^{3}

Scale invariance in the theory of classical mechanics can be induced from the scale invariance of background fields. In this paper we consider the relation between the scale invariance and the constants of particle motion in a self-similar spacetime, only in which the symmetry is well defined and is generated by a homothetic vector. Relaxing the usual conservation condition by the Hamiltonian constraint in a particle system, we obtain a conservation law holding only on the constraint surface in the phase space. By the conservation law, we characterize constants of motion associated with the scale invariance not only for massless particles but for massive particles and classify the condition for the existence of the constants of motion. Furthermore, we find the explicit form of the constants of motion by solving the conservation equations.

The symmetry of external fields can induce the symmetry of a theory on the background, which is embodied as that of the action integral. As is well known, the action invariance under a symmetric continuous transformation leads to a conservation law via Noether’s theorem (see, e.g., Ref. [

In classical particle mechanics with continuous symmetry, a particle has a conserved quantity throughout the motion associated with the symmetry—a constant of motion. In particular, geometrical symmetry in a spacetime can be the origin of symmetry in a particle system. If a spacetime metric admits an isometry, then the generator (i.e., the Killing vector) yields a constant of motion. This symmetry is generalized to spacetime hidden symmetries such as higher-rank Killing–Stackel tensors or Killing–Yano forms (see, e.g., Refs. [

The scale invariance of a metric is a special class of the conformal symmetry and is known as the spacetime self-similarity. In the context of relativity, the self-similar spacetime appears as a critical point of the critical phenomena in gravitational collapse [

The way to solve this inconsistency is to consider that a particle system in gravitational theories is a constraint system. The constraint arises from the reparameterization invariance of the action. We can utilize it to relax the conservation condition for a constant of motion, i.e., we restrict the conservation condition holding only on the constraint surface in the phase space. The purpose of this paper is to characterize the constant of massive particle motion associated with scaling symmetry as the quantity conserved only on the constraint surface.

This paper is organized as follows. In the following section, we reformulate the conservation law for a dynamical quantity in the Hamiltonian formalism by taking into account the constraint condition. In

We summarize our notation in what follows. We denote the symbol

If we identify

Note that this equation holds for any derivative operator. In the case where

We consider classical particle mechanics in a

We should apply the weak equality after evaluating derivatives or Poisson brackets.

In the particle system, we consider a dynamical quantity

Now, we assume that

We find that

Let ^{1}^{2}

In addition, the variation of ^{3}

This equation implies that

In the following, we solve the hierarchical equations (

The integrability condition for

We can replace this first term by using Cartan’s identity^{4}

In the following subsections, we discuss each of the following cases: (A)

Then, there locally exists a solution to Eq. (

Hence, if any one of the following conditions holds:

(i)

(ii)

(iii)

In Case (i), the particle is massive and is not subject to the scalar potential force, so that ^{3}

Therefore, from the point of view in

To seek constants of motion in addition to those in

The quantities

We ignore a constant term of

This condition coincides with the necessary condition (

We focus on

Hence, if the conditions

Next, we focus on the remaining case where

Substituting these results into Eq. (

Note that this is not compatible with Eq. (

In this case we obtain the separated equations from Eq. (^{5}

With these constraints for

Though we obtain two constants of motion corresponding to two linearly independent solutions, we find that these coincide with each other by rescaling the parameter

We solve the remaining equations in the case where

On account of the integrability, there locally exists a solution in the form

The quantities

The Lie derivative of the first equation with respect to

We have considered constants of particle motion associated with scale invariance in classical mechanics on a curved spacetime. The scale invariance can be induced from the scale symmetry of background fields. If both the mechanics and the background fields share continuous scale symmetry, a particle can have a constant of motion associated with it. On the other hand, a conformal Killing vector generating conformal symmetry is related to a constant of massless particle motion in general. Since a homothetic vector generating scale symmetry is classified in the class of conformal symmetry, it was unclear how to characterize a constant of massive particle motion associated with a homothetic vector. In this paper, we have made it clear that, even for massive particles, a constant of motion associated with the homothetic vector exists by virtue of the reparameterization invariance of a particle system. We have utilized the constraint condition relevant to the reparameterization invariance to relax the conservation condition for constants of motion. The conservation law that we have obtained takes the form of hierarchical equations. Solving these equations to particle mechanics in background fields with scale invariance, we have obtained constants of motion associated with the scale invariance and have classified the conditions for the existence by the self-similar weight of the background fields. Consequently, we have found that constants of massive particle motion associated with the scale invariance must depend explicitly on a parameter on the world line.

The author thanks T. Harada, M. Nagashima, and T. Tanaka for useful comments. This work was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities, 2014–2017 (S1411024).

Open Access funding: SCOAP

We briefly review nonrelativistic classical mechanics with scale invariance. Let

In general, Eq. (

^{1} See, e.g., Ref. [

^{2} Since a gauge-invariant quantity is not

^{3} One can deparameterize the theory by solving the constraint for

Hence, we can regard the particle system in Eq. (

^{4} This is the relation between the Lie derivative £

^{5} The potential proportional to the norm of