synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
The Legendre transformation is an operation on convex functions from a real normed vector space to the real line; it is one of the cornerstones of convex analysis. The space of arguments changes accordingly.
Two differentiable functions $f, \tilde f \;\colon\; \mathbb{R} \to \mathbb{R}$ on the real line are said to be Legendre transforms of each other, if their derivatives $D f, D\tilde f \;\colon\; \mathbb{R} \to \mathbb{R}$ are inverse functions of each other:
The main application of and the historical root of the notion of Legendre transform (in differential geometry) is in classical physics and its formalization by symplectic geometry. In classical mechanics, the Hamiltonian function $H$ is a Legendre transform of the Lagrangean $L$ and vice versa.
When one formalizes classical mechanics as the local prequantum field theory given by prequantized Lagrangian correspondences, then the Legendre transform is exhibited by the lift from a Lagrangian correspondence to a prequantized Lagrangian correspondence. For more on this see at The classical action, the Legendre transform and Prequantized Lagrangian correspondences.
In many dimensions, hybrid versions are possible. When the physics of the system is given by the variational principle, then the Legendre transform of an extremal quantity is a conserved quantity. In thermodynamics, we can have some quantities set to be fixed (some candidates: entropy $S$, temperature $T$, pressure $P$, volume $V$, magnetization $M$); this dictates the choice of variables and quantity which is extremized as well as which one takes the role of conserved energy. Some of the standard choices are enthalpy $H$, Helmholtz free energy $F$, Gibbs free energy $G$, internal energy $U$, etc.
See also wikipedia:Legendre transformation and wikipedia:Legendre-Fenchel transformation; the two wikipedia articles have much detail in certain specific approaches and cases, but also miss some of the basic ones to be balanced.
See at prequantized Lagrangian correspondence.
See at multisymplectic geometry – de Donder-Weyl-hamilton equations of motion.
Hamiltonian | $\leftarrow$ Legendre transform $\rightarrow$ | Lagrangian |
---|---|---|
Lagrangian correspondence | prequantization | prequantized Lagrangian correspondence |
The concept is named after Adrien-Marie Legendre.
Reviews include
See also
Discussion of Legendre transformation in the context of Lie algebroids is in:
Paulette Liberman, Lie algebroids and mechanics (ps)
Juan Carlos Marrero et al, A survey of Lagrangian mechanics and control on Lie algebroids and Lie groupoids (pdf)
Juan Carlos Marrero, Nonholonomic mechanics: a Lie algebroid perspective (pdf talk notes)