Non-Leibniz Hamiltonian and Lagrangian formalisms for certain class of dissipative systems

1 INTRODUCTION

It is well known that there is no variational principle without complementary equations that yields to a linear dissipative set of differential equations with constant coefficients.1 By adding an exact differential to the integrand of a variational problem, it is possible to formulate the variational principles for the certain dissipative systems.2 In these cases, a set of solutions of the additional equations can be expressed in terms of the solutions of the original equations. Using these relations, the Lagrangian function expresses through of the dependent variables in the original set of equations.

To define the metriplectic systems, the structures of Hamiltonian and metric systems were combined.3 The phase space for metriplectic systems is equipped with a bracket operator. The bracket operator possesses an antisymmetric Poisson bracket part and a symmetric dissipative part. The combination of the Casimir invariants of the Poisson bracket represents the entropy.

Consider an ensemble of states that occupies a particular volume of phase space in the initial moment. The evolution of the volume of phase space is governed by Hamilton's equations. The flow of phase space may deform the shape of the ensemble, but it does not change its volume. The conservation of the phase space volume is known as the Liouville's theorem. On the other hand, it is well known that, for dissipative systems, the phase space volume is not conserved.4 This means the evolution of the system could not be described by Hamilton's equations in their ordinary formulation. The multiplication of the undamped Lagrangian by an increasing exponent of time leads to a formally correct Hamilton's equation5. Conversely, the modified Hamiltonian is not a sum of potential and kinetic energy of the system. Bloch et al6 showed that the system on Lie algebras cannot have linear dissipative terms of a Rayleigh dissipation type.

For Lagrangian treatment of the dissipative systems, Kaufman7 introduced “dissipative bracket.” For a dissipative bracket, a certain set of observables was associated, for which the dissipative bracket vanishes with any observable. The dissipative invariants, such as energy and momentum, were pointed. Furthermore, this idea was reformulated in the Lorentz-covariant form.8 The evolution of the system was associated with two scalar functionals. One functional was the Lagrangian action, and the second was the global entropy. The introduction of the bracket established the local covariant laws of entropy production and of energy momentum conservation.

Mshelia9 evaluated the probability amplitude for the transfer of collective excitation energy in coupled dissipative systems. The normalization and orthogonalization of eigenvectors corresponding to degenerate eigenvalues were carefully considered.

The Lagrangian with derivatives of fractional order was also proposed.10, 11 The Euler–Lagrange equation of motion for nonconservative forces was obtained in previous studies using the variation of the fractional Lagrangian. Using methods of classical mechanics with fractional and higher-order derivatives, the conjugate moments were defined. The method was applied to the frictional force proportional to velocity. However, the dissipation matrix appears in an unusual, skew-symmetric form. Extending this approach12 implemented the Riemann-Liouville fractional derivative of curves evolving on real space. With the fractional derivative, a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler–Lagrange equations (both in the continuous and discrete settings) was developed. The application of a variational principle and summation of the variations yields the restricted fractional Euler–Lagrange equation, which admits invariant linear change of variables.

The dissipation effects in quantum systems were studied by Tarasov.13 For the construction of quantum theory, the mathematical concepts that are general for Hamiltonian and non-Hamiltonian systems were implemented. Quantum dynamics was described by the one-parameter semigroups and the differential equations on operator spaces and algebras using the Lie–Jordan algebraic structure, Liouville space, and superoperators. This method consistently formulates the evolution of quantum systems and permits the investigation of the dynamics for open, non-Hamiltonian, dissipative, and nonlinear quantum systems.

The conditions for a Lagrangian formulation of dissipative dynamical systems were summarized by Musielak.14 In the previous study, the nonstandard Lagrangians and the corresponding equations of motions were derived.

In the current article, an alternative way is examined. Instead of modifying the Hamilton's function, we look at the alteration of the derivatives. The formalism is based on the generalized differentiation operator with a nonzero Leibniz defect. The Leibniz defect of the generalized differentiation operator linearly depends on one scaling parameter.15 In a special case, if the scaling parameter turns to one, the Leibniz defect vanishes, and a generalized differentiation operator reduces to the common differentiation operator. The generalized differentiation operator allows the formulation of the variational principles and corresponding Lagrange and Hamiltonian equations. In the present article, the alternative definition of the derivatives and the mechanical consequences of this alterative definition are investigated.

The structure of the article is as follows. The first section introduces the generalized differentiation operator with a nonzero Leibniz defect. The relation between common and generalized differentiation operators is established. The expression of the generalized differentiation operator in terms of a common differential operator facilitates the solution of the generalized differentiation equations and the investigation of special functions. The next addressed question is the uniqueness of the derivative definition. All solution operators of the classical Leibniz product rule were determined by König and Milman.16 Their method is extended to the operators with the Leibniz defect, and the uniqueness is confirmed.

In the second section, the solutions of the generalized differential equations of the first and second orders are found. The solutions of the linear equations present the κ-exponential function and κ-trigonometric functions.

In the third section, the κ-derivatives of higher and fractional order are introduced. The integer higher κ-derivative is based on the common binomial formula. Using the fractional binomial formula, the definition of the fractional derivatives follows from the expression of the integer higher κ-derivative.

The κ-antiderivative is defined in the fourth section. This definition is necessary for the integration procedure and is required in the succeeding sections.

The κ-Lagrange function of the systems with a single and multiple degrees of freedom could now be defined. The corresponding procedure is studied in the fifth section. The κ-canonical moments are used instead of common canonical moments, which are defined by the ordinary partial derivatives of Lagrangian.

The κ-Lagrange oscillating systems with a single degree and multiple degrees of freedom are studied later in the sixth section. The closed-form solutions are found for the dynamic equations.

The Poisson bracket and conservation laws in the non-Leibniz mechanics are investigated in the seventh section. The investigation is based on the formal replacement of the common derivative thought its generalized analog.

The eighth section establishes the solutions of the homogeneous linear dissipative wave equation in spaces of different dimensions. The most impressive result is the following. For the nonvanishing defect, the frequency and wavelength depend on the distance from the source. The character hangs on the sign of the defect constant. The frequency fades with the travel distance if κ > 0. Accordingly, the wavelength increases with the distance, giving rise for the “red shift” of the linear κ-wave. Oppositely, if κ < 0, the frequency increases with the travel distance. The wavelength reduces with the distance, and the ultimate radius of the wave propagation is |κ|.

2 GENERALIZED DIFFERENTIATION OPERATOR

2.1 Definition of generalized differentiation operator

The adjoint L^* of a given bounded operator L acting on a certain Hilbert space with an inner product is defined by the following equality17:

Here f and g are arbitrary vectors in the Hilbert space equipped with the corresponding inner product. The role of the inner product in mechanics plays the integration.

If the adjoint operator satisfies the identity

the operator is self-adjoint. For the following study of variational principles, the operators that allow the integration by parts and are anti–self-adjoint are necessary:

For example, the common derivative ∂ is anti–self-adjoint if the end terms in the Lagrangian vanish. This property is used to derive Euler–Lagrange equations.

A κ-differential field is a field F with the generalized derivations Р: F → F, which satisfies the following rules:

(1)

(2)

From 1 to 2, it immediately follows that the operator Ð is linear.

The last term in 2 is referred to as the Leibnitz defect. The value K is a given constant of the inverse dimension of the independent variable. In the limit case K → 0, the Leibniz defect vanishes, and generalized derivative turns into a common derivative f ′ = ∂f, which satisfies the common Leibniz rule:

The common Leibniz ∂ differentiation operator is anti–self-adjoint ∂ =  −∂^*. Namely, the integration by parts could be expressed by the anti–self-adjoint of the common Leibniz derivative. The derivation of the Euler equation in the variational calculus is based on the integration by parts. The anti–self-adjointness of the common Leibniz ∂ differentiation is essential for Hamiltonian and Lagrangian mechanics.

The adjoint operator for Ð is Ð^*. Accordingly, for the application of the integration by parts and for the corresponding variational principles, the operator Ð is required to be anti–self-adjoint:

(3)

2.2 Relation between common and generalized differentiation operators

We look for the operators in the following form:

(4)

In Equation 4, two functions α = α(x), β = β(x) are assumed to be arbitrary. To determine these functions, the conditions 1-2, and 3 are used. The adjoint operator reads

Consequently, the anti–self-adjointness

leads to the following condition:

The substitution of the expression 3 in Equation 2 delivers an equation α′ = 2K.

Each appropriate choice for the pair of the coefficients α = α(x) and β = β(x) leads to a certain operator algebra. The initial conditions for the differential equations specify the functions uniquely. The motivation for the choice α(x = 0) = 1 is the following. With this condition, expression 4 reads Ðx ≡ α ∂x+β x. Both operators Ðx and ∂x match in the point x = 0, if α(x = 0) = 1. In this case, Ðx ≡ α ∂x = ∂x.

With the condition α(0) = 1, the functions turn into

where κ =  −2K.

Suppose that the functions g and f vanish on the ends of the integration interval. As a final point, the anti–self-adjoint operator is

(5)

2.3 Uniqueness of the derivative definition

The next addressed question is the uniqueness of the derivative definition, based on the product rule with the Leibniz defect. All solution operators of the classical Leibniz product rule were determined by König and Milman.16 The method they established could be immediately extended to the rule 2 with the Leibniz defect. Similarly, the only solution of 2 that satisfies the correspondence condition 3 is essentially the κ-derivative. The product rule with the Leibniz defect after the substitution is

(6)

For the operator Ð, the Leibniz rule 2 reads

The application of the method of theorem 2 in the work of König and Milman16 leads for the operator Ð to the functional equation:

(7)

With the substitution H(s) = F(exp(s)) into 7, the product rule with the Leibniz defect reads as follows:

(8)

The only difference between the functional equation 8 and the functional equation that is considered in16 is the last term. The solution of the functional equation 8 delivers18

(9)

The considerations of König and Milman16 are repeated with some minor alterations for the product rule with a similar entropy function that contains the nonvanishing Leibniz defect κ. Thus, all solution operators of the product rule with the Leibniz defect are determined.

3 GENERALIZED DIFFERENTIAL EQUATIONS OF THE FIRST AND SECOND ORDER

3.1 κ-Exponential function

The special functions will be defined through the solution of κ-differential equations. At the first order, we define the κ-exponential function exp_κ(x, ω) as the solution of the linear ordinary κ-differential equation of the first order:

(10)

The solution of Equation 10 is

(11)

(12)

The κ-exponential function 11 reduces to the ordinary exponential function in its limit case to

(13)

3.2 κ-trigonometric functions

At the second order, we define the κ-trigonometric functions. The definition is based on the Euler's formula:

(14)

The corresponding κ-differential equations follow from Equation 10 as its imaginary and real parts:

(15)

From 14 follows the solutions of Equation 15:

(16)

If κ > 0, the solution displays an oscillating behavior with the decreasing frequency. In contrast, if κ < 0, the solution demonstrates oscillatory tendency with the increasing frequency. In the limit case κ → 0,

(17)

4 κ-DERIVATIVES OF HIGHER AND FRACTIONAL ORDER

4.1 κ-derivatives of higher integer order

Expressions for the higher order κ-derivatives follow from the successive use of the solitary generalized derivative:

Formally, the subsequent application delivers the differentiation operator of an integer order n:

The above expression could be transformed to a sum of the common derivatives d^n − kf/dx^n − k of the function by the use of the binomial formula.19

Finally, the application of the common binomial formula delivers the general expression for the κ-derivative of an integer order n:

(18)

with the binomial coefficient19:

4.2 Fractional κ-derivatives

Using the expressions for the higher integer order κ-derivatives, the fractional κ-derivative will be straightforward defined. For this purpose, the representation of κ-derivative in the form 18 is applied.

The derivative of any rational order α of the function f = x^β is20

(19)

The substitution of the expression 19 into 18 with α = n leads to

(20)

The sum 20 could be evaluated in a closed form even for a rational derivative order α:

(21)

with the confluent hypergeometric function of the first kind ₁F₁(a,b; x)21 and

The expression 21 allows the calculation of the derivatives of any rational order.

The solutions of the fractional κ-differential equations could be studied using the expression 21 in closed form, but the corresponding solutions are dispensable for the principal tasks of the current article.

5 INTEGRATION

5.1 κ-antiderivative

Once a generalized derivation Ð is defined, we look for a corresponding indefinite κ-integral. We define now the κ-indefinite integral (κ-antiderivative). The κ-indefinite integral represents a class of functions whose κ-derivative is the integrand:

(22)

Consequently, the integration of the function f (x) to get the antiderivative Φ(x) is equivalent to the solution of the ordinary differential equation of the first order 22:

(23)

5.2 Integral of κ-exponential function

In particular, for the function f (t) = exp_κ(t, ω) from Equation 11, the integral 23 evaluates as

(24)

5.3 Integral of power function

The integral 23 of the function f = x^β, β ≠  −1, is

(25)

Here, is the hypergeometric function.21 In the limit case, the κ-indefinite integral 25 turns into the common indefinite integral:

(26)

The hypergeometric function reduces for integer positive β to elementary functions:

(27)

and for β =  −1,

Since the κ-integration is a linear operation, integrals of polynomials and Taylor series can also be integrated using the rule 25.

6 LAGRANGIAN AND HAMILTONIAN

6.1 κ-Lagrange function

In mechanics, the time t plays frequently the role of the independent variable x and marks the evolution of the system. The basic method in the Lagrange mechanics consists in treating the generalized coordinates q_i, i = 1,…, n, as independent variables. The lover Latin indices are used for generalized coordinates. The time dependence of these variables is determined by the Lagrange equations of the second order. The generalized velocities Ðq_i are all dependent, derived quantities. The initial values for q_i and Ðq_i are determined from the 2n integration constants.

A common mechanical system possesses the kinetic energy

and the potential energy

The velocity ∂q_i is an ordinary derivative of the coordinate q_i. The Lagrangian of the mechanical system is

(28)

The correspondence principle is the following. The ordinary derivatives ∂q_i must be replaced in the expression of the Lagrangian 28 by ∂q_i → Ðq_i:

(29)

This function will be referred to as the κ-Lagrange function of the system. The variation of the action integral reads

If δq_i(t₁) = δq_i(t₂) = 0, the last term disappears, and the vanishing of the variation delivers the Euler–Lagrange equations:

(30)

6.2 Canonical moments

The canonical moments pⁱ(t) are based on partial (functional) derivatives of Lagrangian with respect to Ðq_i:

(31)

The canonical moments pⁱ(t) are used instead of the common canonical moments, which are defined by the ordinary partial (functional) derivatives of the ordinary Lagrangian.4

For canonical moments, the upper Latin indices will be used. Then, the moments pⁱ are raised at the same level with the coordinates q_i(t). Consequently, the set

forms a sequence of 2n independent variables. These variables fulfill a sequence of 2n coupled differential equations of the first order:

(32)

In the following, we imply the convention for a summation about repeated indexes. The variable q or p without an index designates the whole sequence.

7 LINEAR HARMONIC OSCILLATOR

7.1 Linear system with one degree of freedom

Consider the system performing linear oscillations. Namely, consider a system with one degree of freedom in a stable equilibrium position. Let q₀ be the value of the generalized coordinate corresponding to the equilibrium position. When the system is slightly displaced to a position q from the equilibrium position, a force occurs that acts to restore the equilibrium when its potential energy. The potential energy can be written as

The coefficient k represents the value of the second derivative of V(q) for q = 0. To study the dissipative system, we define formally the κ-kinetic energy of a system as

with the mass m and the Hamilton function

with ω² = k/m.

The harmonic oscillator with the mass of the particle equal to one possesses the Lagrangians

(33)

(34)

The equations of motion of the κ-harmonic oscillator reduce to

(34)

From Equation 34 follows the κ-equation of motion for an oscillator with the single degree of freedom:

(35)

The substitution of 5 into 35 transforms the κ-equation of motion to the ordinary differential equation

(36)

In Equation 36, the damped frequency of the κ-oscillator appears as

The solution of the differential Equation 36 reads

with

7.2 Linear system with several degrees of freedom

Previously, we dealt with the oscillations of one mass. In this section, we will look at oscillations involving more than one object. Consider a dynamical system with N degrees of freedom near a minimum of the potential energy. The Lagrangian has a general form

where the mass and stiffness coefficients are symmetric:

If the defects for all degrees of freedom are equal, the Lagrangians can be written in the matrix form as

(37)

Here, M is the mass matrix, K is the stiffness matrix, and X = (q₁, …, q_N).

Lagrange equations of motion lead to a set of N coupled algebraic equations:

(38)

We search for the solution of 40 in the following form:

(39)

Equations 39 describe the evolution of both systems. Using the formulas 38 and 39, we obtain the solitary system of the linear algebraic equations for both differential Equations 38:

(40)

The system of Equations 40 possesses a nontrivial solution if the characteristic equation fulfilled:

(41)

The characteristic of Equation 41 defines the fundamental frequencies ω_i, i = 1,…,N of both systems 38. By transformation to normal coordinates, the quadratic forms for the kinetic and potential energies in Equation 37 can be reduced simultaneously to sums of squares in these coordinates and their derivatives, hence making the coupled oscillator problem separable into independent motions, each with a particular normal frequency. One can prove that the fundamental frequencies of both systems 38 are positive if the potential energy is a positively defined bilinear form, as it is the case for the energy minimum.

8 POISSON BRACKET AND CONSERVATION LAWS IN THE NON-LEIBNIZ MECHANICS

In general, the Poisson bracket of any two dynamical variables f (t, q_i, pⁱ) and g (t, q_i, pⁱ) is defined as

(42)

For arbitrary functions f (t, q_i, pⁱ), g (t, q_i, pⁱ), and h (t, q_i, pⁱ), the following relations are valid:

(43)

(44)

(45)

(46)

(47)

The relations 43 to 46 do not involve the κ-derivatives and are well known from mechanics.22, Ch. VIII The relation 47 contains an additional term. The specifications for the non-Leibniz mechanics identities 47 could be immediately demonstrated with the expressions 2 and 5. From Hamilton equations 44, the κ-derivative of a function F(t, q_i, pⁱ) becomes

(48)

A special case of 48 is

(49)

Equations 49 are referred to as equations of motion 32 in Poisson bracket form.

Another special case is

(50)

The identity [H, H] = 0 is used in 50. If the Hamiltonian does not depend directly on time

the subsequent formula demonstrates the time dependence of the Hamiltonian:

(51)

This condition is the non-Leibniz counterpart to the common conservation law

In accordance with 13, we have

such that the solution of 51 reads

9 THREE-DIMENSIONAL HOMOGENEOUS LINEAR DISSIPATIVE WAVE EQUATION

9.1 Three-dimensional waves

In three-dimensional wave propagation in elastic media, the traveling waves exhibit various modes of vibration including longitudinal and transverse waves. To derive the appropriate equations of motion in continuous media, we need to extend the Hamilton principle by considering the displacement vector φ(x,t). We use symmetric motion given by

The components of particle velocity are

Using this notation and tensor summation convention, the kinetic energy T and the potential energy V are given by

(52)

In the absence of the external forces, the Lagrangian density is given by

(53)

The Lagrangian functional is of the following form:

(54)

The generalized Hamilton principle for a three-dimensional continuum for various modes of wave propagation described by φ(x,t) takes the following form:

This condition leads to the Euler–Lagrange equations of motion:

(55)

In particular, if the Lagrangian is of the form 53-55 gives the homogeneous wave equations for each component of the displacement vector:

(56)

The solution of the homogeneous wave Equations 56 in the case N = 1 with the initial conditions ϕ|_t = 0 = ϕ₁(x), ∂ϕ/∂t|_t = 0 = ϕ₂(x) can be found by means of characteristics method.23

9.2 Radial green functions

The radial Green functions of the homogeneous wave Equation 56:

will be found in spaces of dimensions N = 1,2,3.24 Separation of variables is applied by making a substitution of the following form:

breaking the resulting equation into two independent ordinary differential equations for radius- and time-dependent functions:

(57)

(58)

The frequency Ω plays the role of the separation constant. The Equations 57 and 58 are the homogeneous hypergeometric differential equations.25 The solution of the differential Equation 57 leads to the expression for ρ(r) as the function of radius r to the singular source of waves:

(59)

Here, ₂F₁ are the hypergeometric functions.21

For N = 1, the general solution 59 simplifies to the elementary functions:

(60)

In the limit case κ → 0, the solution 59 reduces to an ordinary solution for the two-dimensional waves in terms of Bessel functions26:

(61)

The solution of the second, time-dependent Equation 58 is in the following form:

(62)

The frequency of the linear wave depends 60 on the distance from the source. For positive κ, the frequency fades with the travel distance. Consequently, the wavelength grows with the distance to the source, resulting in the “red shift” of the linear κ-wave. Contrarily, for negative κ, the frequency grows with the travel distance. It turns to infinity at the point r = |κ|. The wavelength declines with the distance to the source, resulting in the “blue shift” of the κ-wave. The distance r = |κ| could be considered the confinement radius of the wave.

The corresponding ordinary limit case κ → 0 for 62 is

10 CONCLUSIONS

The method for the study of dynamical systems is based on the introduction of a derivative with the Leibniz defect. The differential algebra of this generalized derivative is briefly established. The simple non-Leibniz differential equations and their solutions are solved in closed form in terms of special functions. The introduction of the derivative allows the formulating of the dissipative equations with Lagrange and Hamilton structures. The analytical solution of the equations of dissipative oscillator is found. The most notable result is the frequency distance reliance of the linear dissipative wave. The frequency depends on the distance from the source. For positive κ, the frequency declines with the travel distance. Consequently, the wavelength grows with the distance to the source, resulting in the “red shift” of the linear κ-wave.

For negative κ, the frequency increases with the travel distance and turns to infinity at the distance |κ|. The wavelength reduces with the distance to the source, resulting in the “blue shift” of the κ-wave.