aa r X i v : . [ phy s i c s . c l a ss - ph ] M a r Dissipation in Lagrangian formalism
András Szegleti a) and Ferenc Márkus b) Department of Physics, Budapest University of Technology and Economics,H-1111 Budafoki út 8., Budapest, Hungary (Dated: 10 March 2020)
In this paper we present a method with which it is possible to describe a dissipative systemin Lagrangian formalism, without the trouble of finding the proper way to model the en-vironment. The concept of the presented method is to create a function that generates themeasurable physical quantity, similarly to electrodynamics, where the scalar potential andvector potential generate the electric and magnetic fields. a) Corresponding author: [email protected] b) [email protected] . INTRODUCTION Newtonian mechanics can provide a general description of a physical system, as there are nolimitations on the force terms, that can contribute to the equations of motion. This freedom isactually a drawback of the formalism, as correct equations of motion can be generated using ad-hoc forces, which limits the possibility of gaining predictions from the theory.Lagrangian mechanics is built on a more general principle, Hamilton’s principle (or least actionprinciple), which states, that there is a function L ( q i , ˙ q i , t ) , that describes the physical system, andthe action functional S = t Z t d tL ( q i , ˙ q i , t ) (1)is extremal in case of physical trajectories. The equations of motion (Euler–Lagrange equations)for the system can be calculated from variational principle (functional derivative). This approachstrongly limits the form of the equations one can derive using this formalism, which means thatthe form of Lagrangian describing a system is also restricted.Dissipation, being a statistical phenomenon, could only be described by a Lagrangian contain-ing all degrees of freedom (both for the system and its environment). Using this system-plus-reservoir approach, it is always an important question how the environment should be modelled.For example, using a harmonic bath model , one assumes, the reservoir can be represented as aset of uncoupled harmonic oscillators. By using a different model for the environment, a set oftwo-state systems , the resulting dynamics of the system might be different.Using an explicitly time-dependent Lagrangian, the resulting equations of motion will show adissipation of energy. This may give the idea that there might be a workaround with which it ispossible to describe dissipation phenomena (more generally than dissipation of energy), withoutthe trouble of finding the correct model for the environment. It has been the motive of severalresearches, considering both classical case and quantization throughout the years . One way ofdoing this (at least in principle), is to define a potential (by doubling the degrees of freedom) whichgenerates the physical quantity. II. INTRODUCING AN ABSTRACT POTENTIAL
In this method, potential is a function which not necesarrily carries any physical meaning, butcontains all physical information, thus it generates the observable physical quantity. It can be a2urely mathematical tool to build Lagrangian formalism, that will generate the desired equation ofmotion for the observable in the end. It is possible to define a potential for any quantity describedby any linear differential equation . The idea is analogous to how one deals with observables andpotentials in electrodynamics. The observables (electric field and magnetic flux density) cannotbe handled in Lagrangian formalism, but potentials (scalar potential and vector potential) can bedefined, with corresponding differential equations of higher order.
A. Creating a Lagrangian using potentials
A general linear Euler–Lagrange equation can be written in the form˜ D (cid:26) ∂ L ∂ D { u } (cid:27) = D is a formal linear differential operator and ˜ D is its formal adjoint defined by Z Ω d τ v · D u − Z Ω d τ u · ˜ D v = Z ∂Ω d ν B { u , v } , (3)where B { u , v } is called the bilinear concomitant. This definition provides the possibility to calcu-late ˜ D u through repeated integration by parts. Let’s look at the example of a general differentialoperator of order n acting on a function with a single variable D u = p n d n u d t n + p n − d n − u d t n − + · · · + p d u d t + p u , (4)for which the adjoint operator acting on the function is˜ D u = ( − ) n d n d t n ( p n u ) + ( − ) n − d n − d t n − ( p n − u ) + . . . . (5)We can say that D is self adjoint if D u ≡ ˜ D u .Suppose that a measurable physical quantity u ( t ) is described by the following inhomogeneousequation D u ( t ) = c ( t ) , (6)where c ( t ) is arbitrary function. If D is not self adjoint, it cannot be calculated from variationalprinciple. One can define the potential φ ( t ) through the definition equation u ( t ) = ˜ D φ ( t ) . (7)3ubstitute Eq. (7) in Eq. (6) and D ˜ D φ ( t ) = c ( t ) . (8)is received. By using Eq. (3), it is easy to see that the differential operator D ′ : = D ˜ D is self-adjoint,hence the equation of motion for the potential φ can be calculated from variational principle, so aLagrangian exists, from which the equation of motion (8) can be calculated. This Lagrangian canbe written in the following form L = (cid:0) ˜ D φ ( t ) (cid:1) · (cid:0) ˜ D φ ( t ) (cid:1) − φ ( t ) · c ( t ) . (9)By using Eq. (2), the Euler–Lagrange equation can be calculated, resulting in Eq. (8) B. On the solutions
The potential φ ( t ) contains all physical information, and some excess, non-physical informa-tion can be encoded in it as well. Consider the linear operator D and its adjoint ˜ D , and supposethat D , ˜ D ∈ Lin ( V ) . As it is possible to obtain the original differential equation (6) from Eq. (8), itis safe to say that the kernel of ˜ D contains only non-physical information. By writing the solution φ ( t ) in such a way that φ ( t ) = ϕ ( t ) + λ ( t ) , where ˜ D ϕ ( t ) ∈ Im ( ˜ D ) and λ ( t ) ∈ Ker ( ˜ D ) it is easyto see that the λ ( t ) term can be omitted D u ( t ) = D ˜ D φ ( t ) = D ˜ D ( ϕ ( t ) + λ ( t ))= D ( ˜ D ϕ ( t ) + ˜ D λ ( t )) = D ˜ D ϕ ( t ) . (10)This can be interpreted as a kind of gauge freedom, because by omitting the λ ( t ) ∈ Ker ( ˜ D ) part of the potential, the measurable physical quantity will stay invariant, so one can define thegauge transformation as φ ( x ) → φ ( x ) + Λ ( x ) where Λ ( x ) ∈ Ker ( ˜ D ) . The solution of the adjoint equation ˜ D λ ( t ) = D = N ∑ n = p n d n d t n , D ζ ( t ) = N ∑ n = ( − ) n p n d n ζ ( t ) d t n = . It can be seen, that every odd order derivative changes its sign, and the even order terms areinvariant. By changing the sign of the variable t ( t → − t ), the adjoint equation can be rewritten N ∑ n = ( − ) n p n d n ζ ( − t ) d t n ( − ) n = N ∑ n = p n d n ζ ( − t ) d t n = D ζ ( − t ) = . In such a simple case, it can be clearly seen, that if ζ ( t ) is a solution of ˜ D ζ ( t ) =
0, then itstime reversed is a solution of D ζ ( − t ) =
0. As a consequence, λ ( t ) ∈ Ker ˜ D is related to thetime reversed of v ( t ) ∈ Ker D . Dissipative processes in nature tend to an equilibrium state, sothe time reversed of these solutions are divergent. To obtain a stable solution, the divergent term( λ ( t ) ∈ Ker ˜ D ) should be omitted. C. On initial conditions
In theory it is easy to omit the solutions from Ker ˜ D and for an analytical solution one caneasily perform the correct gauge transformation. Unfortunately, it does not seem possible if wewish to solve the differential equation numerically. A good idea would be to choose initial andboundary value conditions carefully so that the non-physical part λ ( t ) vanishes. The aim is tofind the relation between the initial conditions for the potential and the initial conditions for themeasurable.For the sake of simplicity, let’s deal with only one variable. Firstly write the general solutionfor the inhomogeneous equation Eq. (8) in the form φ ( t ) = N ∑ k = [ a k ϕ k ( t ) + b k λ k ( t )] + ξ ( t ) , (11)where ˜ D ϕ k ( t ) form the basis for the subspace Ker ( D ) and λ k ( t ) form the basis for the subspaceKer ( ˜ D ) and ξ ( t ) is a particular solution of the inhomogeneous equation (so the solution φ ( t ) = ϕ ( t ) + λ ( t ) is expanded on a basis). To solve a differential equation of order 2 N , we need 2 N initial conditions. As the number of initial conditions and the number of coefficients ( a k and b k )are the same, a unique solution exists. Physics provides only half of it, so we have to come up withthe other half in a way that ensures the vanishing of all b k coefficients in Eq. (11). The general5orm of the measurable is u ( t ) = N ∑ k = a k v k ( t ) + w ( t ) , (12)where v k ( t ) = ˜ D ϕ k ( t ) , which is a basis in Ker ( D ) .It is possible to create the initial conditions for the measurable from the initial conditions forthe potential. Let the initial conditions for the potential be φ , n = d n − φ d t n − (cid:12)(cid:12)(cid:12)(cid:12) t = where n = { , , . . . N } , (13)and let the initial conditions for the measurable be u , n = d n − u d t n − (cid:12)(cid:12)(cid:12)(cid:12) t = where n = { , , . . . N } . (14)The initial conditions for the measurable can be obtained by a linear combination of the initialconditions for the potential. It can be proven by straightforward calculation: u , n = d n − d t n − ˜ D φ (cid:12)(cid:12)(cid:12)(cid:12) t = = " d n − d t n − N ∑ i = ( − ) i d i d t i ( p i φ ) t = == " d n − d t n − N ∑ i = i ∑ l = ( − ) i (cid:18) il (cid:19) d l d t l p i · d i − l d t i − l φ t = == N ∑ i = i ∑ l = n − ∑ m = T n , i , l , m φ , n + i − l − m , (15)where T n , i , l , m = ( − ) i (cid:18) il (cid:19)(cid:18) n − m (cid:19) d l + m d t l + m p i (cid:12)(cid:12)(cid:12)(cid:12) t = . (16)Unsurprisingly, this relation cannot be inverted, but it provides a limitation on the configuration ofthe potential initial conditions.One possible (but not effective) way to find correct initial conditions for the potential in a nu-merical simulation is to try random configurations which reproduce the physical initial conditions(this can be checked using Eq. (15)). The closer the system starts in the phase space to the con-figuration that ensures the vanishing of the non-physical part, the slower the divergent part of thesolution will start to dominate. Other than trying, it seems improbable, that there is a method tocreate the desired initial conditions. 6 . Theoretical background of higher order Lagrangian and Hamiltonian mechanics Usually we are dealing with systems that can be described by a Lagrangian function of theform L ( t , q i , ˙ q i ) , so it contains at maximum first order derivatives. If we wish to use the method ofabstract potential, we need to generalize the Lagrangian formalism to Lagrangians depending onhigher order derivatives. We may examine such Lagrangian L ( t , q i , ˙ q i , ¨ q i , . . . ) , in which case wecan generalize the variational principle (Hamilton principle demands that the action functional S is extremal on physical trajectories)0 = δ S δ q = δδ q t Z t L ( t , q i , ˙ q i , ¨ q i , . . . ) d t (17)to obtain equations of motion 0 = N ∑ n = ( − ) n d n d t n ∂ L ∂ (cid:0) d n d t n q i (cid:1) . (18)We can also build a Hamiltonian formalism by correctly choosing canonical coordinate and mo-mentum pairs q i , n : = d n − d t n − q i (19a) p i , n : = N − n ∑ k = ( − ) k d k d t k ∂ L ∂ (cid:0) dd t q i , n + k (cid:1) , (19b)where n = , . . . N . In that case the Hamiltonian function can be calculated from the Lagrangianfunction H = ∑ i (cid:18) p i , d q i , d t + p i , d q i , d t + · · · + p i , N d q i , N d t (cid:19) − L (20)and we can obtain the canonical equations, which take the usual formd q i , n d t = ∂ H ∂ p i , n (21a)d p i , n d t = − ∂ H ∂ q i , n . (21b)As we can see, in the Hamiltonian formalism there are no higher order derivatives, the canonicalequations are first order and the dimension of the phase space M · N , where M is the number ofgeneral coordinates and N is the order of the highest order derivative present in the Lagrangian.7 ield theoretical generalization We could go one step further by generalizing this to Lagrangian densities hence be able todescribe field theories of higher order. In order to do this we first define the N th order Lagrangiandensity: L = Z V d x . . . d x d L (cid:18) φ i , ∂φ i ∂ x k , . . . ∂ N φ i ∂ x k . . . x k N (cid:19) . (22)The x coordinate is used as time. The Euler–Lagrange equation is again calculated from theHamilton principle using variational calculus :0 = N ∑ n = d ∑ α ,... α n = ( − ) n ∂ n ∂ x α . . . ∂ x α n ∂ L ∂φ ( n ) i [ α , α ,... α n ] , (23)where φ ( n ) i [ α , α ,... α n ] = ∂ n φ i ∂ x α . . . ∂ x α n . (24)Now, we introduce the canonical momentum density and canonical field pairs: φ i , n : = ∂ n − ∂ t n − φ i (25a) π i , n : = N − n ∑ k = ( − ) k ∂ k ∂ t k ∂ L ∂ ( ∂∂ t φ i , n + k ) . (25b)The Hamiltonian density can be obtained similarly to Eq. (20): H = ∑ i (cid:18) π i , ∂φ i , ∂ t + · · · + π i , N ∂φ i , N ∂ t (cid:19) − L (26) III. THE DAMPED LINEAR HARMONIC OSCILLATOR (A TOY MODEL)
The damped harmonic oscillator is a really good toy model, to test different methods on it. Theundamped harmonic oscillator is a well-known system both classically, and quantum mechanically,so it provides a good starting point for introducing the damping. The equation of motion for thedamped harmonic oscillator is m ¨ x + m λ ˙ x + m ω x = , (27)where m is the mass, λ is the damping coefficient and ω is the angular frequency.In order to define a potential q for the measurable quantity x , the adjoint equation must becalculated first. As the coefficients are constant, this can be easily done, and the definition equationcan be obtained x = ¨ q − λ ˙ q + ω q . (28)8y following the method, described in section II A, the following Lagrangian is received: L = (cid:0) ¨ q − λ ˙ q + ω q (cid:1) . (29)The method guarantees, that the Euler–Lagrange equation will be (cid:18) d d t + λ dd t + ω (cid:19)(cid:18) d d t − λ dd t + ω (cid:19) q = . (30) A. Underdamped and overdamped cases
As the coefficients are constants in the differential operator, it will commute with its adjoint,which means, that the solution for q ( t ) can be easily calculated q ( t ) = a e − ( λ + γ ) t + a e − ( λ − γ ) t + b e ( λ + γ ) t + b e ( λ − γ ) t , (31)where γ = √ λ − ω . The terms proportional to e λ t are solutions of the adjoint operator, hencethey are non-physical solutions, that will not contribute to the measurable x ( t ) . The effect of theadjoint operator on the other two terms is just a multiplication by a constant value, so they are thetwo independent solutions of the original differential operator.Physics provides the initial conditions for the measurable x ( t = ) = x , (32)˙ x ( t = ) = v . (33)By choosing the initial conditions for the potential q ( ) = λ x + v λ ( λ − γ ) , (34)˙ q ( ) = − x λ , (35)¨ q ( ) = − v λ , (36)... q ( ) = ( λ − γ ) x + λ v λ , (37)the non-physical solutions (the exponentially increasing terms in Eq. (31)) will vanish, so the9oefficients will be a = ( γ − λ ) x − v γλ ( λ + γ ) , (38) a = ( γ + λ ) x + v γλ ( λ − γ ) , (39) b = , (40) b = . (41) B. Critical damping and undamped case
There are 2 interesting cases, when the characteristic equation of the differential equationEq. (30) has repeated roots, λ = ω and λ =
0. For λ = ω the equation of motion isd q d t − ω d q d t + ω q = , (42)for which the general solution and the measurable are q = c e − ω t + c t e − ω t + c e ω t + c t e ω t , (43) x = e − ω t ( c ω − c ω + c ω t ) . (44)Here, the terms proportional to e ω t will not contribute to the measurable, so they will not carryany physical information. This is similar to the previous cases where the exponentially increasingterms were solutions of the adjoint operator. This means, that only the decreasing terms are enoughto construct a potential carrying all physical information. We can choose the initial conditions forthe potential the following way q ( ) = ω x + v ω , (45)˙ q ( ) = − x ω , (46)¨ q ( ) = − v ω , (47)... q ( ) = ω x + v , (48)10t will ensure the vanishing of the non-physical solutions, and will result in the following valuesof the coefficients c i c = ω x + v ω , (49) c = ω x + v ω , (50) c = , (51) c = . (52)Interestingly, something unexpected occurs, if the λ = q d t + ω d q d t + ω q = , (53)for which the general solution and the measurable are q = c e − i ω t + c t e − i ω t + c e i ω t + c t e i ω t , (54) x = − c i ω e − i ω t + c i ω e i ω t . (55)As it can be seen, only the polynomially increasing terms carry physical information. This mightlead to the assumption, that if information is encoded in increasing terms of the general solution,the system is not dissipative. However, the validity of this assumption is a question.In this case, it is also possible to choose the initial conditions, so the non-physical terms willvanish. The correct choice is q ( ) = , (56)˙ q ( ) = − v ω , (57)¨ q ( ) = x , (58)... q ( ) = v , (59)11ith which the coefficients c i are c = , (60) c = − v + i ω x ω , (61) c = , (62) c = − v − i ω x ω . (63) IV. CONCLUSION
By creating a potential, linear differential equations describing a dissipative system can becalculated from a Lagrangian. Using the described method, the potential can be easily constructedto an equation that expresses the dissipative behaviour of a a physical quantity. The problemof properly modelling the environment vanishes, instead the adjoint equation (which defines theconnection between the potential and the measurable) must be solved. Although in concept, it ispossible to omit non-physical solution which result in instabilities, technically there is no way todo that if the equation cannot be solved analytically. Moreover, if initial conditions, that provide azero non-physical part in the solution, are found, during a numerical simulation, numerical errorscan result in an unstable solution. One proper way to stabilize such a simulation is to find a relationthat can be checked throughout the solving procedure and restricts the solution to the physical partonly.The benefit of this method not only lies in the fact, that it provides a way of receiving anequation from a Lagrangian, but it provides the powerful tools of the Lagrangian framework. Ofcourse, it is an open question how handy these tools are on the level of potentials, and how thephysical information is obtained. One highly interesting idea is quantization, whether it is evenpossible through this method. Another exciting utilization is coupling fields . ACKNOWLEDGMENTS
Support by the Hungarian National Research, Development and Innovation Office of Hungary(NKFIH) Grant Nr. K119442 is acknowledged.12
ATA AVAILABILITY STATEMENT
Data sharing is not applicable to this article as no new data were created or analyzed in thisstudy.
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