arXiv:1201.1032v2 [math.DS] 1 May 2012 hsc,tenx tpi ocnie icismd from made circuits consider to memcapacitors. and is of meminductors, understanding step memristors, deep applications, next advancing of in the range role physics, are their broad for their frameworks also for but classical just the these not and important Since (2009) Hamilto- Scherpen therein. and and conven- references Jeltsema of Lagrangian see analysis using methods; nian has and circuits research description electrical of the tional amount to significant devoted a century, been past the references. In state-of-the- of current list extensive the an of (2011) Ventra and overview art Di complete into introduction fairly and an Pershin a for for and (2009) elements, al. memory et the Ventra de- Di memcapacitive See vices. and field and meminductive the memristive using quantum computation using field-programmable in and operations foreseen devices, arithmetic memcapacitive are and several concepts logic Nevertheless, of these limited. of somewhat applications still meminductive and is memcapacitive behavior showing intelligent systems num- to of the ber capabilities, memory adaptive and from learning nano with ranging machines non-volatile behavior, in memristive of applications exhibit many applications that are and systems there realizations While experimental respectively. discovered voltage plane, voltage the and charge plane, in flux versus versus loop current be hysteresis plane, current can pinched versus and their nonlinear by are elements elementsidentified the three which All through evolved. states have past the of re- memory capacitance tain and inductance, circuit resistance, two-terminal whose of elements class important increasingly constitu- an memcapacitors tive and meminductors, Memristors, Abstract: beet ae nti nlg,sml ehncldvcsaepre memcapac are m and Keywords: a devices meminductive, of mechanical memristive, analogue explain simple electrical to some analogy, Kirchho examples the this a integrated didactic as on is of considered Based charge set be absement. time-integrated may a it Although yields theory, s b charges. and circuit follows a element principle fluxes via variational the included node dual of be integrated A can action. the losses of of Memristive dimension fluxes. the time-integrat inte element has of of that the set consist of a charges, terms provide in loop equations Euler-Lagrange usual corresponding co the different the of a natur problem, instead in this circumvent nonconservative that, To are functions. state elements not these are since sense classical .INTRODUCTION 1. erso,Mmnutr ecpctr eoyeeet,Lagra elements, Memory Memcapacitor, Meminductor, Memristor, ef nttt fApidMteais ef nvriyo University Delft Mathematics, Applied of Institute Delft eidcosadmmaaiosd o lo arninformulat Lagrangian a allow not do memcapacitors and Meminductors arninMdln fEetia Circuits Electrical of Modeling Lagrangian eoyEeet:APrdg hf in Shift Paradigm A Elements: Memory eewg4 68C ef,TeNetherlands The Delft, CD 2628 4, Mekelweg Eal [email protected]) (Email: iir Jeltsema Dimitri ihsm nlrmrsi eto 6. 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DEFINITION OF MEMORY ELEMENTS ation with respect to time, yields

q = CM (ϕ)V, (7) From a mathematical perspective, the behavior of a two- where C (ϕ) := dσ/dϕ denotes the incremental capaci- terminal resistor, inductor, and capacitor, whether linear M tance. The memory aspect of a flux-modulated memcapac- or nonlinear, is described by a relationship between two itor stems from the fact that it ‘remembers’ the amount of the four basic electrical variables, namely, voltage V , of voltage that has been applied to it. current I, charge q, and flux ϕ, where t A meminductor (resp. memcapacitor) that depends on q(t)= I(τ)dτ, (1) the history of its flux (resp. charge) can be formulated Z−∞ t by starting from a constitutive relationship of the form ϕ(t)= V (τ)dτ. (2) q =q ˆ(ρ) (resp. ϕ =ϕ ˆ(σ)). Z−∞ In the special case that the constitutive relationship of A resistor is described by a constitutive relationship be- a memristor is linear, a memristor becomes an ordinary tween current and voltage; a capacitor by that of voltage linear resistor. Indeed, in such case (3) reduces to ϕ = Rq, and charge; and an inductor by that of current and flux with constant memristance M (the slope of the line), or linkage. equivalently, V = RI, which precisely equals Ohms law. Based on logical and symmetry reasonings, Chua (1971) Hence it is not possible to distinguish a two-terminal linear postulated the existence of a fourth element that is char- memristor from a two-terminal linear resistor. The same acterized by a constitutive relationship between the re- holds for a linear meminductor and linear memcapacitor, maining two variables, namely charge and flux. This el- where (6) reduces to ρ = Lq, or equivalently, ϕ = LI, and ement was coined memristor (a contraction of memory (7) to σ = Cϕ, or equivalently, q = CV , respectively. and resistor) referring to a resistor with memory. The memory aspect stems from the fact that a memristor ‘re- 3. SELF-ADJOINTNESS OF CIRCUIT DYNAMICS members’ the amount of current that has passed through it together with the total applied voltage. More specifically, The dynamical behavior of any electrical circuit consisting if q denotes the charge and ϕ denotes the flux, then a of conventional, possibly nonlinear, resistors, inductors, charge-modulated memristor is defined by the constitutive and capacitors is basically determined by three types relationship ϕ =ϕ ˆ(q). Since flux is defined by the time of equations: those arising from Kirchhoff’s voltage law integral of voltage V (like in Faraday’s law), and charge is (KVL), those arising from Kirchhoff’s current law (KCL), and the constitutive relationships of the elements. In many the time integral of current I, or equivalently, V =ϕ ˙ and 1 I =q ˙, we obtain cases this leads to differential equations of the form j V = RM (q)I, (3) Aij (x ˙)¨x + Bi(x, x˙)=0, (8) where RM (q) := dϕ/dq is the incremental memristance. where i, j = 1,...,n, and x ∈ Rn represents a column Note that (3) is the definition of a memristor in impedance vector of loop charges and node fluxes. form. The admittance form I = GM (ϕ)V , with incremen- The system of differential equations (8) allow a Lagrangian tal memductance GM (ϕ) := dq/dϕ, is obtained by starting description if we can find a Lagrangian L(x, x˙) satisfying from a constitutive relationship q =q ˆ(ϕ). d ∂L ∂L − ≡ A (x ˙)¨xj + B (x, x˙). (9) In addition to the memristor, it is shown in Di Ventra dt ∂x˙ i ∂xi ij i et al. (2009) that the memory-effect can be associated to In mechanics it is known that the existence of a Lagrangian inductors and capacitors as well. For that, let σ and ρ L(x, x˙) relies on the fact that the system of differential denote the time-integrals of charge and flux, equations (8) is self-adjoint, which for the present form is t tantamount to the following set of integrability conditions σ(t)= q(τ)dτ, (4) (Santilli, 1978): Z−∞ t Aij = Aji, (10) ρ(t)= ϕ(τ)dτ, (5) Z−∞ ∂Aik ∂Ajk = , (11) respectively. Then, a memory inductor, or meminductor ∂x˙ j ∂x˙ i for short, is a two-terminal element defined by a constitu- ∂B ∂B 1 ∂ ∂B ∂B i − j = i − j x˙ k, (12) tive relation ρ =ρ ˆ(q). Indeed, differentiation of the latter ∂xj ∂xi 2 ∂xk  ∂x˙ j ∂x˙ i  with respect to time yields ∂Bi ∂Bj ϕ = LM (q)I. (6) + =0. (13) ∂x˙ j ∂x˙ i Since LM (q) := dρ/dq relates flux with current, its values clearly correspond to the units of inductance. Bearing 3.1 Conventional Conservative Circuits in mind that charge is the time integral of current, the memory aspect of a charge-modulated meminductor stems From a Lagrangian perspective, it is well-known that from the fact that it ‘remembers’ the amount of current for a large class of circuits that are made from current- that has passed through it. controlled inductors and charge-controlled capacitors, and Dually, a flux-modulated memcapacitor is defined by a 1 Throughout the document we adopt the summation convention of constitutive relationship σ =σ ˆ(ϕ), which, after differenti- repeated indices. consisting of n loops, the differential equations (8) can be To circumvent this problem, a so-called content function, written as which is a nonlinear multi-domain generalization of the d ∂L ∂L − =0, (14) Rayleigh dissipation function, is usually introduced. dt ∂q˙i ∂qi As an example, suppose the inductor in Fig. 1 possesses a where qi represents the loop charge associated to the i- i constant series resistance R. In that case, the equation of th loop, andq ˙ represents the loop current circulating motion extends to ϕ′(q ˙)¨q + Rq˙ + q/C = 0, so that A(q ˙)= in the i-th loop. For this particular case, the Lagrangian ϕ′(q ˙) and B(q, q˙) = Rq˙ + q/C. For R = 0, the conditions L(q, q˙) equals the total magnetic co-energy stored in the ∗ (10)–(13) are clearly satisfied, but if R =6 0, condition (13) inductors, T (q ˙), minus the total electric energy stored in is violated and no Lagrangian can be associated to the the capacitors, U(q). circuit. On the other hand, the introduction of a function 1 2 On the other hand, if the inductors are flux-controlled and of the form D(q ˙)= 2 Rq˙ , satisfying the capacitors are voltage-controlled, we need to start from d ∂L ∂L ∂D a node analysis yielding a so-called co-Lagrange equation − + =0, (16) dt ∂q˙ ∂q ∂q˙ associated to each node in the network. Hence, ifϕ ˙ j represents the potential of the j-th node together with solves the problem. However, the variational character of its time-integral, the node flux ϕj , one obtains the dynamics that makes the Lagrangian formalism so d ∂L∗ ∂L∗ appealing is clearly lost. − =0, (15) dt ∂ϕ˙ j ∂ϕj Note that the existence of a content function relies on the where the co-Lagrangian L∗(ϕ, ϕ˙) equals the total electric fact that the resistors in the circuit are current-controlled. ∗ The inclusion of voltage-controlled resistors requires the co-energy stored in the capacitors, U (ϕ ˙), minus the total ∗ magnetic energy stored in the inductors, T (ϕ). introduction of a co-content function D (ϕ ˙). As an illustration, consider a circuit consisting of a nonlin- ear current-controlled inductor, with constitutive relation 4. CIRCUITS WITH MEMORY ELEMENTS ϕ =ϕ ˆ(I), and a linear capacitor C as shown in Fig. 1. 4.1 Problem 1: Path-Dependence I ϕ Now consider a circuit consisting of a meminductor (6) and a linear capacitor shown in Fig. 2. Application of KVL L C yieldsϕ ˙ + q/C = 0, or equivalently, q d q ′ 2 q L (q)I + = L (q)I˙ + L (q)I + =0. (17) dt M C M M C
In order to derive the dynamics using a Lagrangian for- Fig. 1. Circuit with a nonlinear conventional inductor. mulation, one is tempted to start from a Lagrangian that equals the magnetic co-energy stored in the meminductor Let q denote the loop charge andq ˙ = I the associated loop minus the electric energy stored in the capacitor, i.e., current, then the Lagrangian for the circuit reads 1 2 1 2 L(q, q˙)= LM (q)q ˙ − q . (18) 1 2 L(q, q˙)= ϕˆ(q ˙)dq ˙ − q , 2 2C Z 2C However, the latter is clearly not a proper state function which, upon substitution into (14), yields the equation of since it depends on the path q. motionϕ ˆ′(q ˙)¨q + q/C = 0. Note that the latter constitutes the KVL for the circuit. I Dually, if the nonlinear current-controlled inductor is re- placed by a nonlinear flux-controlled inductor, we have to consider LM C ∗ 1 2 L (ϕ, ϕ˙)= Cϕ˙ − Iˆ(ϕ)dϕ, q 2 Z resulting in the equation of motion Cϕ¨ + Iˆ(ϕ) = 0, which constitutes the KCL for the circuit. Fig. 2. Circuit with a meminductor. 3.2 Conventional Non-Conservative Circuits Furthermore, substitution of the Lagrangian (18) into (14) 1 ′ 2 Dissipation due to conventional resistors in the circuit yields the equation LM (q)¨q + 2 LM (q)q ˙ +q/C = 0, where, leads to non-conservative dynamics that is not self-adjoint. in comparison to (17), we observe the appearance of an 1 Resistors can therefore not be included using a (standard) erroneous factor 2 . Lagrangian function. 2 2 In some cases the dynamics can be rendered self-adjoint by looking the usual interpretation of stored energy anymore (Ray, 1979). for an integrating factor (Santilli, 1978). However, the form of the Furthermore, it remains unclear how to apply this method in case of differential equations is altered and the Lagrangian does not have complex circuits consisting of many loops. 4.2 Problem 2: Self-Adjointness latter. It seems therefore more natural to consider, instead of the stored magnetic co-energy, which was obtained in The reason for this discrepancy is that the differential (18) as the integral of (6) with respect to the current, a equation (17) is not self-adjoint. To see this, let us consider function of the form ∗ the general form of a circuit consisting of memristors, me- T¯ (q) := ρˆ(q)dq. (26) minductors, memcapacitors, and their, possibly nonlinear, Z conventional counterparts, independent voltage sources, When plotted in the q-versus-ρ plane, (26) represents and independent current sources, given by the area above the curve associated to the constitutive j Aij (x, x˙)¨x + Bi(x, x˙)=0, (19) relation. On the other hand, its complementary part, the function T¯, is defined to be the area below the curve and where, in comparison to (8), it is observed that Aij now also depends on the x-coordinates. The necessary and takes the form sufficient conditions for the existence of a Lagrangian for T¯(ρ) := qˆ(ρ)dρ. (27) this case extend to Z Under the assumption that ρ =ρ ˆ(q) and q =q ˆ(ρ) are Aij = Aji, (20) invertible (i.e., one-to-one), the two state functions can be related via the Legendre transform ∂Aik ∂Ajk = , (21) ∗ d d ∗ ∂x˙ j ∂x˙ i T¯ + T¯ = qρ, q = T¯(ρ), ρ = T¯ (q). dρ dq ∂B ∂B 1 ∂ ∂B ∂B i − j = i − j x˙ k, (22) ∂xj ∂xi 2 ∂xk  ∂x˙ j ∂x˙ i  In a similar fashion, for a flux-modulated memcapacitor, we propose a function of the form ∂Bi ∂Bj ∂Aij k + =2 x˙ . (23) ∗ ∂x˙ j ∂x˙ i ∂xk U¯ (ϕ) := σˆ(ϕ)dϕ, (28) Z Returning to the differential equation (17), it is directly representing the area above the curve associated to the ′ 2 verified that, with A(q)= LM (q) and B(q, q˙)= LM (q)q ˙ + constitutive relation, whereas for an integrated charge- q/C, the circuit is not self-adjoint, and therefore does not modulated memcapacitor allow a Lagrangian formulation. U¯(σ) := ϕˆ(σ)dσ, (29) A possible solution to compensate for the erroneous term Z is to add contra terms to the right-hand side of the representing the area underneath the curve. differential equation; see Pesce (2003). This is tantamount Furthermore, we have to introducing a content function of the form ∗ d d ∗ 1 ′ 3 U¯ + U¯ = ϕσ, ϕ = U¯(σ), σ = U¯ (ϕ). D(q, q˙)= LM (q)q ˙ , dσ dϕ 6 ¯ ¯∗ ¯ ¯ ∗ like in (16). However, as argued before, the variational Note that T , T , U, and U all serve as a state function and character is lost. all exhibit the units of energy times time-squared (Joule- second-squared [Js2]), which is equivalent to action times time. 4.3 Conservation of Flux and Charge 4.5 Meminductor and Memcapacitor Circuits The conventional Lagrangian formalism applied to electri- cal circuits essentially codes the KVL and KCL in terms Consider again the circuit of Fig. 2. First, we select, instead of energy storage via the Lagrangian. On the other hand, of a loop charge q, the integrated loop charge σ, withσ ˙ = suppose that we integrate both Kirchhoff laws with respect q, as the configuration variable. Secondly, it is observed to time. This would result in the laws of conservation of that the conventional capacitor can be considered as a flux and charge around a particular loop and node linear memcapacitor with constitutive relation ϕ = σ/C. t In terms of the state functions proposed in the previous ϕi(t)=0, ϕi(t)= Vi(τ)dτ, (24) Z−∞ subsection, let us define a Lagrangian of the form Xi σ˙ 1 2 t L¯(σ, σ˙ )= ρˆ(q)dq − σ . qj (t)=0, qj (t)= Ij (τ)dτ, (25) Z0 2C Z−∞ Xj Then, it can be demonstrated that, invoking Hamilton’s respectively, which state that flux and charge can neither principle of least action and considering variations in terms be created nor be destroyed (Chua, 2009). of σ, we get the Lagrangian type of equation d ∂L¯ ∂L¯ In the sequel, we will refer to (24) as the iKVL and to (25) − =0, (30) as the iKCL. dt ∂σ˙ ∂σ which, in turn, generates the nonlinear differential equa- ′ 4.4 Memory State Functions tionρ ˆ (σ ˙ )¨σ + σ/C = 0. Differentiating the latter with respect to time yields ′ ... ′′ 2 In Section 2, we have seen that the most fundamental ρˆ (σ ˙ )σ +ρ ˆ (σ ˙ )¨σ +σ/C ˙ =0, relationship of a charge-modulated meminductor is given which, after a change of variablesσ ˙ = q, and recognizing ′ by ρ =ρ ˆ(q), so that (6) is rather a consequence of the that the incremental meminductance LM (q) :=ρ ˆ (σ ˙ ), clearly coincides with the correct equation of motion (17). RM Furthermore, it is easily seen that (30) constitutes the iKVL for the circuit. Thus, we have rendered the dynamics self-adjoint by considering the circuit characteristics from the perspective of flux conservation. L CM R 1 2 In general, the Lagrangian equations for circuits contain- σ σ ing charge-modulated meminductors, integrated charge- modulated memcapacitors, and their linear conventional counterparts, take the form Fig. 3. Circuit with a memristor and a memcapacitor. ¯ ¯ d ∂L ∂L The associated Lagrangian equations take the form i − i =0, (31) dt ∂σ˙ ∂σ L¯ L¯ D¯ ∗ ∗ d ∂ ∂ ∂ with as Lagrangian L¯(σ, σ˙ )= T¯ (σ ˙ ) − U¯(σ), where T¯ (σ ˙ ) i − i + i =0, i =1, 2, ¯ dt ∂σ˙ ∂σ ∂σ˙ and U(σ) are now representing the sums of the individual which provide the iKVL for loop 1: memory storage functions associated to the meminductors 1 1 2 1 and the memcapacitors in the circuit, respectively. Lσ¨ +ϕ ˆCM (σ − σ )+ϕ ˆRM (σ ˙ )=0, and the iKVL for loop 2: Naturally, the dual version of (31) reads 1 2 2 −ϕˆCM (σ − σ )+ Rσ˙ =0. d ∂L¯∗ ∂L¯∗ The KVL’s are obtained by differentiating the latter equa- j − j =0, (32) dt ∂ρ˙ ∂ρ tions with respect to time. with the co-Lagrangian L¯∗(ρ, ρ˙)= U¯ ∗(ρ ˙) − T¯(ρ). 5. ABSEMENT: A MECHANICAL ANALOGY OF 4.6 Including Memristors INTEGRATED CHARGE

Although memristors are dynamical elements they also In science and engineering, the ideas and concepts devel- behave as resistors. For that reason they cannot included oped in one branch of science and engineering are often using a Lagrangian or co-Lagrangian alone, and therefore transferred to other branches. One approach to transfer- we need to introduce, at the cost of a underlying varia- ring these ideas and concepts is by the use of analogies. tional principle, a second pair of state functions that play Classically, voltage is commonly considered as the electri- a similar role as the content and co-content functions for cal analogue of force, and current is the electrical analogue conventional resistors (see Subsection 3.2). of velocity. Consequently, flux can be considered as the analogue of momentum or impulse, and charge as the For a charge-modulated memristor, we propose the state analogue of position or displacement. function In mechanics, displacement and its various derivatives D¯(q) := ϕˆ(q)dq, (33) define an ordered hierarchy of meaningful concepts. The Z first derivative of displacement is velocity, the second which, when plotted in the ϕ-versus-q plane, represents derivative is acceleration, the third derivative is jerk, the area underneath the constitutive relation. the fourth derivative is jounce, etc.. On the other hand, For a flux-modulated memristor, we introduce the comple- recently also the integral of displacement over time is mentary function introduced to model the essential behavior of flow-based ∗ musical instruments, such as the hydraulophone discussed D¯ (ϕ) := qˆ(ϕ)dϕ, (34) in Mann et al. (2006). This quantity is called absement, Z a contraction of absence and displacement, which, in SI representing the area above the curve. Note that (33) and units, is measured in meter times seconds [ms]. One meter- (34) exhibit the units of energy times time (Joule-second second corresponds to being absent one meter from an [Js]), which is equivalent to action. origin or other reference point for a duration of one second. As an illustration, consider the circuit depicted in Fig. 3 Thus, integrated charge can be considered as the electrical consisting of a charge-modulated memristor RM , a conven- analogue of absement. tional linear inductor L, an integrated charge-modulated memcapacitor CM , and a conventional linear resistor R. 5.1 Absement Related to Memcapacitors From a Lagrangian perspective, it is necessary to consider a loop analysis. For that, we select as configuration vari- To gain some intuition of absement in the context of the ables the integrated loop charges σi, with i = 1, 2. The memory elements, consider the commonly used analogy of Lagrangian for the circuit reads an electrical capacitor and a bucket that can be filled with σ1−σ2 water. If current were the flow rate of water from a tap, ¯ 1 2 1 1 1 2 then charge would be the amount of water in the bucket. L(σ , σ , σ˙ )= L(σ ˙ ) − ϕˆCM (σ)dσ, 2 Z0 As the flow rate is proportional to how far open the tap is, whereas the memristive and resistive action is given by the rate of flow increases as the tap is opened up further. σ˙ 1 Now, if we consider the displacement of the tap handle ¯ 1 2 1 2 2 D(σ ˙ , σ˙ )= ϕˆRM (q)dq + R(σ ˙ ) . from its rest position, then the amount of water in the Z0 2 bucket is approximately proportional to the time-integral of the handle’s displacement, i.e., the handle’s absement, order the bring the sources to the iKVL or iKCL level, which is a measure of how ‘absent’ (how far and for how we need to consider their time-integrals. For example, long) the handle is from its closed position. Equivalently, suppose we add a voltage source to the first loop of since displacement is the time-derivative of absement, the the circuit of Fig. 3. Then we need to extract either position of the handle is the time-derivative of how much a term −σ1ϕe(t) from the Lagrangian or add a term water is accumulated in the bucket. −σ˙ 1ϕe(t) to the memristive action function, with t From an electrical perspective, the absement of a capacitor ϕe(t)= e(τ)dτ, is a measure for the amount of charge that is needed in a Z−∞ particular time interval to charge the capacitor to a certain where e denotes the source voltage. level. For a memcapacitor this process involves both a • The amount of integrated charge (electrical abse- charging and a shaping process of the capacitor and its ment) can be used as a measure of how much charge capacitance. The amount of absement may then be used and time is needed to write the memory of a mem- as a measure of how much charge and time is needed to ristor, meminductor, or memcapacitor. Although not write the memory of a memcapacitor. addressed in the present paper, a similar interpreta- tion should apply to integrated flux, which in the 5.2 Absement Related to Memristors and Meminductors mechanical domain corresponds to integrated mo- mentum measured in Newton times second-squared A mechanical example that exhibits meminductive phe- [Ns2] or kilogram-meter [kg·m]. nomena concerns the elementary problem of a heavy cable • Although we have restricted our attention to circuits that is deployed from a reel. As argued in Jeltsema and that can be modeled via either a loop or a node analy- D`oria-Cerezo (2010), under some reasonable assumptions sis, mixed formulations are of course also possible; see on the geometry, this system admits a nonlinear constitu- e.g., Jeltsema and Scherpen (2009) for a discussion in tive relationship between integrated angular momentum p the context of conventional circuits. (the mechanical analogue of integrated flux) and angular displacement (the mechanical analogue of charge) σ of the REFERENCES form ρ =ρ ˆ(θ), withρ ˙ = p. The amount of cable deployed from the reel equals Rθ, where R is the diameter of the Chua, L. (1971). Memristor, the missing circuit element. reel. The absement in this case could be a measure for the IEEE Trans. on Circuit Theory, 18(5), 507–519. amount of cable that needs to be deployed from the system Chua, L. (2009). Introduction to Nonlinear Network in a particular time interval to empty the reel. Theory. McGraw-Hill, NY. Di Ventra, M., Pershin, Y., and Chua., L. (2009). Circuit The simplest physical example of a mechanical memristor elements with memory: memristors, memcapacitors and is a tapered dashpot. This type of dashpot is a mechan- meminductors. Proc. of the IEEE, 97(10), 1717–1724. ical resistor whose resistance (i.e., its friction coefficient) Jeltsema, D. and D`oria-Cerezo, A. (2010). Mechanical depends on the relative displacement of its terminals. Al- memory elements: Modeling of systems with position- though a physical electrical passive two-terminal memris- dependent mass revisited. In Proc. 49th IEEE Confer- tive device was constructed only recently by Strukov et al. ence on Decision and Control (CDC), 3511–3516. (2008), a tapered dashpot was already brought forward Jeltsema, D. and Scherpen, J. (2009). 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Advances in minductors, memcapacitors, and their conventional linear Physics, 60(2), 145–227. counterparts, the Lagrangian equations can be obtained Pesce, C. (2003). The application of Lagrange equations from Hamilton’s principle of least action. Memristive and to mechanical systems with mass explicitly dependent linear resistive elements can be included via the intro- on position. Journal of Applied Mechanics, 70, 751–756. duction of an action function that plays a role similar to Ray, J. (1979). Lagrangians and systems they describe: the Rayleigh (co-)dissipation function or (co-)content. We how not to treat dissipation in quantum mechanics. Am. conclude the paper with the following remarks. J. Phys., 47(7), 626–629. Santilli, R. (1978). 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