1.1. The Meaning of Time Reversal

In the Newtonian “force” formulation of classical mechanics, time reversal is simply the reversal of the order of events in a trajectory . That is, if is the curve describing the position x of a particle in space at every moment in time t, then the time-reversed motion is given by the curve , which describes the same positions occurring in the reverse order.

In the Hamiltonian formulation of classical mechanics, reversing the order of events in a trajectory is not enough. Since time reversal transforms a rightward-moving body into a leftward-moving body, the momentum of an instantaneous state is reversed, .

The operator T appearing in the first part of the transformation is referred to as the time reversal operator. One can speak quite generally about time reversal operators, in a way that applies to both the Hamiltonian and the Newtonian formulations. In general, the time reversal operator is a bijection on a theory’s space of states, whatever that space may be. In the Hamiltonian description of a particle on a string, it is an operator on phase space, . In the Newtonian description, it is a transformation of physical space, . The latter is easy to miss because it is simply the identity transformation . The former is more conspicuous because it is in general not the identity. When we say time reversal, we mean the application of the time reversal operator on states, together with the reversal of the order of states in trajectories.

1.2. Time Reversal Invariance

Invariance of a law under a transformation means that if one motion is possible according to the law, then the transformed motion is possible as well. More precisely:

We say “the time reversal operator with respect to ” because, at this level of generality, one cannot say much more than that about T. Its meaning depends on physical facts about the degrees of freedom that the space of states represents. If represents the location of a particle in space, T is the identity operator. However, if represents the position and linear momentum of a particle, T is not the identity.

1.3. Two Useful Facts

Let me conclude this section by stating two useful (although not novel) facts, which will facilitate the identification of time reversal invariance in the remainder of our discussion.

Proof (⇒): Suppose . Let satisfy Hamilton’s equations with . Since these equations hold for all t, we can substitute to get and . The former implies , and the latter implies , by simply pushing negative signs around and by our hypothesis that . But this just says that the time-reversed trajectory satisfies Hamilton’s equations.

(⇐): Suppose that (a) and (b) are both solutions. Substituting into Hamilton’s equations with a gives and . Hamilton’s equations with b give and . Combining, we find that

This implies that , for some function f such that . But the only such function is a constant function, so (for some ). QED

The proof of this statement is very similar to that of lemma 1, and so I omit it.

2.1. Conservative but Not Time Reversal Invariant, Part I

Velocity-dependent forces can be conservative in a natural sense but still fail to be time reversal invariant. To see this, we will make use of a textbook definition of “conservative” in the Newtonian formulation. Consider a segment of a smooth curve through configuration space, with an initial point and a final point . The work required to transport the system from to is given by the work integral

where . We say (cf. Goldstein, Poole, and Safko Reference Goldstein, Poole and Safko2002, 3) that a system is conservative if, whenever the curve is a closed loop with , it follows that the work done around the loop is zero. For example, if a particle sets out along a path through space that ends where it started, then no work is accrued during the round-trip journey. Work is “conserved” along such a loop, in the sense that nothing is added or lost.

Although many conservative forces depend only on position, this definition of a conservative system makes perfect sense when a force depends on velocity as well. For example, consider a particle in a force field , which exerts a force on the particle when it has position and velocity . Defining the work integral as before, we can again take a conservative system to be one in which work is conserved on a closed loop through configuration space . We must only take care that some other well-known definitions of a conservative system, such as one involving path independence, may not be mathematically well defined in this context.Footnote ²

Here is a simple example of a system that is conservative in this sense but not time reversal invariant.Footnote ³ Take a particle in three spatial dimensions, with position . As a shorthand, we will write and thus denote the particle’s velocity by . Suppose the particle is subject to a force field defined by the cross-product

This describes a force on a particle that is orthogonal to both its position and velocity vectors, which is strange but easy to model.Footnote ⁴

This system is “conservative” on the definition above. The reason is that the cross-product is orthogonal to . So, the integral characterizing work W along a section of any curve vanishes:

No work is ever done along any path whatsoever, and the system is trivially conservative. There are also less trivial conservative forces of this kind, such as,

This is a less trivial force, in that the work done is not always zero. However, it is still a conservative force because it is the sum of conservative forces.

Nevertheless, both systems fail to be time reversal invariant. To verify this formally, we simply observe that . So, , and time reversal invariance fails by lemma 2.

2.2. Conservative but Not Time Reversal Invariant, Part II

There is another natural definition of a “conservative” system in the context of Hamiltonian mechanics. Namely, since we interpret the Hamiltonian h to represent a system’s total energy, “conservative” can naturally be taken to mean that h is a conserved quantity, .

However, there are many conservative systems of this kind that violate time reversal invariance. A simple example is the system for which at every point . Another is . Since each satisfies , each system is conservative in the required sense.Footnote ⁵ However, since , lemma 1 implies that each system violates time reversal invariance.

2.3. ‘Strong’ Conservative Implies Time Reversal Invariance

In the context of Newtonian force mechanics, Arnold (Reference Arnold1989, 22) defines a conservative system to be one in which all forces have a particular functional form:

for some scalar field , which (crucially) depends only on position. We might refer to this as “strong” conservativeness. On this definition, Newton’s equation is manifestly time reversal invariant because the force has no time or velocity dependence. Thus, , and we have time reversal invariance by lemma 2.

This is one way to guarantee time reversal invariance. But it would be nice to determine physical conditions that guarantee time reversal invariance, in a way that is more informative than just demanding forces have a certain functional form. It is certainly the case the many familiar forces appear to have a form that is time reversal invariant. But is it possible to explain why this is the case?

The question has an analogue in the Hamiltonian formulation. If the Hamiltonian h has its “common” form , then , and we are guaranteed time reversal invariance. But what reason do we have to think that the Hamiltonian must have this functional form? One would like to go beyond the superficial fact that many classical systems happen to be that way and ask why familiar classical Hamiltonians (or classical force fields) tend to have a form that is time reversal invariant. In the next section, I will point out two ways to answer this question.

3.1. Velocity-Momentum Proportionality and Time Reversal Invariance

Suppose a classical system is described by a 2n-dimensional phase space and a Hamiltonian function . Such a system is only time reversal invariant if , according to lemma 1. Nevertheless, time reversal invariance is guaranteed in the presence of velocity-momentum proportionality.

Indeed, part of what makes many classical systems familiar is that they have their momentum pointed in the same direction as their velocity. Such systems, it turns out, are always time reversal invariant. This can be shown by the following simple calculation.

Proof. Taking partial derivatives of with respect to gives . So, we may substitute in for the appearing in Hamilton’s equations,

Now multiply by on both sides, so that we can integrate for h to get

for some function of q alone. This Hamiltonian obviously satisfies , and so we have time reversal invariance by lemma 1. QED

This provides a first step toward understanding the extent to which classical mechanics is time reversal invariant. It may be summarized as follows. Classical mechanics does allow a variety of “anomalous” systems that are not time reversal invariant, even among those systems that conserve energy. But, if the momentum of a particle is proportional to its velocity, then none of these anomalous systems are allowed. Time reversal invariance is guaranteed.

3.2. Galilei Invariance and Time Reversal Invariance

Another aspect of familiar classical systems is that they have certain symmetries. For example, one may be used to the idea of obtaining the same results when the same experiment is set up in two different locations in space. This is known as invariance under spatial translations. Similarly, one might think that if an experiment were done in the cabin of a boat moving with constant velocity across the sea, then we would find the same result as if the experiment were set up on shore. This is known as invariance under Galilei boosts. There is a precise sense in which systems that satisfy these familiar properties are guaranteed to be time reversal invariant as well. A rough statement of this fact may be summarized as follows, although it is no substitute for the more precise expression given in proposition 2 below.

Settling this statement requires a precise language in which to talk about symmetries in the Hamiltonian formulation. Unfortunately, talk of symmetries in classical mechanics is often reduced to imprecise language involving “preserving the form” of an equation of motion. In order to avoid risking the pitfalls of that language and of coordinate-based approaches in general, I have chosen in what follows to adopt a more precise geometric formalism, in which claim 4 can be clearly stated and settled. The price is that a fair amount of mathematical machinery must be introduced along the way. I proceed hoping that the precise result might be worth the cost.

3.2.2. Defining Invariance under Translations and Boosts

We noted that the presence of certain symmetries, such as invariance under spatial translations and Galilei boosts, may make a classical system look “familiar.” In this section, we will motivate and give precise expression to those symmetries. The beautiful generality of the Hamiltonian formulation is that a state in phase space can represent anything at all, from the position of a particle to the angle of a swinging pendulum. But we must now talk about spatial translations and Galilei boosts. This requires introducing additional structure to guarantee we are talking about position.

Following Woodhouse (Reference Woodhouse1991, sec. 4.5), we will talk about position in terms of what is called a “maximal orthogonal set” (or sometimes a “real polarization”) on .

Definition 2. A maximal orthogonal set for a 2n-dimensional manifold is a set of n smooth functions such that (i) for each , and (ii) if f is another smooth function satisfying for all i, then is a function of the .

For example, if we think of positions as points in and represent phase space by the cotangent bundle , then the analogues of those positions in form a maximal orthogonal set.Footnote ⁸ This particular maximal orthogonal set allows us to talk about position on the manifold . A maximal orthogonal set in the abstract allows us to talk about position in classical mechanics for any arbitrary manifold .

Now that we have a structure we can refer to as “position,” we can define “velocity” or instantaneous change in position over time. Since change over time is given by the phase flow generated by h, the velocity of a function q is given by . In what follows, we will make use in particular of the initial velocity of a classical system, given by

With a definition of position and velocity in hand, we may now define what spatial translations and Galilei boosts mean.

Definition 3 (Translations and Boosts). We take a translation and boost group for a classical system to be a 2n-parameter family of diffeomorphisms , which forms a representation of , and such that

1. 1.

2. 2. ,

where is a maximal set of orthogonal functions, and is the corresponding initial velocity. We define two associated diffeomorphism groups and and refer to them as the translation group and the boost group, respectively. When these groups have a generator, we denote those generators by and , respectively, and ask that they satisfy for all , to guarantee the translations and boosts are defined in n independent directions.

The notion of “invariance” under translations and boosts will be similar to what we have discussed so far: if a trajectory is possible, then so is the transformed trajectory. We may state this precisely as follows. Let be the vector field for a given Hamiltonian h, which represents a set of possible trajectories. Let be some transformation, and let us use a starred and to denote its pullback and push forward. For a classical system to be invariant under means that transformed trajectories are also a set possible dynamical trajectories with respect to some Hamiltonian function ; that is, for some smooth .

One can show that if and only if is symplectic, meaning that it preserves the symplectic form: (Marsden and Ratiu Reference Marsden and Ratiu2010, proposition 2.6.1). So, requiring classical systems to be invariant under translations and boosts can be expressed by the requirement that translations and boosts be symplectic. This motivates the following simple definition.

Definition 4 (Translation and Boost Covariance). A classical system is invariant under translations and boosts if there exists a translation and boost group on such that each element of the group is symplectic, in that for all .

3.2.3. Establishing Time Reversal Invariance

With these definitions in hand, we may now formulate our main result. We take the time reversal operator to be a transformation such that and . The time reverse of a classical system with Hamiltonian vector field is then the transformation that takes each integral curve of to .

Proposition 2. If a classical system is invariant under translations and boosts, then it is time reversal invariant in that if is an integral curve of the Hamiltonian vector field generated by h, then so is .

The proof of this proposition makes use of a lemma, which shows that translation and boost invariance places strong constraints on the form of the Hamiltonian and of q and (for a proof, see Roberts Reference Roberts2013).

Lemma 3. If is translation and Galilei boost invariant with respect to a maximal orthogonal set , then q and form a local orthonormal coordinate chart for some (nonzero) , and for some function of q alone.

From this lemma our result follows straightforwardly.

Proof of Proposition 2. The lemma shows that q and form a local orthonormal coordinate chart. This implies that the symplectic form can be expressed as the product . Let be the mapping such that and . Then,

Moreover, since the lemma guarantees that , we have . But if and , then itfollows from proposition 4.3.13 of Abraham and Marsden (Reference Abraham and Marsden1978, 308) that is time reversal invariant in the sense we have stated. QED