----------------------------------------- S1p. Dissipative dynamics and Lagrangians ----------------------------------------- Any system of ordinary differential equations can be brought into an artificial Lagrangian form, by first rewriting it in first order form F(q,q')=0 doubling the degrees of freedom by introducing conjugate variables p, and then considering the Lagrangian L(p,q)= p^T F(q,q'). In particular, this provides a Lagrangian formulation of dissipative systems, such as the damped harmonic oscillator m q'' + c q' + k q = 0 (m,c,k >0) Unfortunately, the Hamiltonian in such a formulation has nothing to do with the physical energy E = (m q'^2 + k q^2)/2 The same holds for various other representations for the damped harmonic oscillator found in the literature. Lagrangians for the damped harmonic oscillator go back to H. Bateman, Phys. Rev. 38, 815-819 (1931); the treatise P.M. Morse and H. Feshbach, Methods of Theoretical Physics MacGraw-Hill, Boston 1953 discusses the procedure in Chapter 3 in terms of 'mirror images' = additional dynamical variables needed to absorb the missing energy, and remarks on p 313: ''The introduction of the mirror image ... is probably too artificial a procedure to expect to obtain much of physical significance from it.'' And indeed, the book doesn't make use of it anywhere. Having a formal Lagrangian or Hamiltonian is no virtue in itself. In particular, for a _quantum_ system, the Hamiltonian _must_ be the energy. Playing around with alternative Lagrangians and Hamiltonians may be amusing, but does not produce relevant physics. Since dissipative equations (like the diffusion equation or the damped harmonic oscillator) describe open systems (where energy is lost to an unspecified environment), they cannot be described by a Schroedinger equation. Classically, dissipative systems are described by stochastic processes describing dynamical semigroups. These can be classified into stochastic differential equations (and their equivalent deterministic Fokker-Planck equations), master equations, or a combination of both. See the differential Chapman-Kolmogorov equation in Gardiner's Handbook of stochastic methods - (3.4.22) in the second edition. One of the simplest of such models - the diffusion equation - is the particular case of a Fokker-Planck equation for Brownian motion. Quantum mechanically, dissipative systems are described by stochastic Schroedinger equations or, corresponding to the Fokker-Planck level, by quantum dynamical semigroups, expressed using quantum Liouville equations with Lindblad terms. This gives correct physics in a dissipative environment. Many quantum optical systems are directly modeled on the Lindblad level, where the terms have an understandable and experimentally verifiable meaning independent of any underlying more microscopic model. An important recent example is that of photons on demand, M. Keller, B Lange, K Hayasaka, W Lange and H Walther, A calcium ion in a cavity as a controlled single-photon source, New Journal of Physics 6 (2004), 95. There is no trace of a Lagrangian in the modeling, and indeed, a useful Lagrangian formulation does not exist - unless one extends the dynamics and explicitly includes the environment. Of course, in theory, a dissipative system is thought to be a contracted version of a bigger conservative system which includes the envoironment, and in simple situations, this theoretical view can indeed be substantiated. If one models the dissipative environment explicitly, on gets a bigger conservative system, not a dissipative system. Of course, this conservative system has a Hamiltonian or Lagrangian description, but it does not describe the dissipative system alone. When one contracts it to the degrees of freedoms of the original system, one gets an integro-differential equation with memory, which is no longer described by a physically meaningful Hamiltonian or Lagrangian framework. The reduced dynamics takes the exact form m x''(t) + k x(t) = int_0^t G(s) x(t-s) ds + F(t). with functions F(t) (the noise caused by the environment) and G(s) (the memory kernel) that depend on the state of the environment. If the interaction is of the usual, dissipative nature then both F(t) and G(s) are extremely oscillating, even for intervals short compared to the inverse frequency T of the oscillator. But the short time averages of the memory Kernel have an exponentially decaying bound on their size and become negligible after some relaxation time tau << T. Thus it suffices in a good approximation to take the integral from s=0 to s=tau only. This allows us to expand x(t-s) in a second order Taylor expansion (valid since s<=tau<0, recovering the traditional equation for the damped harmonic oscillator, including a stochastic force term. (Its size can be related to the damping coefficient and the temperature of the environment, a relation known as the fluctuation-dissipation theorem.) A thorough discussion of the reduction of microscopic conservative large systems to dissipative subsystems of interest is given in H Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer Tracts in Modern Physics, 1982 at a much more general level that also applies for many other dissipative systems. There are cases where one needs to model the memory to capture the essence of the reduced dynamics. But in many cases, a simpler, memory-free description is possible and adequate. One can remove the memory by employing a Markov approximation, and gets again a differential equation, which defines the Lindblad (or, classicallally, the Focker-Planck) dynamics. Again, this is no longer described by a Hamiltonian or Lagrangian framework. In the extended formulation with explicit environment or with memory, already a simple damped harmonic oscillator becomes a huge and unwieldy dynamical system which is no longer equivalent to the damped harmonic oscillator, but includes unwanted environment terms or memory terms. In cases where one really needs to model the memory, the system therefore is no longer a damped harmonic oscillator. The latter is described by a simple linear constant coefficient second order differential equation for a single function, and has no memory. Its analysis is very simple, and compared to that any more detailed description is unwieldy. In practice, the dissipative formulation therefore stands by itself (apart from lip service paid to a hypothesized more fundamental conservative description). The situation is similar to that in fluid dynamics. In theory, the Navier-Stokes equations (which are dissipative) should be derivable from a Lagrangian. Indeed, such derivations have been given, but only for very simple model problems such as an ideal gas. However, there is no microscopic derivation of the Navier-Stokes equations in the practically interesting case of water at room temperature...