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Concept# Hamilton–Jacobi equation

Summary

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
The Hamilton–Jacobi equation is the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, Schrödinger's equation, as described below; for this reason, the Hamilton–Jacobi equation is considered the "closest approach" of classical mechanics to quantum mechanics. The qualitative form of this connection is called Hamilton's optico-mechanical analogy.
In mathematics, the Hamil

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This thesis deals with applications of Lie symmetries in differential geometry and dynamical systems. The first chapter of the thesis studies the singular reduction of symmetries of cosphere bundles, the conservation properties of contact systems and their reduction. We generalise the results of [15] to the singular case making a complete topological and geometrical analysis of the reduced space. Applying the general theory of contact reduction developed by Lerman and Willett in [33] and [57], one obtains contact stratified spaces that lose all information of the internal structure of the cosphere bundle. Based on the cotangent bundle reduction theorems, both in the regular and singular case, as well as regular cosphere bundle reduction, one expects additional bundle-like structure for the contact strata. The cosphere bundle projection to the base manifold descends to a continuous surjective map from the reduced space at zero to the orbit quotient of the configuration space, but it fails to be a morphism of stratified spaces if we endow the reduced space with its contact stratification and the base space with the customary orbit type stratification defined by the Lie group action. In this chapter we introduce a new stratification of the contact quotient at zero, called the C-L stratification (standing for the coisotropic or Legendrian nature of its pieces) which solves the above mentioned two problems. Its main features are the following. First, it is compatible with the contact stratification of the quotient and the orbit type stratification of the configuration orbit space. It is also finer than the contact stratification. Second, the natural projection of the C-L stratified quotient space to its base space, stratified by orbit types, is a morphism of stratified spaces. Third, each C-L stratum is a bundle over an orbit type stratum of the base and it can be seen as a union of C-L pieces, one of them being open and dense in its corresponding contact stratum and contactomorphic to a cosphere bundle. The other strata are coisotropic or Legendrian submanifolds in the contact components that contain them. We also describe the relation between contact vector fields and the time dependent Hamilton-Jacobi equation. The reduction of contact systems and time dependent Hamiltonians is mentioned. In the second chapter we study geometric properties of Sasakian and Kähler quotients. We construct a reduction procedure for symplectic and Kähler manifolds using the ray preimages of the momentum map. More precisely, instead of taking as in point reduction the preimage of a momentum value μ, we take the preimage of ℝ+μ, the positive ray of μ. We have two reasons to develop this construction. One is geometric: non zero Kähler point reduction is not always well defined. The problem is that the complex structure may not leave invariant the horizontal distribution of the Riemannian submersion πμ : J-1(μ) → Mμ. The solution proposed in the literature is correct only in the case of totally isotropic momentum (i.e. Gμ = G). The other reason is that it provides invariant submanifolds for conformal Hamiltonian systems. They are usually non-autonomous mechanical systems with friction whose integral curves preserve, in the case of symmetries, the ray pre-images of the momentum map. We extend the class of conformal Hamiltonian systems already studied and complete the existing Lie Poisson reduction with the general ray one. As examples of symplectic (Kähler) and contact (Sasakian) ray reductions we treat the case of cotangent and cosphere bundles and we show that they are universal for ray reductions. Using techniques of A. Futaki, we prove that, under appropriate hypothesis, ray quotients of Kähler-Einstein or Sasaki-Einstein manifolds remain Kähler or Sasaki-Einstein. Note that it suffices to prove the Kähler case and the compatibility of ray reduction with the Boothby-Wang fibration. In the last chapter, we prove a stratification theorem for proper groupoids. First we find an equivalent way of describing the same result for a proper Lie group action, way which uses the theory of foliations and can be adapted to the language of Lie groupoids. We treat separately the case of free and proper groupoids. The orbit foliation of a proper Lie groupoid is a singular Riemannian foliation and we show this explicitly.

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In this thesis we describe a path integral formalism to evaluate approximations to the probability density function for the location and orientation of one end of a continuum polymer chain at thermodynamic equilibrium with a heat bath. We concentrate on those systems for which the associated energy density is at most quadratic in its variables. Our main motivation is to exploit continuum elastic rod models for the approximate computation of DNA looping probabilities. We first re-derive, for a polymer chain system, an expression for the second order correction term due to quadratic fluctuations about a unique minimal energy configuration. The result, originally stated for a quantum mechanical system by G. Papadopoulos (1975), relies on an elegant algebraic argument that carries over to the real-valued path integrals of interest here. The conclusion is that the appropriate expression can be evaluated in terms of the energy of the minimizer and the inverse square root of the determinant of a matrix satisfying a certain non-linear system of differential equations. We then construct a change of variables, which establishes a mapping between the solutions of the aforementioned non-linear Papadopoulos equations and a matrix satisfying an initial value problem for the classic linear system of Jacobi equations associated with the second variation of the energy functional. This conclusion is trivial if no cross-term is present in the second variation, but ceases to be so otherwise. Cross-terms are always present in the application of rod models to DNA. We therefore can conclude that the second order fluctuation correction term to the probability density function for a chain is always given by the inverse square root of the determinant of a matrix of solutions to the Jacobi equations. We believe this conclusion to be original for the real-valued case when the second-variation involves cross-terms. Similar results are known for quantum mechanical systems, and, in this context, a connection between the so called Van-Hove-Morette determinant, which involves partial derivatives of the classical action with respect to the boundary values of the configuration variable, and the Jacobi determinant have also been established. We next apply the formula described above to the specific context of rods, for which the configuration space is that of framed curves, or curves in R3 × SO(3). An immediate application of our theory is possible if the rod model encompasses bend, twist, stretch and shear. However the constrained case, where the rod is considered to be inextensible and unshearable, is more standard in polymer physics. In this last case, our results are more delicate as the Lagrangian description breaks down, and the Hamiltonian formulation must be invoked. It is known that the unconstrained local minimizers approach constrained minimizers as the coefficients in the shear and extension terms of the energy are sent to infinity. Here we observe that the Hamiltonian form of the unconstrained Jacobi system similarly has a limit, so that the fluctuation correction in the path integral can still be expressed as the square root of the determinant of a matrix solution of a set of Jacobi equations appropriate to the constrained problem. As in reality DNA or biological macromolecules are certainly at least slightly shearable and extensible, the limit of the fluctuation correction is undoubtedly physically appropriate. The above theory provides a computationally highly tractable approach to the estimation of the appropriate probability density functions. For application to sequence-dependent models of DNA the associated systems of equations has non-constant coefficients, which is of little consequence for a numerical treatment, but precludes the possibility of finding closed form expressions. On the other hand the theory also applies to simplified homogeneous models. Accordingly, we conclude by applying our approach in a completely analytic and closed-form way to the computation of the approximate probability density function for a uniform, non-isotropic, intrinsically straight and untwisted rod to form a circular loop.

Ludovica Cotta-Ramusino, John Maddocks

We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e. g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end points. Our approximation formula for the looping pdf is of quite general applicability as, in contrast to most prior approaches, no assumption is made of either uniformity of the elastic chain, nor of a straight intrinsic shape. If the chain is uniform the Jacobi system evaluated at certain minimum energy configurations has constant coefficients. In such cases our approximate pdf can be evaluated in an entirely explicit, closed form. We illustrate our analysis with a planar example of this type and compute an approximate probability of cyclization, i.e., of forming a closed loop, from a uniform elastic chain whose intrinsic shape is an open circular arc.

2010