What is Legendre transformation in thermodynamics?
A Legendre transform converts from a function of one set of variables to another function of a conjugate set of variables. Both functions will have the same units.
What is the Legendre transform of internal energy?
In thermodynamics, the internal energy U can be Legendre transformed into various thermodynamic potentials, with associated conjugate pairs of variables such as temperature-entropy, pressure-volume, and “chemical potential”-density.
What is importance of Legendre transformation?
The Legendre transform shows how to define a function that contains the same infor- mation as F x but as a function of dF/dx. is a strictly monotonic function of x because this character- ization also permits us to treat functions whose negative is convex.
What is Hamilton equation of motion?
A set of first-order, highly symmetrical equations describing the motion of a classical dynamical system, namely q̇j = ∂ H /∂ pj , ṗj = -∂ H /∂ qj ; here qj (j = 1, 2,…) are generalized coordinates of the system, pj is the momentum conjugate to qj , and H is the Hamiltonian.
Is Legendre transform linear?
The Legendre transform of a convex function is convex.
How do you find Hamilton equations?
Now the kinetic energy of a system is given by T=12∑ipi˙qi (for example, 12mνν), and the hamiltonian (Equation 14.3.7) is defined as H=∑ipi˙qi−L. For a conservative system, L=T−V, and hence, for a conservative system, H=T+V. If you are asked in an examination to explain what is meant by the hamiltonian, by all means …
How do you solve Hamiltonian equations?
The generalized coordinate and momentum do not explicitly depend on time, so H = E. (c) Hamilton’s equations are dp/dt = -∂H/∂q = -ωq, dq/dt = p∂H/∂q = ωp. Solutions are q = A cos(ωt + Φ), p = A sin(ωt + Φ), A and Φ are determined by the initial conditions, ω = (k/m)½.
What is Q in Lagrange equation?
Lagrange’s Equations In this case qi is said to be a cyclic or ignorable co-ordinate. Consider now a group of particles such that the forces depend only on the relative positions and motion between the particles.
What is the difference between Lagrangian and Hamiltonian?
Lagrangian mechanics can be defined as a reformulation of classical mechanics. The key difference between Lagrangian and Hamiltonian mechanics is that Lagrangian mechanics describe the difference between kinetic and potential energies, whereas Hamiltonian mechanics describe the sum of kinetic and potential energies.
Which is the form required for a Legendre transformation of the basis variables?
The Legendre transform is linked to integration by parts, pdx = d(px) − xdp. The function -g(p, y) is the Legendre transform of f(x, y), where only the independent variable x has been supplanted by p. This is widely used in thermodynamics, as illustrated below.