Search results for "Legendre transformation"
showing 6 items of 6 documents
The Principles of Canonical Mechanics
2010
Canonical mechanics is a central part of general mechanics, where one goes beyond the somewhat narrow framework of Newtonian mechanics with position coordinates in the three-dimensional space, towards a more general formulation of mechanical systems belonging to a much larger class. This is the first step of abstraction, leaving behind ballistics, satellite orbits, inclined planes, and pendulum-clocks; it leads to a new kind of description that turns out to be useful in areas of physics far beyond mechanics. Through d’Alembert’s principle we discover the concept of the Lagrangian function and the framework of Lagrangian mechanics that is built onto it. Lagrangian functions are particularly …
Real symplectic formulation of local special geometry
2006
We consider a formulation of local special geometry in terms of Darboux special coordinates $P^I=(p^i,q_i)$, $I=1,...,2n$. A general formula for the metric is obtained which is manifestly $\mathbf{Sp}(2n,\mathbb{R})$ covariant. Unlike the rigid case the metric is not given by the Hessian of the real function $S(P)$ which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains $S$, its Hessian and the conjugate momenta $S_I=\frac{\partial S}{\partial P^I}$. Only in the one-dimensional case ($n=1$) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor.
Mixed Phases, Phase Transitions, Stability of Matter
2016
Phase mixtures and phase transitions are two major themes of thermodynamics. A third one, related to the former, is the stability of macroscopic matter around us. Mixed phases can be analyzed and illustrated in a nice geometric way. Phase transitions are dealt with from the point of view of classical thermodynamics as well as in the framework of models of statistical mechanics.
Multifractal electronic wave functions in disordered systems
1992
Abstract To investigate the electronic states in disordered samples we diagonalize very large secular matrices corresponding to the Anderson Hamiltonian. The resulting probability density of single electronic eigenstates in 1-, 2-, and 3-dimensional samples is analysed by means of a box-counting procedure. By linear regression we obtain the Lipschitz-Holder exponents and the corresponding singularity spectrum, typical for a multifractal set in each case. By means of a Legendre transformation the mass exponents and the generalized dimensions are derived. Consequences for spectroscopic intensities and transport properties are discussed.
Entropy function from toric geometry
2021
It has recently been claimed that a Cardy-like limit of the superconformal index of 4d $\mathcal{N}=4$ SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographic dual AdS$_5$ rotating black hole. Here we study this Cardy-like limit for $\mathcal{N}=1$ toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data. Furthermore, for some families of models, we compute the Legendre transform of the entropy function, comparing with similar results recently discussed in the literature.
An alternative formulation of Classical Mechanics based on an analogy with Thermodynamics
2013
We study new Legendre transforms in classical mechanics and investigate some of their general properties. The behaviour of the new functions is analyzed under coordinate transformations.When invariance under different kinds of transformations are considered, the new formulation is found to be completly equivalent to the usual Lagrangian formulation, recovering well established results like the conservation of the angular momentum. Furthermore, a natural generalization of the Poisson Bracket is found to be inherent to the formalism introduced. On the other hand, we find that with a convenient redefinition of the Lagrangian, $\mathcal{L}^{\prime}=-\mathcal{L}$, it is possible to establish an …