legendre transformatio.pptx

Legendre transformation detail presentation

1. Presentation Topic Legendre Transformation

2. Group Members Ayesha Anwar Iqra Bashir Benish Bibi Fatima Irshad

3. Abstract Legendre transform ation appear in two places in a standard undergraduate physics curriculum: in classical mechanics when one switches from Lagrangian to Hamiltonian dynamics in thermodynamics to motivate the connection between the internal energy, enthalpy, and Gibbs and Helmholtz free energ ies.

4. History The transform ation is named after the French mathematician Adrien-Marie Legendre (1752–1833). He is also noted for establishing the modern notation for partial derivatives . which was subsequently adopted by Carl Jacobi in 1841, as well as for work on his eponymous differential equation and polynomials.

5. Defination of Legendre transformation The Legendre transformation connects two ways of specifying the same physics, via functions of two related (“conjugate”) variables.

6. What is legendre tranformation used for? Legendre transform ation 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. Example#1: Convert from the Lagrangian L(x, x!) to the Hamiltonian T he velocity x!and the linear momentum p are conjugate variables, and both L and H have units of energy.

7. Example 2 Convert between the internal energy U, enthalpy H, Helmholtz free energy F, and Gibbs free energy G. The two conjugate pairs of variables are pressure P and volume V, and temperature T and entropy S. All of these thermodynamic potentials have units of energies

8. Hamiltionian Mechanics Describes the system motion in terms of n generalized coordinates q j & n generalized momenta p i . It gets 2n 1 st order, time dependent equations of motion. Recall that by DEFINITION: The generalized Momentum associated with t he generalized coordinate q j : p i  ( ∂ L/ ∂ q i ) (q,p)  “conjugate” or “canonical” variables .

9. Hamiltonian formualtion of Mechanics Hamiltonian Mechanics: A fundamentally different picture! Describes the system motion in terms of 1 st order, time dependent equations of motion. The number of initial conditions is, of course, still 2n. We must describe the system motion with 2n independent 1 st order , time dependent, differential equations expressed in terms of 2n independent variables . We choose n of these = n generalized coordinates q i . We choose the other n = n generalized (conjugate) momenta p i .

10. Lagrange Eq uation of motion: n degrees of freedom (d/dt)[ ( ∂ L/ ∂ q i )] - ( ∂ L/ ∂ q i ) = 0 (i = 1,2,3, … n) n 2 nd order , time dependent, differential equations.  The system motion is determined for all time when 2n initial values are specified: n q i ’s & n q i ’s We can represent the state of the system motion by the time dependent motion of a point in an abstract configuration space (coords = n generalized coords q i ). PHYSICS : In the Lagrangian Formulation of Mechanics, a system with n degrees of freedom = a problem in n independent variables q i (t). The generalized velocities, q i (t) are simply determined by taking the time derivatives of the q i (t) . The velocities are not independent variables. Lagrange Equation of Motion

11. Explaination Physically, the Lagrange formulation assumes the coordinates q i are independent variables & the velocities q i are dependent variables & only obtained by taking time derivatives of the q i once the problem is solved! Mathematically , the Lagrange formalism treats q i & q i as independent variables. e.g., in Lagrange’s equations, ( ∂ L/ ∂ q i ) means take the partial derivative of L with respect to q i keeping all other q’s a d also all q’s constant. Similarly ( ∂ L/ ∂ q i ) means take the partial derivative of L with respect to q i keeping all other q’s also all q’s constant.

12. Consider a function f(x,y) of 2 independent variables (x,y) The exact differential of f : df  u dx + v dy Obviously: u  ( f/x) v  ( f/y) Now, change variables to u & y , so that the differential quantities are expressed in terms of du & dy . Let g = g(u,y) be a function defined by g  f - ux Change from f(x,y)  df  u dx + v dy u  ( f/x) v  ( f/y) To g(u,y)  f - ux. The exact differential of g : dg  df - u dx - x du = v dy – x du Obviously: v  ( g/x) x  - ( g/u) This is a Legendre Transformation .

13. Change from the Lagrange to the Hamilton formulation.  Changing variables from (q,q,t) (q,q, independent) to (q,p,t) (q,p independent) is a Legendre Transformation. However, it’s one where many variables are involved instead of just 2. Consider the Lagrangian L = L (q,q,t) (n q’s, n q’s) The exact differential of L (sum on i): d L  (  L /q i )dq i + (  L /q i )dq i + (  L /t)dt (1) Canonical Momentum is defined: (d/dt)[ ( ∂ L/ ∂ q i )] - ( ∂ L/ ∂ q i ) = 0 p i  (  L/ q i )  p i  (  L/ q i ) (2) Put (2) into (1):  d L = p i dq i + p i dq i + (  L /t)dt (3) Changing from lagrange to the Hamilton formulation

14. d L = p i dq i + p i dq i + (  L /t)dt (3) Define the Hamiltonian H by the Legendre Transformation: (sum on i ) H(q,p,t)  q i p i – L (q,q,t) (4)  dH = q i dp i + p i dq i – d L (5) Combining (3) & (5) :  dH = q i dp i - p dq i - (  L /t)dt (6) Since H = H(q,p,t) we can also write: dH  (  H /q i )dq i + (  H /p i )dp i + (  H /t)dt (7) Directly comparing (6) & (7)  q i  (  H /p i ), - p i  (  H /q i ), - (  L /t)  (  H /t)

15. Hamiltonian. Hamiltonian H: (sum on i) H(q,p,t)  q i p i – L (q,q,t) (a)  q i  (  H /p i ) (b) - p i  (  H /q i ) (c) - (  L /t)  (  H /t) (d) (b) & (c) together  Hamilton’s Equations of Motion or the Canonical Equations of Hamilton 2n 1 st order , time dependent equations of motion replacing the n 2 nd order Lagrange Equations of motion