To Square Root the Lagrangian Or Not: an Underlying Geometrical Analysis on Classical and Relativistic Mechanical Models

Open Access Journal of Mathematical and Theoretical Physics

Review Article Open Access To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models

Abstract Volume 3 Issue 1 - 2020 The geodesic has a fundamental role in physics and in mathematics: roughly speaking, it represents the curve that minimizes the arc length between two points on a manifold. We Rizzuti BF, Vasconcelos Júnior GF, Resende analyze a basic but misinterpreted difference between the Lagrangian that gives the arc MA length of a curve and the one that describes the motion of a free particle in curved space. Department of Physics, Federal University of Juiz de Fora, Brazil They are taken as equivalent but it is not the case from the beginning. We explore this difference from a geometrical point of view, where we observe that the non-equivalence is Correspondence: Mateus Antanio Resende, Department of nothing more than a matter of symmetry. The equivalence appears, however, after fixing Physics, Federal University of Juiz de Fora, MG, Brazil, a particular condition. As applications, some distinct models are studied. In particular, we Email: explore the standard free relativistic particle, a couple of spinning particle models and also Received: January 14, 2020 | Published: March 04, 2020 the forceless mechanics formulated by Hertz.

Introduction mechanics of Hertz4,12 H Hertz12 has provided a particular description of Newtonian classical mechanics without forces. His clever idea was Differential geometry is extensively used as the basic tool for to insert one more nonphysical dimension into the game, together with description of many distinct areas of physics, after all it is widely the introduction of the Newtonian potential inside the metric. In this accepted the space-time possesses the very structure of a differentiable case, second Newton law appears as the geodesic equation of a free 1 manifold. Besides that, it goes from special and general relativity particle in a curved manifold. These applications were chosen because 2 3 theories, till electrodynamics and gauge theories. We would like to they represent very clearly the difference between systems described focus our attention to its application even in the less advanced topic, S S by 1 and 2 . Finally, Section 6 is left for conclusions. Throughout all though not less important, of classical mechanics. One may find an the paper, Einstein notation of sum over repeated indexes is adopted. extensive literature on the subject, see 4,5 and references therein. The central idea of these notes is to draw attention to an underlying fact, L or L ? which is often misinterpreted: given a (semi-)riemannian manifold Let us consider a (semi-)riemannian m-dimensional manifold 1 (, g ) , the action functionals: m m ( , g ) and a chart (ϕ,U ) . U is an open set U ⊂  and S = ∫ dT g (,) v v (1) 1 mm1 ϕϕ:U→⊂ A( open )  P ( P) : =( x( P) ,..., x( P)) . (3) and 1 We write collectively s 2 = ∫ dt g ( v , v ) (2) 2 ϕii(P) = xPi( ); = 1,..., m (4) are not necessarily equivalent. In the literature, this fact is usually ignored. However, it is a subtle result that cannot be taken for granted. and elements of m will be denoted by capital letters. This problem will become clear along the text and it is intimately m Let X( ) be the set of all smooth vector fields on m , that related to what is known as pseudo-classical mechanics,6 that is, is, the set of mappings V which assigns to each point P on m models whose number of configuration variables is less than the  a tangent vector V in the tangent space T m . In coordinatesϕ number of physical degrees of freedom. It turns out that these models P P ( ) ,VV=i ∂ . We also fix the notation for the (pseudo-)metric g in the are useful to describe quantum phenomena even before quantization, i coordinatesϕ , see7,8 justifying, thus, our presentation. The paper is divided as follows. i j ij In Section 2, we will show the non-equivalence between S1 and S2 L:,= gVV( ) = gV( ∂i , V ∂= j) gVV ij . (5) and under what conditions they become equivalent. Sections 3 and 4 are dedicated to examples on the lack of equivalence between actions For the case, there is a unique connection mm m S1 and S2 . In particular, we discuss in the former the free relativistic D :,XX()×→( ) X( ) (6) particle, while the latter contains examples of classical models of spinning particles. These consist of an example of a semiclassical known as the Levi-Civita connection, such that inϕ , it reads 9 model whose quantization leads to the Pauli equation and another k i1 il D∂ (∂j) =Ã ij ∂ k ; Ã jk =gg( ∂j kl +∂ k g jl −∂ l g jk ). (7) model for the relativistic rotator, where the particle is considered i 2 to resemble a moving clock with the pointer characterizing its gil denotes the elements of g −1 . A particle shall be denoted by the spin.10,11 Then, we devote the entire Section 5 to explore the forceless corresponding curve it traces2 1Here V is a tangent vector.

Submit Manuscript | http://medcraveonline.com Open Acc J Math Theor Phy. 2020;3(1):8‒15. 8 ©2020 Rizzuti et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic Copyright: 9 mechanical models ©2020 Rizzuti et al.

m we consider the case of the quantity xgx as a function ofτ . Then, we α:.I ⊂→  τ ατ( ) (8) treat it as a constant with respect toτ . I is an open interval on. In this sense, xi (τ ) corresponds to the  Let us consider xgx as a function of the parameterτ . Thus, the representation of α in the coordinatesϕ , time derivative in the first term of Eq. (15) implies the equation ii (ϕ ατ)( ) :,= x ( τ) (9) ¨ j Π+ i j kl = (16) d j xÃkl xx 0, as well as xi (τ ) are the components of the tangent vector in  dτ the point ατ( ) ik ik iixgjk x i xgjk x where Π=jjδ − , or, by denoting Λ=j , d xgx xgx |:=xi (τ ) ∂ | . (10) τ ατ( ) i ατ( ) d Πiii =δ −Λ jjj (17) i dx j We have denoted xi ≡ . Since the state of motion is intuitively It follows that the vector x , whose components are given by x dτ , is in the null space of the operator associated with the matrix Ð . described by the corresponding position together with its velocity, we Thus, the null space is not trivial and Ð is not invertible, implying are naturally led to the notion of a tangent bundle, that det Π=0. Besides that, the Hessian matrix is proportional to mm TM ( ) = U T ( M ) (11) the matrix Ð . pM∈ p ∂2L ii ∼ i whose elements are parametrized by xx,  . ijÐ.j (18) ( ) ∂∂xx There are two common ways to find the equations of motion of Therefore, we are dealing with a singular Lagrangian theory. It the particle on the manifold. Although both of them are based on can be shown that this is intimately related with the reparametrization the principle of least action, they are fundamentally different. The invariance of this functional, which we will cover below. Before we equations of motion the distinct actions only become equivalent after consider the case where xgx does not depend ofτ , we can also a particular condition. The main objective of this work is to clarify understand the ambiguity of the singular system in terms of the this point. Let us consider the two actions, already presented in the algebraic properties of the operators Ð and Ë .4 We point out that Introduction, which are linear functionals defined on the space they are projectors because of the following: mm 22 F(T()) ::= { f T ( ) → ; f is smooth } . (12) Π=ΠΠ=Π, ΠΛ = 0 We then take two different smooth functions to write (19) Π+Λ= ττ22  =ττ ≡ ij S1 ∫∫ d L d gij ( x) xx (13) ττ11 So, given an arbitrary vector v , it is uniquely decomposed as v=Π v +Λ vv: = ⊥+ v  (20) and ττ i 2211 If we fix the vector x that specifies the projector Ð , then =ττ ≡ ij S 2 ∫∫ d L d g ij ( x ) xx (14) ττ22 iig( xv, ) 11 v= x; g( xv ,⊥ ) = 0. (21)  g( xx, ) In a sense, S1 has a geometrical meaning, as it describes the arc We observe that v is a projection of v on the direction of xi length of the curve α connecting the points ατ( 1 ) andατ( 2 ) . On the   other hand, S2 is analogous to the usual action in classical mechanics whilst v⊥ belongs to the corresponding orthogonal complement. of a free particle, for we are integrating a kinetic energy term. Let us Considering the vector v defined by discuss separately both actions, explaining why one or the other shall v:= xi +Γ ixx  jk  be elected for further investigations. jk (22)

S1 : Geometric point of view our variational problem has led us to the equation of motion (16), which is just the projection Ð0vv=⊥ = . This restricts v⊥ , but v Adopting the standard notion that S1 is related to the arc length  remains arbitrary. This is the ambiguity exposed before. of α connecting ατ( 1 ) andατ( 2 ) ,the extremization of S1 provides the following differential equation for the curve which has minimum Now, we work with xgx as if it were a constant with respect to length τ . Then, it follows that Eq. (15) turns in to i x 1 i jk +Γ = i  jk xx 0 (15) ¨ xgxxgx  i jk  x+=Ãjk xx 0. (23) ij in which we have used the short notation xgx := gij x  x . The This is the well known equation of a geodesic. The problem is coefficients Ãwere given in Eq. (7). The subsequent analysis of this the meaning of the evolution parameter. In fact, S1 is invariant under equation relies on how the quantity xgx relates with the parameterτ reparametrizations, . We shall do it by separating this analysis in two distinct cases. First,

ττ→=' f ( τ) (24) the Aharonov-Bohm effect.15 In our case, the coordinates xi are not all observables because they are altered by the local symmetry the for any smooth function → . This picture is understood as f :  model presents, see (29). It is, though, possible to find the physical a lack of physical meaning ofτ , which, in turn, implies no physical sector of the model and we shall do it for the particular case of a free meaning for the solution xi τ of Eq. (??). This fact is known in the ( ) relativistic particle, see Section 3. literature as pseudo-classical mechanics: the number of configuration 6 variables is less than the number of physical degrees of freedom. S2 : Kinetic point of view The origin of reparametrization invariance is clarified if we interpret Starting now with the action S , one finds the following equations ij 2 L= gij xx as Lagrangian and use the Dirac-Bergmann algorithm of motion 4,13 of hamiltonization to constrained systems. Defining the conjugate ¨ i i i jk momenta to the variables x x+=Ãjk xx 0, (33) j ∂ L gxij  also after a least action principle. For this case, we have no local πi :,= = (25) ∂xi xgx symmetry and there is no arbitrariness as the one posed by (24). Thus, is interpreted as the evolution parameter. On the contrary of S1 , if one immediately finds the constraint one writes the conjugate momenta ij g ππij= 1. (26) ∂ 1 j πi =L = gxij  , (34) ∂ i Invoking the Dirac conjecture that first class constraints generate x 2 a local symmetry,14 we can write a local infinitesimal transformation then (34) may be interpreted as an algebraic equation to obtain the i i that leaves the Lagrangian L invariant (modulo a total derivative maximum number of velocities x in terms of x and π j . It turns out term) that we may find all of them once g is invertible, i ij i 1 i ij xg = π j (35) δx= ε( τ){ xg, ππij− 1|} π (xx, ) . (27) 2 and no constraint, such as gijππ = 1, appears. In this case, all ε ij In the above formula, the factor 1/2 is set by convenience, coordinates xi of a particle have deterministic evolution given by τ is an arbitrary smooth function of and {,} is the Poisson bracket (33). Naturally, the equations of motion for both actions S and defined by 1 S2 (??) and (33) are different. However, once xgx is fixed, the ∂∂AB ∂∂ AB = − equations of motion coincide, see (23) and (33). Hence, S1 turns {AB,:} ii. (28) ∂∂xx∂∂ππii out to be equivalent to S2 . It is possible to see the meaning of τ in

Thus, (27) reads S = =∫τ2 = ττ − 1 as well. With xgx 1 , s1 τ1 dt 21, that is the action xi δεxi = . (29) is proportional toτ , which in this case may be called the evolution xgx parameter 16 A direct calculation shows that We will consider in the next sections some examples to show the relevance of one or the other formulation. ij dεε xx δ Lg= ij , (30) dτ xgx xgx Free relativistic particle   The free relativistic particle is the canonical example where the as stated. reparametrization invariance plays a central role.17 For the case, let The situation here is analogous to electrodynamics.14 In the former us consider Minkowski space-time  as our manifold. It is isometric case, equations of motion are derived from a least action principle 4 to the semi-riemannian manifold (1 ,η ) . The index 1' means that applied to the Lagrangian −η is positive defined in a subspace of dimension 413−= . We chose 1 µν  = − FFµν (31) its signature to beη = diag ( +−−− 1, 1, 1, 1) , providing the known 4 Einstein-Weyl space-time causal structure. Let us take coordinates xµ ∂ . As usual, Greek letters µ , ν , etc run the values 0 , 1 , 2 and 3 . where F=∂ AA −∂ and ∂=: . Clearly, the model 0 i µν µ ν ν µ µ µ x is related to the time coordinate while x , i = 1, 2, 3 are the spatial possesses the local symmetry ∂x ones.2 We begin our analysis by writing the action

δψAµµ= ∂ , (32) 1 µ v S av = ∫ d τη µ xx (36) where ψ is an arbitrary function of space-time variables. Since 2 Aµ changes under the transformation (32), it is not an observable of The principle of least action provides the deterministic evolution the theory. The physical sector is obtained by looking for combinations ¨ µ µ µµ of the field Aµ and its derivative that remain unchanged by (32). For x=⇒=+0. x(ττ) VX (37)   example, δ Fµν = 0 and δ (∮ A.0 dl) = . Fµν are just components of 2 4   Actually,  is a set of events. 1 is the linear space with  being the the electromagnetic field whereas ∮ A . dl manifests itself through corresponding affine space.

µ V and X µ are constants. We have then a straight line, with no of a spinning particle model from a variational problem. To do so, physical interpretation though. In fact, the speed of the particle has we must work with a classical mechanics system endowed with no superior bound. It would be the case, if one sets a restriction of position variables xi and additional degrees of freedom. The latter µν the formηµν xx= 1 , to impose, in turn, a maximum value to the are necessary to form the inner space associated with the particle, modulus of the physical three velocity. So, we write which, in turn, is described by a point in a world line.18 Once again,

1 µ v e the reparametrization invariance plays a central role, as it will become S bv = ∫+ d τη  µ xx  (38) 22e clear along the discussion. The particular model presented here can be found in Refs.19, 20 In this case, is an auxiliary variable. Its equation of motion is given by Consider a phase space equipped with canonical Poisson ii ˆ 11µν µν bracket, for instance,{ωπ, jj} = δ . Spin operators Si act on the −ηηµν xx + =⇒=0. eµν xx (39) 2e2 2 two-component wave function as 22×−matrices, according to the definition This last expression shows that e(τ ) has no independent  µ ˆ = σ dynamics. It is completely described by the evolution of x . If we Sii:. (44) 2 µν substitute ηµν xx back into the action Sb we finally get This shows that the spin operators are proportional to the Pauli matrices = τη µν S c ∫ d µν xx (40) 01  0−i  1 0  σσ12= , =  , σ 3=  . 10 i 0   0− 1  which has the same structure of S1 . As discussed before, Sc has invariance under reparametrizations. In this case xµ (τ ) has no Interesting properties follow through this definition. Most importantly, physical interpretation. Sc has, on the other hand, global invariance ˆˆ= ε ˆk under Lorentz transformation SSi, j i ijk S (45) µ' µµv x → x =Λ v x ; Λ∈ SO ( 1, 3 ) (41) and 32 Actually, this is one of the main reasons why we work with non- SSˆˆ⋅=2 ss( +1.) = (46) physical degrees of freedom: it is possible to carry a fully Lorentz 4 In addition to that, spin operators form SO 3 − algebra with covariant model along the way, knowing that not all of the variables ( ) are physical ones. respect to the commutators (45). Noting that this is different from the variables’ algebra and that we have distinct numbers of variables and Now, the principle of least action provides the following equations spin operators, we follow the common procedure of defining the spin of motion from Sc , i as a vector product between coordinates of a phase space, say ωπ, j ν . Thus, ηµν x =π µ = const . (42) = ε ωπjk xxη Si:. ijk (47)

µν S together with the constraintη ππµν= 1. The physical sector Poisson brackets for i yield k can be obtained if we eliminate the τ -dependence of the game. It {SSi,. j} = ε ijk S (48) may be done by declaring x0 as the evolution parameter. Then, the i 0 Then, the definition (47) implies SO 3 -algebra for the spin deterministic evolution of spatial variables x in terms of x is given ( ) by variables. Besides that, we impose constraints to our variational ii i i problem, dx x ππ = = = . (43) 00 0 i 2 iii22 dx x π 1()+ π ππiii=ba, ωω = , πω = 0. (49) The last equality comes from the constraint: ππ02=1() + i Here, a and b are constants. This way, the number of degrees of freedom for the spinning particle coincides with the one chosen here . Thus, we have a particle with bounded three-velocity walking, 19 as expected, in a straight line. All the structure presented above is in (49), see. Our next step is to write a Lagrangian which gives, apart commonly used in different contexts. On the next Section, we will from dynamical equations, the imposed constraints (49). Consider discuss how it appears for description of classical spinning particles. 12 11 2 22 Lspin =()ωωi +−gb (( i ) −a ) . (50) 22g φ Spinning particles g (τ ) and φτ( ) are auxiliary degrees of freedom. Firstly, we In this Section we discuss two more examples where invariance observe that the two initial terms above resemble those in (38). under reparametrization is present. In the first case, we discuss a So, one expects a local symmetry due to the presence of first class particular model whose canonical quantization leads to the Pauli constraints. In fact, the model is invariant under the following gauge equation. The second treats a relativistic rotator, known as the transformations Staruszkiewicz model. .  δωii= ξω , δgg = ( ξ ) , δφ = ξφ − ξφ , (51) Classical spinning particle where ξ is an arbitrary function ofτ . The variables ωi and πi Let us now see how this formalism is applied to the construction , which compose the spin, are affected by these local symmetries

kkη (generated by the constraints), while the quantities Si are left =∫τη +−2 S dm xx l 2 invariant under them. The case here is similar to what was discussed ( kx η  ) (58) in the subsection 2.2. Si is gauge-invariant, therefore, representing a possible observable quantity of the model. Variation of Lspin with The -variables have usual meaning as a vector formed by pairs of respect to auxiliary variables gives events on Minkowski space, while k is a null direction. According to the Wigner’s idea that quantum systems can be classified according δ L 1 ()ω 2 spin ==−0bg2 ()ω 22 ⇒=i , (52) to irreducible representations of the Poincaré group, the parameters δ 22i g gb m and l were introduced by A. Staruszkiewicz and label the representations. In fact, defining the conjugate momenta pLµµ= ∂ x δ Lspin 1 22 22 ==0 ((ωωii ) −⇒aa) () = . (53) π µµ= ∂ L δφ φ 2 and k , we can verify that the following Casimir invariants are satisfied From (52) we see that g (τ ) has no independent dynamics, while 1 µµ= 2 = − 42 i p pµµ m, W W ml. (59) (53) implies that ωω i = 0 . Computing the conjugate momenta we get 4 i µ ∂Lspin ω Here, W is the Pauli-Lubanski pseudovector πi = = . Substitution of g (τ ) into the last equality gives ∂ω i g µ1 µναβ W= − ò Mpνα β (60) i 2 ω ππ= ⇒=22 iibb. (54) and M are the components of the total angular momentum, ()ω 2 να i given by Thus, we have obtained the desired constraints through variation of Mνα=−+− xp ν α xp α ν k νππ α k α ν . (61) Lspin . This Lagrangian, however, can be presented in a more compact The action (58) has a structure similar to that of S and when form, structurally close to the one in S1 , and still provide the same 1 we take l = 0 , the expression for free relativistic particle (40) is ()ω 2 2 = i recovered. This is one among many examples where reparametrization constraints. Since g 2 , we can write b invariance leads to pseudo-classical mechanics, with constraints in 1 the corresponding Hamiltonian formulation. This particular model Lb=−−()ωω 2 ( ) 22a . (55) spin i φ ( i ) was used here because of its property of having Casimir invariants as parameters, instead of just constants of motion. A complete analysis We could go even further though. If we use the constraint of the model may be seen in.25 22 ()ωi = a to exclude φ of the Lagrangian, and one of the ω ’s, say, ω3 , the model would lose its manifest rotational invariance, since Forceless echanics of hertz ωω= ∈ 'iR ij j , Rij SO(3) . However, it becomes fully reparametrization Here we turn our attention to the opposite direction we have invariant, been following. We shall discuss one classical example where S2 αβ is explicitly used, while invariance under reparametrization is not Lspin = bgαβ(ω γ ) ωω; αβγ , ,= 1,2. (56) present, the forceless mechanics of Hertz. It is a very interesting ωω proposal because it can be seen as the first example where, through the =δ + αβ Here, gαβ αβ 2 abare the metric components over usage of extra degrees of freedom, the force concept was substituted a − δαβ ωω by the curved space itself. One observes a similar methodology in a sphere21 and this form of Lagrangian has the same very structure as general relativity, proposed by A. Einstein. Curiously enough, the model is of S2 -type but we can kill the extra degree of freedom the one presented in S1 . At this point, we would like to highlight one though. We use geometrical arguments and physical reality instead of the main properties of the model above. It admits interaction with of invoking local symmetries to characterize the observable sector of electromagnetic fields. Incorporating both position space variables the theory. and the spinning ones, together with the corresponding interaction, we find Introducing the model  m 11eb2 S= dt  x 2 + eA x i −+ eA  ωω  − ò v jk B ) 2 + g − 22 − a (57) Heinrich Hertz, who is greatly known for his experimental expertise, ∫  i i 0 i ijk 2 φ ( i )  22gm2a worked on a non-Newtonian form of mechanics. Philosophically, it A computation shows that the canonical quantization leads to the seems that Hertz did not find satisfactory the idea that two distinct Pauli equation.20 Below, we discuss a more complex spinning particle quantities, “mass” and “force”, were necessary to explain the motion model, in which the invariance can also be accomplished. of a massive object .28 His idea, thus, was to describe the motion of any mechanical system as the motion of a free particle in a curved Staruszkiewicz model space. The curvature would appear in the model as a consequence Let us now discuss another classical model that describes of inserting auxiliary degrees of freedom to the system, which, in a relativistic rotator.11 It is one of a series of papers concerning turn, would alter the space’s metric. The idea here is close related the description of spinning particles before and after canonical to what Einstein proposed latter, in his general relativity theory. quantization, see.8,10,22–27and references therein. For this particular Gravitational interactions happen between mass objects in space- case, once again we have a reparametrization invariant formulation, time due to the curve structure of the latter. This way, the concept with action given by of force is expendable. In the Sections above, we have discussed

Citation: Rizzuti BF, Vasconcelos JGF, Resende MA. To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models. Open Acc J Math Theor Phy. 2020;3(1):8‒15. DOI: 10.15406/oajmtp.2020.03.00056 To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic Copyright: 13 mechanical models ©2020 Rizzuti et al. some models whose dynamical equations are given by geodesics. were able to hide the potential  into Ã( g ) , allowing one to interpret However, the models had invariance under reparametrization and, the classical potential as the cause of the curvature. because of that, the functions xi (τ ) had no physical meaning. Physical sector of the model Here, the goal is to present a model which resembles the action S2 , being the forceless mechanics of Hertz the chosen one. The system Despite the fact this formalism allows us to work with a free will have a well defined evolution parameter and, consequently, the particle, in a possibly more onerous space structure, we would still dynamical equations shall describe a proper physical solution, except like to obtain the physical description of the system. In other words, for the additional spacial dimension introduced by Hertz. We provide while xa (τ ) describes a free particle in a curved space equipped with a modern detailed geometrical analysis of the model, in which the the metric g , the physical sector is still described by the functions removal of the extra degree of freedom is naturally obtained. Our xi (τ ) , in the m manifold, with the Euclidean metric δ . Let us main result in this Section is the geometrical proof that guarantees the now present some arguments to kill the auxiliary variable X , after removal of the auxiliary degree of freedom. We begin by considering constructing an appropriate model, to obtain the equivalent physical the standard Newtonian mechanics description of a particle in system. First, we consider the classical approach of imposing initial m δ i riemannian m -dimensional manifold ( , ) , represented by x (τ ) i i conditions. Here, our generalized coordinates x and X obey , subject to a potential  ( x ) . One can promptly write ii= ii= 1 xx (0) 00 , xv ( 0) , (68) = τδij− i S ∫ d  ij xx  U ( x ) (62) 2  i XX(0) = 00 , X( 0) = 2 ( x) . (69) and, through variation of (62), obtain the equations of motion Equation (67) can be written as i ¨ dX x = −∂i . (63) =⇒=0Xk 2. (70) dτ 2 Here, the indices i,, jkgo through 1 to m . Where k is a constant. To respect the initial condition of X , we To avoid the Newtonian force approach, the procedure developed must have k = 1 . Substitution of X = 2 into (66) gives by Hertz leads us to insert one extra function X (τ ) in the set of functions representing the particle. Then, we begin to work with a ¨¨ii1 m+1 + ∂ = ⇒ +∂ = (71) m +1 -dimensional differentiable manifold  . Given a chart x2 ii XX0 x 0. a 4 (ϕ,U ) and considering the indices ab,= 1,..., mm , + 1, x (τ ) is defined as That is the result of the initial formulation, independent of X . Here, we followed the tradition in Newtonian mechanics, which is to a (ϕα ) :,= x ( τ) (64) insert manually initial data, after we found how the system behaves where, as introduced before, α is the curve corresponding to the dynamically. m+1 X (0) particle in the  manifold Another way to attack this problem is to set k = . i m+1 20 ( x ( )) α : I ⊂→  Instead of fixing conditions that restrict k = 1 , we write this constant τ ατ( ). in a way that it is now essentially part of the dynamical equations. The definition ':= k gives ij1 g=δ dx ⊗+ dx dX ⊗ dX i By using the (pseudo-)metric ij i ¨ 2 ( x ) x = −∂  '. (72) i , we consider the action functional The parameter k can be seen as a controller of the potential’s 1 112 strength. = ττab=i +  2 S ∫∫ d g ab xx d ( x  ) X (65) 2 24U At this point, we would like to emphasize that the formalism Hence, we obtained with (65) the action of a free particle in the developed by Hertz has a non-uniqueness.29 In effect, consider m+1  manifold. We point out that the particle that (65) describes 111i 22 2 i Lx=++() X X. (73) is now immersed in a curved space, as the metric is explicitly x 2412 4 -dependent. Equations of motion for this particle assume the well- 12 ¨ a Equations of motion give a bc known geodesic form x+=Ã0bc xx , reading separately. ¨ i 22 xk=−∂−1122ii k ∂ (74) ¨ i 1 +∂ = x2 i  XX 0 (66) 2 4 =−∂k11i ( +γ 2), (75) and X 0 ¨ 1 l ( ) k2 2 i  where Kl = , l = 1, 2 and γ = (). X−∂i  xX =0. (67) i  20l ( x ( )) k1 The above equations allow us to relate the potential  with the i 1 Alternatively, one can write connection coefficients Ã( g ) , such as Ã mm++ 1, 1 = ∂i and 4 2 m+1 1 11i 2 Ã im ,+ 1 =−∂i . Thus, by extending our configuration space, we Lx=() + X (76)  2 12+ γ

and obtain the same dynamical description for xi , as we got from having a resultant force on the system, these rotations do not affect (73). Thus, we may have different mechanical systems on the extended the isotropic properties of the space. These results should suffice in space described by the same equations of motion. These calculations showing that the direction of ∂ X is privileged, advocating towards indicate that, indeed, X may not represent a physical variable. As we our previous claim that the auxiliary variable X should not be part of have pointed out, when we extend our space, the particle behaves as if our physical dynamical equations. Therefore, we have a geometrical it were on a curved surface. Regarding X (τ ) and its initial condition proof that different trajectories on the extended space project on the i X 0 , we see that equations for both X and x have been separated. same physical trajectory. As a consequence, the system’s dynamics is not altered by different choices of X 0 . Our next goal is to give geometrical proof to this claim. Conclusion Killing vector fields In this work we have made explicit the fundamental difference between two actions describing particles lying on a manifold. While, We may regain the particle’s physical trajectory by projecting the in the first case, one minimizes an arc length, in the second we extreme geodesic line described in the extended space on the m − dimensional a kinetic energy term. In the initial action, due to reparametrization space. Geometrically, we can prove this by showing that ∂ X is one invariance, the evolution parameter has no physical interpretation. of the metric’s Killing vector fields, that is, the vector fields which This implies that not all degrees of freedom are the physical ones. It represent the direction of the symmetry of a manifold. In addition to is reflected by the presence of a local symmetry generated by what that, we can show that ∂ X , under rotations, represents a privileged is called a first class constraint in Dirac’s terminology. On the other direction, which is not desirable if the isotropy of the space is taken hand, even though in the second action the equations of motion read for granted. We begin our discussion with the very equation the same as in the previous one, the time evolution parameter does  g = 0, (77) have physical interpretation, as it cannot be changed arbitrarily. There K is no constraint into the formulation and all coordinates have physical that represents the Lie Derivative of the metric tensor with respect interpretation in the sense that their dynamics is independent from one to the Killing vector field K . It tells us that the metrical properties of another. The two descriptions, however, turn out to be equivalent as the manifold m+1 do not get altered by this vector field. Indeed, we fix the condition xgx= const. Denoting g= g xia dx⊗ dx b, where, as before, i, jk ,∈…{ 1, , m} ab ( ) We exposed the subtle difference of both actions with distinct and abc, ,∈…{ 1, , mm , + 1} , we have examples. For the first case, we explored the action of a free relativistic particle with no internal structure and its natural continuation for a ab Kg= c( g ab dx ⊗= dx ) particle with spin. In all actions, invariance under reparametrizations K ∂c plays a central role as fully global Lorentz invariance is explicit. It c a b ac b ba c =∂Kc g ab dx ⊗ dx +∂ gab c K dx ⊗ dx +∂ gab c K dx ⊗ dx = is worth mentioning that one may describe the free particle, with or without spin, by using physical coordinates. However the price you c c ca b pay is to represent the Lorentz group non linearly. Our last Section =(K ∂c g ab +∂ g cb a K +∂ g ac b K) dx ⊗ dx =0. (78) was designed to present and analyze the forceless mechanics of Hertz. 1 Following Hertz’s procedure, we wrote the action of a free particle Since g ++= , the Killing equations are mm1, 1 2 in curved space. Although force becomes an unrequired quantity to i describe the system, as Hertz desired, we are still required to map the −K X ∂iX +∂K =0, (79) solution on the extended space to the physical one. Following this 2 idea, we gave a few arguments to kill the auxiliary degree of freedom.

1 Initially, it was shown that the extra variable vanishes with proper ∂KKXi +∂ =0, (80) 2 iX choices of initial conditions. It was also shown that proper adjustments of the parameter controlling the potential led to an analogous result. ji ∂ijKK +∂ =0. (81) After this, the non-uniqueness of the Hertz formalism was highlighted. One of the paper’s main results was then presented. Using Substituting K1 =0. ∂+iX 1. ∂ in to the equations (79,80,81) one differential geometry, an elegant proof was given to justify the verifies that they are satisfied. In other words, K1 is a Killing vector removal of the auxiliary degree of freedom. Analyzing the vector field of the metric g which generates translations along the ∂ X direction. fields which broke the space’s isotropy, we were able to determine that ∂ , the vector field associated with the auxiliary variable, generates Our next analysis consists of taking the vector field X translations along its own direction. Thus, different choices for X ii Rm+1 =−∂+ xX Xi ∂ (82) would map into the same physical solution, giving substantial basis for the projection argument. Additionally, it was exposed that rotations and inserting it in the Killing equations. It is clear that Eq. (79) is around fixed axes, which include the direction of ∂ , are not Killing ∂≠ X not satisfied, since, generally, i  0 . This reveals that the rotation vector fields. As a consequence, a curious fact was pointed out. The ∂ around fixed axes, including the subspace generated by X , breaks presence of force in the physical sector implies a symmetry break on the space’s isotropy. Curiously enough, this shows that a resultant i the extended space, even though we describe our system as a free force different than zero in the physical system implies Rm+1 is not a m+1 particle on it. Particularly, it reveals that the extra degree of freedom Killing vector field in the  manifold. However, there, the system should not take part on the physical dynamical solution of the system. is represented by a free particle, where the potential is hidden in the ∂ kinetic term. Thus, we could argue that i  is intrinsically affecting Acknowledgements the extended system and it is responsible for the symmetry break in i This work is supported by Programa Institucional de Bolsas de Rm+1 ’s direction. Meanwhile, it is easy to see that rotations given ij Iniciação Científica - XXVII PIBIC/CNPq/UFJF-2018/2019, project by ∂iKxx2 =−∂+ ji ∂ obey the Killing equations. Regardless of

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