On Geometrical Couplings of Dissipation and Curl Forces
aa r X i v : . [ m a t h - ph ] J a n On Geometrical Couplings of Dissipation and Curl Forces
January 29, 2021
O˘gul Esen Department of Mathematics,Gebze Technical University, 41400 Gebze, Kocaeli, Turkey.Partha Guha Department of Mathematics, Khalifa UniversityP.O. Box 127788, Zone -1 Abu Dhabi, UAE.Hasan G¨umral Department of Mathematics,Yeditepe University, 34755 Ata¸sehir, ˙Istanbul, Turkey.
Dedicated to Sir Michael Berry on his 80th birthday with great respect and admiration.
Abstract
In this paper, we present several geometric ways to incorporate gyroscopic and dissipativeforces to curl forces. We first present a proper metriplectic geometry. Then, using the Herglotzprinciple and generalized Euler-Lagrange equation, we propose a formulation of dissipativeradial curl forces. Finally, we extend our result to azimuthal curl force using Galley’s method,which leads to a natural formulation for Lagrangian and Hamiltonian dynamics of genericnon-conservative systems.
MSC classes:
PACS numbers:
Keywords:
Curl forces; Herglotz principle; Contact Geometry; Damped curl forces; Gal-ley’s method; Gyroscopic forces.
Contents E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Hamiltonian Realization of Curl Forces 3
In a series of papers [4, 6, 5], Berry and Shukla have introduced Newtonian dynamics driven byforces, depending only on position, whose curl is non-zero. Assuming unit mass for convenience,we record the dynamics generated by so called a curl force as¨ r = F ( r ) , ∇ × F ( r ) = 0 . (1.1)Note that, the motion governed by a curl force is non-conservative, that is the work done by F depends on the path. But, apart from velocity-dependent frictional forces, curl forces are notdissipative. If there exists no attractors, it is immediate to realize that, a flow generated by acurl force preserve the volume in the position-velocity phase space ( r , v ). The theory of curlforces has found profound applications in optics, laser physics and anisotropic Kepler problem[6, 10, 2, 59, 25, 11].Independently, the concept of curl force has appeared in some works [57, 42, 43] of Link¨opingSchool by Rauch-Woiciechowski, Marciniak and Lundmark. Practically, Link¨oping School studiedthe most general version of the construction of curl forces.Curl forces neither can be derived from a scalar potential nor, in general, can be examined in2he realm of Hamiltonian Mechanics. Nevertheless, a large class of such non-conservative forces(though not all) can be generated from Hamiltonians of a special type, in which kinetic energyis an anisotropic quadratic function of momentum. Berry and Shukla [5] listed various permittedclasses of curl forces which are kinetically quadratic and anisotropic. Our investigation in thiswork starts precisely at this point.In the present work, our goal is to extend linear (Hamiltonian) curl force formalism to a generalizedtheory by incorporating dissipative and gyroscopic forces. In order to couple such dissipative termsto the system (1.1), we employ pure geometric/algebraic techniques available in the literature, suchas double bracket dissipations, contact Hamiltonian formalism and Galley’s approach. To fix thenotation and for the sake of the completeness of this work, in Section 2, we shall recall some basicson Hamiltonian formulations of curl forces. It is Section 3 where the first trial through the couplingof dissipative terms to curl forces will be presented by means of a proper metriplectic bracket. Tohave this geometrically, we shall realize the canonical Hamiltonian formalism as a coadjoint flowthen apply the double bracket dissipation approach to this realization. In this section, we shallalso address symplectic two-forms involving magnetic terms. This leads to couple gyroscopic forcesto the curl force model. In Section 4, we shall provide contact Hamiltonian realization of the curlforce model and, particularly, study dissipative curl forces using the generalized Euler-Lagrangeequations formulated by Gustav Herglotz. This will lead us to analyse the variational aspects,as well as Lagrangian formulation, of the dissipative terms. An alternative method to employdissipative Lagrangian dynamics is Galley’s approach. In Section 5, after briefly summarizingGalley’s approach for dissipative Lagrangian-Hamiltonian dynamics. We shall incorporate thisapproach to the case of curl force models. We conclude this paper with a modest outlook. Let P be a Poisson manifold equipped with a Poisson bracket {• , •} , [37, 62] a skew-symmetricalgebra on the space of real valued functions satisfying the Jacobi and the Leibniz identities. For agiven Hamiltonian function H , the dynamcs of an observable F are determined through Hamilton’sequations ˙ F = { F, H } = X H ( F ) , (2.1)where X H is the Hamiltonian vector field. Skew-symmetry of Poisson bracket reads that Hamilto-nian function is a conserved quantity. In some classical models, Hamiltonian function is consideredto be the total mechanical energy, so that the equality ˙ H = 0 states the conservation of energy.3his manifests the reversible character of Hamiltonian dynamics.Alternatively, a Poisson structure can be introduced by means of a bivector field. To have thisformulation, start with a Poisson bracket, and define a bivector field Λ according to the identityΛ( dF, dH ) := { F, H } (2.2)for all real valued functions F and H on P . Here, dF stands for the exterior (de-Rham) derivativeof F . Accordingly, a Poisson manifold can be defined by a tuple ( P , Λ). In this case, the Jacobiidentity is manifested in terms of the Schouten-Nijenhuis bracket [3] that is,[Λ , Λ] = 0 . (2.3)Inversely, for a bivector field commuting with itself under the Schouten-Nijenhuis algebra, one candefine a bracket of functions {• , •} satisfying the requirements of being a Poisson bracket. This issimply achieved by referring to the identity (2.2) but from the reverse order.Consider a Poisson manifold P . The characteristic distribution, that is the image space of allHamiltonian vector fields is integrable. This reads (symplectic) foliation of P , [63]. In this foliation,on each leaf, the Poisson bracket turns out to be non-degenerate and determines a symplectic two-form. This latter argument manifests that every symplectic manifold is a Poisson manifold. Inthe present work, we are interested in the following particular instance. Assume a 2-dimensionalmanifold Q equipped with a coordinate chart r = ( x, y ). Being cotangent bundle, P = T ∗ Q is asymplectic manifold hence a 4-dimensional Poisson manifold [24]. Here, by taking the momenta p = ( p x , p y ), the induced Poisson bracket, for two functions F and H , is computed to be { F, H } ( q , p ) = ∂F∂x ∂H∂p x + ∂F∂y ∂H∂p y − ∂F∂p x ∂H∂x − ∂F∂p y ∂H∂y . (2.4)Recalling from [4], in the present work, we are interested in Hamiltonian functions in the form H ( q , p ) = 12 ( p x − p y ) + U ( q ) . (2.5)The Hamiltonian function (2.5) generates a Newtonian dynamics induced by a curl force. Beforeexamine this, we encounter two different types of curl forces according to the functional structureof the potential U ( q ). Then a direct calculation proves that the force field is curl free if the mixedsecond partial derivative of the potential function is non vanishing.4 .2 Radial Curl Forces We first consider the Hamiltonian function H in (2.5) where the potential energy U is a functionof a saddle with equal principal curvatures [21]. That is, we consider U as a function of ξ =(1 / x − y ). Let us record this particular instance for future reference as follows H = 12 ( p x − p y ) + U (cid:0)
12 ( x − y ) (cid:1) . (2.6)The Hamilton’s equation can be computed through (2.1). Accordingly, by properly coupling thedifferential equations one can easily obtain Newton’s equations as¨ x = ˙ p x = − xU ′ (cid:0)
12 ( x − y ) (cid:1) = F x , ¨ y = − ˙ p y = − yU ′ (cid:0)
12 ( x − y ) (cid:1) = F y . (2.7)In this case, the curl of the generating force field F = ( F x , F y ) is computed to be ∇ × F = − xyU ′′ (cid:0)
12 ( x − y ) (cid:1) k (2.8)and, evidently, it is non-vanishing as long as the second derivative U ′′ ( ξ ) is not zero. To explorethe nature of the curl force, we employ the polar coordinates x = r cos θ and y = r sin θ withcorresponding unit directions given by ( r , φ ). In this realization, the force field has the followingappearance F = − rU ′ ( 12 r cos 2 φ ) r (2.9)showing that it is radially directed. This formulation of the force field manifests that the angularmomentum is a constant of motion. The conservation of the angular momentum reads that there isno torque about the origin for the radial curl forces. So that, there are two conserved quantities, theenergy and the angular momentum. Hence, we can argue that the radial curl forces are integrable.It is known for almost a century that the equilibrium position of the particle becomes stable if thesurface rotates around the vertical axis sufficiently fast. This can be demonstrated in the followingsense. Consider the saddle force field (cid:0) F x = x, F y = − y (cid:1) , make it time-dependent by rotatingeach vector counterclockwise with an angular velocity, which describes the motion of the unitpoint mass in this force field [22]. Earnshaw’s theorem states that trapping of a charged particlecannot be achieved with a static potential, because any static potential, which fulfills Laplace’slaw, lacks potential minimal. Microscopic particle traps usually use time dependence to workround Earnshaw’s theorem. The trapping of the particle can be achieved by rotating the saddlepotential around the axis with a suitable angular frequency, this was idea of Wolfgang Paul [55].Paul’s idea was to stabilize the saddle by “vibrating” the electrostatic field, by analogy with the5o called Stephenson-Kapitsa [60, 29] pendulum. Nobel Prize in physics was awarded to W. Paulfor his invention of the trap. A mathematically formal analysis of stable trapping by using theanalysis of the Mathieu equation allows to formulate the limitations of the effective pseudopotentialdescription. The Mathieu stability limit can be reached for a few charges in the trap. RecentlyKirillov and Levi [33, 32] demonstrated that rotating saddle potentials exhibit precessional motiondue to a hidden Coriolis-like force. Dynamics driven by curl forces has been studied by engineersand applied mathematicians, it was not so popular among physicists, although it has very beautifulapplications in central force and nonlinear dynamics [6, 16, 19]. As a second particular case of the Hamiltonian function (2.5), we take that the potential is afunction of ξ = xy . In this case, we write the Hamiltonian function as H = 12 ( p x − p y ) + U ( xy ) . (2.10)In this case, Newton’s equations turn out to be¨ x = − yU ′ ( xy ) , ¨ y = xU ′ ( xy ) . (2.11)Such forces are called azimuthal curve forces. To justify this labelling it is enough to see that, inthe polar coordinates, the force field admits only angular unit vector F ( r ) = rU ′ ( 12 r sin(2 φ )) φ . (2.12)In general, the energy, that is the Hamiltonian function (2.10), is the only conserved quantity. Sothat, this class of azimuthal forces is non-integrable. This is connected to optical vortex curl forcefor U ( xy ) = xy . This case possesses the rotational symmetry. One must note that for variousnon-central forces F ( r ) = F φ ( r ) φ do not fall into this Hamiltonian class for general F φ ( r ). Remark 1
There exists another type of curl forces which are directed in one direction in theCartesian coordinates, but depending on both of the coordinates. These are known as the shearcurl forces. The special classes of shear forces can be obtained from the Hamiltonians H in form (2.5) where the potential is U = U ( x ± y ) . This force fields are out of our scope in this work. .4 Kapitsa-Merkin Non-conservative Positional Forces The curl forces picture is related to the theory of Kapitsa-Merkin non-conservative positionalforces. The linearised dynamics of a rotating shaft formulated by Kapitsa [28] is given by¨ x + ay + bx = 0 , ¨ y − ax + by = 0 . (2.13)The corresponding characteristic equation shows that the addition of a non-zero non-conservativecurl force ( i.e. a = 0) to a stable system with a stable potential energy makes it unstable. Thisis connected to Merkin’s result [47, 48], which states that the introduction of non-conservativelinear forces into a system with a stable potential and with equal frequencies destroys the stabilityregardless of the form of non-linear terms. It is worth mentioning that the positional force, i.e.the terms ay and − ax are proportional to ω , where ω is the rotation rate of the shaft. Let uswrite (2.13) as ¨ x ¨ y ! + b b ! xy ! + a − a ! xy ! = ! . (2.14)where the potential part, say B , is a diagonal matrix with equal eigenvalues b whereas non-conservative part, say A , is an skew- symmetric matrix. It is easy to see from the correspondingcharacteristic equation det (cid:0) ( λ + b ) I + A (cid:1) = 0 that λ + b is imaginary, thus we say that λ isunstable.It is possible to recast the dynamical equation in (2.13) as a Hamiltonian curl force system asfollows. Consider the following Hamiltonian function H = 12 ( p x − p y ) + 12 b ( x − y ) + axy. (2.15)See that, since the mixed second partial derivative of the potential function does not vanish unless a = 0, the Hamilton’s equation generated by H in (2.15) determines a curl force system. Further,a direct calculation proves that the Hamilton’s equation (2.1) reads precisely (2.13). A saddlepotential is associated with the equation appears in the Euler-Lagrange formulation also. Afteran (inverse) Legendre transformation, it is straight forward to check that the Lagrangian L = 12 ( ˙ x − ˙ y ) − b ( x − y ) − axy (2.16)yields the model in (2.13) as well. A symmetric saddle surface can be described by ˜ U = ( b/ x − y ), where b is a geometrical parameter that specifies the curvature of the saddle. An ion trappotential is a rotating saddle surface on which a ball can be trapped. The ponderomotive potentialof an ion trap is that of a saddle in 2 D but the time evolution of potential is one that flaps up and7own. The potential of the system is U ( x, y, t ) = 12 ( x − y ) cos(2 ωt ) − xy sin(2 ωt ) , (2.17)where ω is the angular drive frequency spinning saddle. The system acquires Coriolis and centrifu-gal forces when written in rotating frame and exhibits many interesting mechanics and geometry[33, 32]. Linear algebraic dual K ∗ of a Lie algebra admits (Lie-)Poisson structure, [26, 40, 41, 46]. For twofunctions F and H , the Lie-Poisson bracket is defined to be { F, H } ( z ) = D z , (cid:20) δFδ z , δHδ z (cid:21) E (3.1)where δF/δ z is the gradient of F . Here, the bracket on the right hand sider is the Lie algebrabracket on K . Note that, we assume the reflexivity condition K ∗∗ = K . The dynamics is thencomputed to be˙ F = { F, H } ( z ) = D z , (cid:20) δFδ z , δHδ z (cid:21) E = D z , − ad δH/δ z δFδ z E = D ad ∗ δH/δ z z , δFδ z E . (3.2)Here, ad x x ′ := [ x , x ′ ] for all x and x ′ in K is the (left) adjoint action of the Lie algebra K on itselfwhereas ad ∗ is the (left) coadjoint action of the Lie algebra K on the dual space K ∗ . Notice that ad ∗ x is defined to be minus of the linear algebraic dual of ad x . Then, we obtain the equation ofmotion governed by a Hamiltonian function H as˙ z − ad ∗ δH/δ z z = 0 . (3.3) Coordinate realizations.
Consider a Poisson manifold P with a local chart ( z i ). Then thePoisson bivector Λ = [Λ ij ] determines Poisson bracket as { F, H } = Λ ij ∂F∂z i ∂H∂z i , (3.4)where the summation convention is assumed over the repeated indices.8ssume a finite dimensional Lie algebra K admitting a basis { k i } with structure constants C kij satisfying [ k i , k j ] = C kij k k , (3.5)On the dual basis { k i } of K ∗ , we denote an element by z = z i k i . In this setting, we compute the(Lie-)Poisson bivector as Λ ij = C mij z m , whereas the Lie-Poisson bracket (3.1) as { F, H } = C kij z k ∂F∂z i ∂H∂z j . (3.6) Metriplectic bracket.
Referring to the (Lie-)Poisson bivector field, define a symmetric bracket,literarily called double bracket, for two functions F and H as( F, H ) ( D ) = G il ∂F∂z i ∂H∂z l = X j Λ ij Λ lj ∂F∂z i ∂H∂z l = X j C rij C slj z r z s ∂F∂z i ∂H∂z l , (3.7)see [9, 52]. Now we define a metriplectic bracket [30, 31, 50, 51] on K ∗ as the sum of Lie-Poissonbracket and the double bracket (3.7). Note that, the metriplectic bracket (3.8) is an example ofa Leibniz bracket [54]. In this case, the dynamics of an observable, generated by a Hamiltonianfunction H and an entropy like function S , is˙ F = [ | F, H | ] ( D ) = { F, H } + a ( F, S ) ( D ) , (3.8)for a real number a . Metriplectic systems are particular instances of GENERIC (an acronym forGeneral Equation for Non-Equilibrium Reversible-Irreversible Coupling), [17, 18, 56]. We computethe equation of motion as ˙ z j − C mij z m ∂H∂z i = a X i C rji C nmi z r z n ∂S∂z m (3.9)where on the left hand side we have the reversible Lie-Poisson dynamics whereas the dissipativeterm is located at the right hand side.It is not straight forward to derive the dissipative azimuthal curl foces due to mixed potential term U ( xy ). We can not use straight away the Herglotz principle and the generalized Euler-Lagrangeequations to derive dissipative azimuthal curl forces due to non-separable potential term. Weovercome this difficulty by using Galley’s formalism to study this generalized system and proposean alternative derivation of dissipative curl force.Note that the addition of a dissipative force destabilizes the system regardless of its stability underthe action of gyroscopic forces. This extension overlaps with the program of Krechetnikov and9arsden [34, 35]. They proposed a rigorous mathematical framework for the motion of a mechan-ical system under the influence of various types of forces, namely, gyroscopic forces, dissipativeforces, potential forces and nonconservative positional forces. We use Galley’s formalism to studythis generalized system and propose an alternative derivation of dissipative curl force. Let V be an n − dimensional vector space equipped with a basis { e i } and its dual V ∗ with thedual basis { e i } . Then consider a (2 n + 1) − dimensional (Heisenberg) Lie algebra g equipped witha basis { e i , e j , f } where i and j run from 1 to n . Here, the bracket operations are defined to be[ e i , e j ] = h e i , e j i f = δ ij f (3.10)and the rest is zero. Here, the second term in (3.10) is simply the dualization between V ∗ and V . See that, we can argue the coefficients of this bracket definition as the canonical symplecticform on the symplectic space V ⊕ V ∗ . On the dual space g ∗ we consider the respected dual basis { e i , e j , l } . We denote a Lie algebra element ξ and a dual element α as ξ = ξ i e i + ξ i e i + ξ f , µ = µ i e i + µ i e i + µ l , (3.11)respectively. Obeying the definition presented in (3.6), the Lie-Poisson bracket of two functions F and H on g ∗ is computed to be { F, H } ( µ ) = µ (cid:18) ∂F∂µ i ∂H∂µ i − ∂H∂µ i ∂F∂µ i (cid:19) . (3.12)Referring to this calculation, we write the Lie-Poisson dynamics (3.3) generated by a Hamiltonianfunction H as follows ˙ µ i − µ ∂H∂µ i = 0 , ˙ µ i + µ ∂H∂µ i = 0 , ˙ µ = 0 . (3.13)Here, the last relation gives that µ is a constant. In particular, if we choose µ i = q i , µ i = p i , and µ = 0 in the equation (3.13) of motion then we arrive at the canonical Hamilton’s equations in itsvery classical form ˙ q i = ∂H∂p i , ˙ p i = − ∂H∂q i (3.14)but as a coadjoint flow. This naive realization permits us to define a dissipative term to theclassical reversible Hamiltonian dynamics by means of a double bracket given in (3.7). Let us10epict this geometry. Double bracket dissipation.
Referring to the structure constant of the Lie algebra g given in(3.10), and in the light of the double bracket definition (3.7), we compute the following symmetricbracket ( F, S ) ( D ) ( µ ) = µ (cid:18) µδ ij ∂S∂µ i ∂F∂µ j + µδ ij ∂S∂µ i ∂F∂µ j (cid:19) . (3.15)We add the Lie-Poisson bracket in (3.12) and the symmetric bracket in (3.15) and arrive at ametriplectic bracket. The metriplectic dynamics generated by a Hamiltonian function H and anentropy like function S is˙ µ i = µ ∂H∂µ i + µδ ij µ ∂S∂µ j , ˙ µ i = − µ ∂H∂µ i + µµδ ij ∂ S ∂µ j , ˙ µ = 0 . (3.16)We wish to record here two interesting particular instances of the metriplectic dynamics (3.16).For this, once more, we choose µ i = q i , µ i = p i , and µ = 1. (1) Let us take the Hamiltonian function H = (1 / δ ij p i p j + V ( q ) to be the total energy of thesystem and S = S ( q ) then the system (3.16) reduces to a second order differential equation witha dissipative term ¨ q i − ∂ S∂q i ∂q j ˙ q j − δ ij ∂V∂q j = 0 . (3.17)We cite [49] for a more elegant geometrization of the second order ODE (3.17) in terms of theGENERIC framework. Let us now consider that the dimension n = 2 and consider the Hamiltonianfunction (2.6) with S = − γ ( t ) δ ij q i q j then the dynamical system (3.16) reduces to˙ x = p x − γ ( t ) x, ˙ y = − p y − γ ( t ) y, ˙ p x = − xU ′ , ˙ p y = yU ′ (3.18)where we consider that q = ( q , q ) = ( x, y ). The corresponding Newton’s equation is computedto be ¨ x + γ ( t ) ˙ x + xU ′ (cid:0)
12 ( x − x ) (cid:1) = 0 , ¨ y + γ ( t ) ˙ y + yU ′ (cid:0)
12 ( x − x ) (cid:1) = 0 . (3.19)This pair yields dissipative radial curl forces and determine the Bateman pair of equation. (2) As another application of the dissipative system (3.16), we consider a Hamiltonian function H and choose S = (1 / aδ ij p i p j for a scalar a , then a fairly straight-forward calculation gives that(3.16) becomes ˙ q i = ∂H∂p i , ˙ p i = ap i − ∂H∂q i . (3.20)11his is the conformal Hamiltonian dynamics as described in [45]. See also [23]. Consider thevector field X generating the dynamics (3.20) and observe that, if the canonical symplectic two-form Ω = dq i ∧ dp i , L X Ω = d (cid:0) dH + ap i dq i (cid:1) = adq i ∧ dp i = a Ω (3.21)where L denotes the Lie derivative. This syas that X preserves the symplectic two-form up to someconformal factor a . Particularly, for the Hamiltonian function (2.6) in dimension 2 the dynamicalsystem reduces to conformal curl force system˙ x = p x , ˙ y = − p y , ˙ p x = − γ ( t ) p x − xU ′ , ˙ p y = − γ ( t ) p y + yU ′ (3.22)where we take a as − γ ( t ). A direct calculation shows that Newtonian realization of this system isexactly the one in (3.19). Thomson and Tait classify non-potential forces into three categories [61]. In n − dimensions, con-sider the power F · ˙ q of a given system. In the classification, a force F is called gyroscopic forceif the power vanishes identically. A force field is said to be dissipative if the power is non-positivewhereas it is accelerating if the power is non-negative.If, further, gyroscopic force is linear then it admits a skew-symmetric coefficient functions s ij sothat the components of the force field becomes F i = s ij ˙ q j . On the contrary, a linear dissipativeforce admits a symmetric structure g ij ˙ q j , where the coefficient functions g ij are symmetric. In thiscase, one can derive the force field by means of a Rayleigh dissipative function R = (1 / g ij ˙ q i ˙ q j . Ifa Rayleigh function exists the force field is computed to be the gradient of the Rayleigh functionwith respect to the velocity variables that is ∇ ˙ q R . Coupling with a Gyroscopic Term.
It is possible to capture gyroscopic forces into a Hamilto-nian formalism in a proper Poisson geometry [46]. See also [7]. To have this, one needs to introducea non-standard Poisson bracket { F, H } gyro ( q, p ) = ∂F∂q i ∂H∂p i − ∂F∂p i ∂H∂q i − s ij ∂F∂p i ∂H∂p j , (3.23)where the gyroscopic skew-symmetric term involving γ ij appears in the “magnetic extension” ofthe bracket. In this case, the dynamics governed by a Hamiltonian function reads˙ q i = ∂H∂p i , ˙ p i = − ∂H∂q i − s ij ∂H∂p j . (3.24)12or a Hamiltonian curl force dynamics, similar to the one presented in (2.5), one can arrive atforce field whose coefficient functions are computed to be linear curl force F i = − Ω ij q j for a skew-symmetric matrix Ω ij = − Ω ji . All though, due to skew-symmetric matrix, this looks similar togyroscopic force theory, it is not the same. In this case, instead of velocities, the force depends onthe positions. These forces, particularly, are called pseudo gyroscopic of radial corrections coinedby Ziegler in 1953.It is interesting to examine the Hamiltonian function (2.5) in the realm of the Poisson bracket(3.24). For this, we once more refer to the local coordinates ( x, y ) on 2 − dimensional base manifold Q . In this case, we take the magnetic term s = s in the Poisson bracket (3.23). Then theHamilton’s equation (3.24) generated by the Hamiltonian function (2.6) turns out to be˙ x = p x , ˙ y = − p y , ˙ p x = − xU ′ (cid:0)
12 ( x − y ) (cid:1) + sp y , ˙ p y = yU ′ (cid:0)
12 ( x − y ) − sp x . (3.25)By collecting all these first order equation in the form of Newton’s equation, we conclude that¨ x + s ˙ y + xU ′ (cid:0)
12 ( x − y ) (cid:1) = 0 , ¨ y − s ˙ x + yU ′ (cid:0)
12 ( x − y ) = 0 . (3.26)In this case, the dissipative terms in the system (3.26) is given by a skew-symmetric form as amanifestation of the magnetic extension of the Poisson bracket. We can conclude that Hamiltonianof a curl force remains Hamiltonian if the linear gyroscopic force is added. Coupling with a Dissipative Term.
The system (3.26) includes a curl force and a gyroscopicterm. As a further step, we now add a dissipative term to this system. As manifested in theprevious section, one can achieve this by employing metriplectic bracket. So we recall the doublebracket in (3.15) on a coadjoint orbit and consider only the terms involving partial derivatives withrespect to momenta. This time we couple the symmetric bracket with the Poisson bracket (3.23)involving a magnetic term. Accordingly, we introduce the following metriplectic bracket[ | F, H | ] mb = { F, H } gyro + c ( F, H )= ∂F∂q i ∂H∂p i − ∂F∂p i ∂H∂q i − s ij ∂F∂p i ∂H∂p j + c ij ∂F∂p i ∂H∂p j , (3.27)where s ij is skew-symmetric whereas c ij is symmetric. We refer [12] for various couplings of Poissonand symmetric brackets.For two dimensional curl force system, we recall the Hamiltonian function in (2.15). We denotethe coefficients of the skew-symmetric quantity by s = s and the coefficients of the symmetric13uantity by c = c = c while the rest is zero. Then metriplectic equation˙ z = [ | z , H | ] mb = { z , H } gyro + c ( z , H ) . (3.28)After adding dissipative forces, by means of a symmetric bracket, to the setting, we get the mostgeneral set of system of equations. Explicitly, we compute the metriplectic equations as˙ x = p x , ˙ y = − p y , ˙ p x = − bx − ay + sp y + cp x , ˙ p y = by − ax + sp x − cp y (3.29)whereas, in the form of Newton’s equations, we have that¨ x + s ˙ y − c ˙ x + bx + ay = 0 , ¨ y + s ˙ x + c ˙ y + by − ax = 0 . (3.30)This pair of equations involve the motion of mechanical system under influences of various forceswhich is closely connected to the work of Krechetnikov and Marsden [35], where they defineand study a notion of dissipation instability in a system ODEs. Krechetnikov and Marsden [35]proposed a rigorous mathematical framework for the motion of a mechanical system under theinfluence of various types of forces, namely, gyroscopic forces, dissipative forces, potential forcesand nonconservative positional forces. Two famous physical examples follow from reductions ofLagrangian top ( b = 0) ¨ x + s ˙ y − c ˙ x + ay = 0 , ¨ y + s ˙ x + c ˙ y − ax = 0 . (3.31)Another case is the one where s = 0. This corresponds dynamics without gyroscopic force term.For this case, the particular instance a = b = 1 can be written as Euler-Lagrange equations bymeans of the following Lagrangian function L = ˙ x ˙ y − xy + 12 ( x − y ) + 12 c ( ˙ xy − x ˙ y ) . (3.32)The case s = 0 and c = 0 manifests dynamics with no gyroscopic or dissipative force. So thiscase reduces to the Kapitsa model in (2.13). One must note that the addition of non-conservativepositional forces (or curl forces) to a system with a stable potential energy makes it unstable. Theorigin of this force lies in the friction between the rotating shaft and hydrodynamic media.In 1879 Thomson and Tait [61] showed that a statically unstable conservative system which hasbeen stabilized by gyroscopic forces could be destabilized again by the introduction of small damp-ing forces. More generally, they consider conservative and nonconservative linear forces. Note thatthe instability pops up for a tiny bit of damping is added to the structure. This destabilizationparadox is related to the Whitney umbrella singularity. See [15, 36, 53, 58] for some works onWhitney umbrella. 14 Radial Curl Forces in Contact Geometry
The Generalized Variational Principle, proposed by Herglotz in 1930, generalizes the classical vari-ational principle by defining the functional, whose extrema are sought, by a differential equation.The generalized variational principle gives a variational description of nonconservative processes.This method provides a link between the mathematical structure of control and optimal controltheories and contact transformations. The contact transformations, which can always be derivedfrom the generalized variational principle, have found applications in thermodynamics. We useHerglotz principle and the generalized Euler-Lagrange equations to derive dissipative radial curlforces.
Let M be a (2 n + 1) − dimensional manifold with a contact one-form σ ∈ Λ ( M ) satisfying dσ n ∧ σ = 0. A contact form determines a contact structure which, locally is the kernel of thecontact form σ , [1, 8, 44, 38]. There is a distinguished (Reeb) vector field satisfying ι R σ σ = 1 , ι R σ dσ = 0 . (4.1)For a Hamiltonian function H , the Hamiltonian vector field X H is the one defined to be ι X H σ = − H, ι X H dσ = dH − ( ι R σ dH ) σ. (4.2)Being a non-vanishing top-form we can consider dσ n ∧ σ as a volume form on M . It is important tonote that the Hamiltonian motion does not preserve the volume form. In this realization, contactPoisson (or Lagrange) bracket of two smooth functions on M is defined by { F, H } = ι [ X F ,X H ] σ, (4.3)where X F and X H are Hamiltonian vectors fields determined through (4.2). Darboux’ Coordinates.
There are Darboux’ coordinates on M given by ( q i , p i , z ). In thisrealization, the contact one-form is σ = dz − p i dq i and the Reeb vector field is R σ = ∂/∂z . For aHamiltonian function H , the Hamiltonian vector field, determined in (4.2), is computed to be X H = ∂H∂p i ∂∂q i − (cid:0) ∂H∂q i + ∂H∂z p i (cid:1) ∂∂p i + ( p i ∂H∂p i − H ) ∂∂z , (4.4)15hereas the contact Poisson bracket (4.3) is { F, H } = ∂F∂q i ∂H∂p i − ∂F∂p i ∂H∂q i + (cid:0) F − p i ∂F∂p i (cid:1) ∂H∂z − (cid:0) H − p i ∂H∂p i (cid:1) ∂F∂z . (4.5)We obtain the contact Hamiltonian system˙ q i = ∂H∂p i , ˙ p i = − ∂H∂q i − p i ∂H∂z , ˙ z = p i ∂H∂p i − H. (4.6) The generalized variational principle, proposed by Herglotz, defines the functional whose extremaare obtained by a differential equation rather than by an integral. The Herglotz principle yieldsa variational description of nonconservative as well as conservative processes involving one inde-pendent variable. The Herglotz principle is defined by the functional z ( q ; τ ) through a differentialequation of the form [20, 27, 39] ˙ z = L ( t, q i , ˙ q i , z ) , ≤ t ≤ τ. (4.7)Here q belongs to the space of C curves q defined on [0 , τ ] satisfying the boundary condition q (0) = q , q ( τ ) = q τ . So z is a solution of the Cauchy problem of Herglotz action with z (0) = z .Let L be a Lagrangian function defined on T Q × R with coordintes ( q i , ˙ q i , z ). Herglotz showed thatthe value of this functional attains its extremum if q ( t ) solution of the generalized Euler-Lagrangeequations (the dissipative Lagrange system) ∂L∂q i − ddt (cid:16) ∂L∂ ˙ q i (cid:17) + ∂L∂z ∂L∂ ˙ q i = 0 , (4.8)for all t ∈ [0 , τ ] and z is a solution of the Cauchy problem (4.7) which depends on q . It isimportant to notice that (4.8) represents a family of differential equations since for each function q ( t ) a different differential equation arises, hence z ( t ) dependes on q ( t ). Without the dependenceof z , this problem reduces to a classical calculus of variations problem. If the functional z definedin (4.7) is invariant with respect to translation in time, then the quantity I = exp (cid:16) − Z t ∂L∂z dθ (cid:17)(cid:16) L ( x, ˙ x, z ) − ∂L∂ ˙ x k ˙ x k (cid:17) (4.9)is conserved on solutions of the generalized Euler-Lagrange equations for regular Lagrangians. The Legendre Transformation.
For a regular Lagrangian function L we define the fiber deriva-16ive is F L : T Q × R −→ T ∗ Q × R , ( q i , ˙ q i , z ) ( q i , ∂L∂ ˙ q j , z ) (4.10)A direct calculation shows that the Legendre transformation (4.10) maps the generalized Euler-Lagrange equations in (4.8) to contact Hamiltonian dynamics (4.6).The flow of a contact Hamiltonian system preserves the contact structure, but it does not preservethe Hamiltonian. Instead we obtain dHdt = − H ∂H∂z .
It can be readily checked, in the light of the integral invariant (4.9), that I ( x, p, z ) = H ( x ( t ) , p ( t ) , z ( t )) exp (cid:16) Z t ∂H∂z dθ (cid:17) (4.11)is constant along the flow of X cH defined by (4.6) with the autonomous contact Hamiltonian H ( x, p, z ). Consider two-dimensional anisotropic damped system with the potential energy ( x − x ) andchanging-sign damping coefficient. The contact Hamiltonian is recasted as H = 12 ( p x − p y ) + U (cid:0)
12 ( x − y ) (cid:1) + γ ( t ) z (4.12)and leads to the equations ˙ x = p x , ˙ y = − p y , ˙ p x = − x U ′ (cid:0)
12 ( x − x ) (cid:1) − γ ( t ) p x , , ˙ p y = x U ′ (cid:0)
12 ( x − x ) (cid:1) + γ ( t ) p y , ˙ z = 12 ( p x − p y ) − U (cid:0)
12 ( x − y ) (cid:1) − γ ( t ) z. (4.13)Notice that the first two lines of equations are precisely the dissipative dynamics in (3.22). Sothat, they are equal to the Bateman pair of equations in (3.19). In order to investigate the lastequation in (4.13), we first recall the inverse Legendre transformation F H : T ∗ Q × R −→ T Q × R , ( q i , p i , z ) ( q i , ∂H∂p i , z ) . (4.14)17vidently, the Hamiltonian function (4.12) has indeed a regular (invertible) Legendre transforma-tion. The first set of equations in (4.13) is the realization of the inverse Legendre trasnformation.Further, it is immediate to see that by imposing the last equation in (4.13) as a differential Herglotzprinciple for the Lagrangian function L ( x, y, ˙ x, ˙ y ) = 12 ( ˙ x − ˙ y ) − U (cid:0)
12 ( x − y ) (cid:1) − γ ( t ) z. (4.15)Then, in the light of the inverse Legendre transformation ˙ x = p x and ˙ y = − p y , the first twolines of the system (4.13) are the generalized (dissipative) Euler-Lagrange equations in (4.8). Thisobservation manifests the variational aspect of the Bateman’s pair (3.19) due the radial curl forceswith a dissipation. Kapitsa-Merkin Model with Dissipation.
Recall the Hamiltonian realization (2.15) intro-duced for the Kapitsa model in (2.13). To generalize this discussion to the contact Hamiltonianframework, introduce the following contact Hamiltonian function H = 12 ( p x − p y ) + 12 b ( x − y ) + axy + γ ( t ) z. (4.16)A direct computation gives that the first two sets of equations in the contact Hamilton’s equations(4.6) can be written in the form of Newton’s equation¨ x + γ ( t ) ˙ x + bx + ay = 0 , ¨ y + γ ( t ) ˙ y + by − ax = 0 . (4.17)The last equation in (4.6) becomes the differential equation˙ z = 12 ( ˙ x − ˙ y ) − b ( x − y ) − axy − γ ( t ) z (4.18)determining the Herglotz principle (4.7). Hamilton’s action principle is not suitable for non-conservative systems. Having observed thatGalley [13, 14] proposed a method based on a new variational principle. This required allowingone to break the time-symmetry manifest in the action. The formalism given in [13] correspondsto a variational principle specified by initial data contrary to Hamilton’s that fixes configurationof system at initial and final times. We first depict this geometry then employ this approach tocurl force theory. 18 .1 Galley’s Formalism
Galley has developed a consistent formulation of Hamilton’s principle that is compatible with initialvalue problems [13, 14]. In the framework of Galley, the degrees of freedom are formally doubledto facilitate the nonconservativity, so that q and q are decoupled from each other. Accordingly,one defines an action, a functional of the coordinates q and q , as S ( q , q ) = Z t f t i dt L ( q , q , ˙q , ˙q ) , (5.1)where the Lagrangian is determined in the form of L ( q , q , ˙q , ˙q ) = L ( q , ˙q ) − L ( q , ˙q ) + K ( q , q , ˙q , ˙q ) , (5.2)where K ( q , q , ˙q , ˙q ) encodes nonconservative nature of a dynamical system. It is immediateto see that K must be skew-symmetric with respect to the relabelling q ↔ q . For K = 0 thetwo paths q ( t ) and q ( t ) get coupled with each other and L describes a non conservative system.The Euler-Lagrange equations for L are not necessarily physical until we take the physical limit(PL) wherein the histories are identified, i.e., q = q = q , after completing all variations andderivatives. A more convenient parametrization of the coordinates is exhibited in [13] as q − = q − q , q + = q + q . (5.3)It is evident that, after taking the physical limit PL, one has that q − → , q + → q = q = q . (5.4)This leads to the dissipative Euler–Lagrange equations ddt (cid:18) ∂L∂ ˙q + h ∂K∂ ˙q − i P L (cid:19) = ∂L∂ q + h ∂K∂ q − i P L . (5.5)Consider a regular Lagrangian function L = L ( q a , ˙ q a ) for a = 1 ,
2. In terms of the Legendretransformation p a = ∂L/∂ ˙q a , we define the canonical Hamiltonian function H ( q a , p a ) = p a · ˙q a − L .This reads the following collective Hamiltonian function H = H ( q , p ) − H ( q , p ) − K ( q , q , p , p ) . (5.6)It is possible to write the Hamiltonian function in terms of the variables in (5.3). So that we cancompute the momenta p + and p − so that we can write the Hamiltonian function with respect to19 coordinates that is H = H ( q − , q + , p − , p + ). After taking the physical limit PL in (5.4), thecanonical Hamilton’s equations reduce to˙ q = ∂H∂ p − h ∂K∂ p − i P L , ˙ p = − ∂H∂ q + h ∂K∂ q − i P L . (5.7)Non-conservative character of this system can be easily observed by the time derivative of theHamiltonian function dHdt = − q · h ddt ∂K∂ ˙ q − − ∂K∂ q − i P L . (5.8) In this section, we employ the dissipative decoupling method presented in the previous subsectionto the Hamiltonian functions governing radial curl forces.
Decoupling Bateman’s Pair.
We start with the Hamiltonian function given in (2.6) where thepotential energy is particularly considered to be U = (1 / x − y ). Decompose this Hamiltonianfunction as H = 12 ( p x − p y ) + 12 ( x − y ) = H ( x, p x ) − H ( y, p y ) , (5.9)where H = (1 / p x + (1 / x . We now couple the Hamiltonian function H in (5.9) with a function K = κp + x − where the plus minus notation is adopted through (5.3) so that x + = (1 / x + y )and x − = x − y . Explicitly, we have that H ( x − , x + , p − , p + ) = H ( x − , x + , p − , p + ) − K ( x − , x + , p − , p + ) = p − p + + x − , x + − κp + x − , (5.10)where H is the one in (5.9) with the plus minus coordinates. The non-conservative Hamilton’sequation in (5.7) read that ˙ x = p x , ˙ p x = − x + κp x (5.11)whereas the Newtonian form of the dynamics is¨ x − κ ˙ x + x = 0 . (5.12)This observation explores that variational underlying the individual dynamics in the Bateman’spair is computed through the coupled Lagrangian L ( x, y, ˙ x, ˙ y ) = 12 ( ˙ x − ˙ y ) −
12 ( x − y ) −
12 ( ˙ xx − ˙ xy + ˙ yx − ˙ yy ) . (5.13)according to the nonconservative Euler-Lagrange equations (5.5).20 issipation added to Curl Forces. In this case we apply Galley’s strategy to couple dissipationto curl forces. Start with the following Hamiltonian function, on the cotangent bundle of two-dimensional Euclidean space, H = 12 ( p x − p y ) + 12 b ( x − y ) + axy, (5.14)where a and b are real constants. Then, we couple the coordinates ( x, y, p x , p y ) with ( u, v, p u , p v ),and define the plus minus notation q − = ( x − u, y − v ) , q + = (cid:0)
12 ( x + u ) ,
12 ( y + v ) (cid:1) , p − = ( p x − p u , p y − p v ) , p + = (cid:0)
12 ( p x + p u ) ,
12 ( p y + p v ) (cid:1) . (5.15)In order to add a dissipation to the Hamiltonian dynamics, we introduce K = − κ p + · q − + f ( t ) · q − , (5.16)where f ( t ) stands for an external force. Evaluating at the physical limit, the dissipative Hamilton’sequation in (5.7) reads the equation of motion for a forced damped linear curl system in theHamiltonian form˙ x = p x , ˙ y = − p y , ˙ p x = − bx − ay − κp x + f x ( t ) , ˙ p y = by − ax − κp y + f y ( t ) . (5.17)whereas in the Newtonian form one arrives at¨ x + κ ˙ x + bx + ay = f x ( t ) , ¨ y + κ ˙ y + by − ax = f y ( t ) . (5.18)The same result will be exhibited from Galley’s equation by adding, for example a term if form p + · q + term since such terms do not contribute anything to the Galley’s equation (5.7). But thisform is clearly connected to the Bateman term appears in the class of quadratic Hamiltonians ofdamped curl forces Observe that, if the the external force is zero and κ vanishes then one arrivesprecisely at the Kapitsa model in (2.13). We have explored several sets of ideas centered around linear Hamiltonian curl forces as proposedby Berry and Shukla. We have used double bracket dissipation method, contact Hamiltonianformalism and Galley’s method to derive the dissipative extension of the curl force. It is noteworthy21hat the simplecticity is also destroyed by an addition of non-conservative positional forces. Doublebracket formalism is manifested in a metriplectic dynamics. The addition of nonzero dissipativecurl force can be described via metriplectic structure. For the linear case, we have investigatedthe steps starting from linear curl force through the Krechetnikov-Marsden [35] type equation byincorporating dissipation and gyroscopic forces.
Acknowledgements
We would like to express our sincere appreciation to Professors Sir Michael Berry, Anindya Ghose-Choudhury, Jayanta Bhattacharjee, Pavle Saksida for their interest, encouragement and valuablecomments in various stages of our work. This work was done while (PG) was visiting Departmentof Mathematics, Gebze Technical University under T¨ubitak visiting professorship. (PG) wouldlike to express his gratitude to the members of the department for their warm hospitality,