescription $$L(v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2) = v_1^2 + v_2^2, \tag{$\Delta$}$$ then that is true for the standard orthogonal coordinate basis, $\mathbf{e}_1 = \mathbf{e}_x$ and $\mathbf{e}_2 = \mathbf{e}_y$, or in usual column vector notation $$\mathbf{e}_x = \begin{pmatrix} 1 \\ 0\end{pmatrix}, \qquad \mathbf{e}_y = \begin{pmatrix} 0 \\ 1\end{pmatrix},$$ but it wouldn't work for the non-orthogonal basis $\mathbf{e}_1 = \mathbf{e}_x$ and $\mathbf{e}_2 = \mathbf{e}_x + \mathbf{e}_y$. You could still resolve this by adding extra terms, namely $$L(v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2) = L((v_1 + v_2)\mathbf{e}_x + v_2 \mathbf{e}_y) = (v_1 + v_2)^2 + v_2^2 = v_1^2 + 2 v_1 v_2 + 2v_2^2,$$ but that would not be of the form $(\Delta)$.

In the example, you must require the coordinate basis to be orthogonal with respect to the inner product. In Lagrangian mechanics, you must require the coordinate system to correspond to some inertial frame of reference. After that, you can still use the typical transformations to a non-inertial frame of reference and the additional terms will come out the way they are supposed to. But starting in a non-inertial reference frame will not yield the correct additional terms.

Answer 3

The statement by Symon is not specific for using the lagrangian formulation of classical dynamics. The statement extends to any use of the concept of kinetic energy in theory of motion.

The equivalence class of inertial coordinate systems expresses the physical properties of inertia.

In order to formulate a theory of motion one must grant the existence of inertia. Conversely, if one does not grant the existence of inertia one cannot formulate a theory of motion at all.

Mathematically one can use a coordinate system that is accelerating (in one form or another), the most frequently used example is of course a rotating coordinate system.

If you use a rotating coordinate system then the equation of motion includes terms with the angular velocity of the rotating coordinate system with respect to the inertial coordinate system. That is, you can use a rotating coordinate system if and only if you continue to use the inertial coordinate system as the underlying reference of motion.

Generalizing: a self-consistent theory of motion obtains if and only if the equivalence class of inertial coordinate systems is used as the reference of motion, because it is the equivalence class of inertial coordinate systems that is correlated with physical inertia.

Returning to the quote from Symon: "we use the kinetic energy with respect to a coordinate system at rest"

For the Lagrangian formulation of classical dynamics the same thing applies as for the newtonian formulation: a self-consistent theory obtains if and only if the equivalence class of inertial coordinate systems is used as the reference of motion. Hence one must use the kinetic energy with respect to the inertial coordinate system.

Inertia

In the following I will discuss the evolution of the concept of inertia throughout the centuries. Inertia has been reconceptualized profoundly, so in order to answer this question discussing inertia is an absolute necessity.

In the history of theories of motion we have so far three distinct theories of motion, each one subsuming the predecessor.

Newtonian dynamics
Special relativity
General relativity

In the following I will describe properties of newtonian dynamics using concepts that were introduced with the advent of relativistic physics. That is, I will reassess newtonian dynamics using the benefit of hindsight. Also, out of abundance of caution I will recapitulate a lot, so as to minimize room for misunderstanding.

Newtonian dynamics starts with assuming that inertia is uniform everywhere. An object in inertial motion will cover equal intervals of distance in equal intervals of time.

Special relativity introduces the concept of Minkowski spacetime. Depending on how you move through Minkowski spacetime the amount of proper time that elapses for you is different.

Example: a fleet of spaceships separates, individual spaceships accelerate away from each other in various directions. The crewmen onboard the spaceships are proficient in application of special relativity. Each spaceship has a travel plan that returns the spaceship to a pre-planned rendez-vous point. As the fleet of ships rejoin they confirm that for the spaceship that traveled the longest spatial distance the least amount of proper time has elapsed, in accordance with the Minkowski metric. One spaceship has just coasted from starting point to rendez-vous point; the amount of proper time that has elapsed for that ship is the maximum possible.

What that means is that in terms of the Minkowskian formulation of theory of motion the metric of Minkowski spacetime constitutes a theory of the physical properties of inertia.

Minkowski spacetime is a spacetime that has a physical effect upon entities moving through it. At the same time Minkowski spacetime is itself not thought of as subject to physical effect. Minkowski spacetime itself has the following in common with newtonian spacetime: Minkowski spacetime itself is not thought of as subject to any form of change.

General relativity introduces a spacetime that is subject to physical effect. GR-spacetime affects matter moving through it, and is affected by gravitational source in it. In terms of GR inertia is a dynamic entity because in terms of GR spacetime itself is a dynamic entity.

In GR the following from SR is carried forward: the metric of the spacetime describes the physical inertia.

This recognition of spacetime being subject to physical effect is a fundamental move away from the Newtonian/Minkowskian concept of a spacetime that is not subject to change.

For an astronomer studying the solar system: the reference of motion (for objects in the solar system) is the common center of mass of the bound system: the celestical bodies of the solar system.

For the astronomer studying the Galaxy: the reference of motion (of the objects that constitute the Galaxy) is the common center of mass of the bound system: the Galaxy.

For the astronomer studying clusters of Galaxies: etc.

As we know, the center of mass of the solar system is orbiting the center of mass of the Galaxy. In that sense the local inertial coordinate system of the solar system is in accelerated motion with respect to the local inertial coordinate system of the Galaxy.

The point is: all these inertial coordinate systems do not rotate with respect to each other.

(I am aware, of course, of the confirmation (by way of the Gravity Probe B experiment) of the phenomenon of frame dragging, in accordance with GR, but that excruciating level of detail is beyond the scope of this answer.)

Here is why I presented the above:
For the purpose of considering rotation the picture in terms of GR does not depart much from the picture in terms of Newtonian spacetime.

All the local inertial coordinate systems have in common that they do not rotate with respect to each other.

All the local coordinate sytems of the Galaxy do not rotate with respect to each other.

The question is not 'Is rotation absolute?'. The proper question is: 'Is there a universally coordinated zero value of rotation rate?' The answer to the latter question is yes. All available astronomical data corroborate that all of the local inertial coordinate systems of the Galaxy do not rotate with respect to each other. All the local inertial coordinates sytems of the Galaxies and clusters of Galaxies that are accessible to astronomic observation do not rotate with respect to each other.