Housekeeping. No announcement HW #7: 3.19

Transcription

1 1 Housekeeping No announcement HW #7: 3.19

2 2 Configuration Space vs. Phase Space 1 A given point in configuration space q q prescribes fully the 1,, n configuration of the system at a given time t. The specification of a point in this space does NOT specify the time evolution of the system completely! (a unique soln for a n-dim 2 nd order ODE needs 2n ICs) Many different paths can go thru a given point in config space q P Different paths crossing P will have n n q 1 q 1 the same set of but diff q 1

3 3 Configuration Space vs. Phase Space 2 To specify the state AND time evolution of a system uniquely at a given time, one needs to specify both AND or equivalently, q, p q q q p The 2n-dim space where both and are independent variables is called phase space. Thru any given point in phase space, there can only be ONE unique path! p GOAL: to find the EOM that applies to points in phase space. q

4 4 Hamiltonian Formulation - Instead of using the Lagrangian,, we will introduce a new function that depends on,, and t: L Lq, q, t q p H H q, p, t - This new function is call the Hamiltonian and it is defined by: L H q L q - Plugging in the definition for the generalized momenta: H p q L (Einstein s Convention: Repeated indices are summed) ( sum) Hq p t Lq, q, t -,, is the Legendre Transform of p L q q q q p with,,

5 5 Hamiltonian Formulation Crank crank crank and dot dot dot, we get: H q H p p q and H t L t These are called the Hamilton s Equations of Motion and they are the desired set of equations giving the EOM in phase space.

6 6 Hamiltonian Formulation Summary of Steps: 1. Pick a proper set of and form the Lagrangian L 2. Obtain the conugate momenta by calculating 3. Form 4. Eliminate from H using the inverse of so as to have 5. Apply the Hamilton s Equation of Motion: q H pq L q H H( q, p, t) H q H p and q p p L q As you will see, this formulation does not necessary simplify practical calculations but it forms a theoretical bridge to QM and SM. p L q

7 7 Hamiltonian Equations in Matrix (Symplectic) Notation The Hamilton equation can then be written in a compact form: 2n 2n H η J η if we define a anti-symmetric matrix, 0 I J I 0 This is typically referred to as the matrix (or symplectic) notation for the Hamilton equations. J n n where I is a identity matrix 0 is a n n zero matrix Note that the transpose of J is its own inverse (orthogonal): J T 0 I I 0 and T T I 0 JJJJ 0 I

8 8 Connection to Statistical Mechanics - The Hamilton s Equations describe motion in phase space q, - a point in phase space uniquely determines the state of the system p AND its future evolution. - points nearby represent system states with similar but slightly different initial conditions. - One can imagine a cloud of points (ensemble of systems) bounded by a closed surface S with nearly identical initial conditions moving in time. q0, p0 q', p' S t q, p q', p' t t

9 9 Liouville s Theorem One can show as a direct consequence of the Hamilton Equation of Motion that the phase space volume of this cloud of points (ensemble of system) is conserved! dv dt 0! This is the Liouville s Theorem: collection of phase-space points move as an incompressible fluid. Phase space volume occupied by a set of points in phase space is constant in time.

10 10 Canonical Transformation Recall (from hw) that the Euler-Lagrange Equation is invariant for a point transformation: Q Q( q, t) L d L q dt q i.e., if we have, 0, L d L Q dt Q then, 0, Now, the idea is to find a generalized (canonical) transformation in phase space (not config. space) such that the Hamilton s Equations are invariant! Q P Q ( q, p, t) P ( q, p, t) (In general, we look for transformations which are invertible.)

11 11 Canonical Transformation We need to find the appropriate (canonical) transformation Q Q ( q, p, t) and P P ( q, p, t) such that there exist a transformed Hamiltonian KQPt (,, ) with which the Hamilton s Equations are satisfied: K Q and P P K Q (The form of the EOM must be invariant in the new coordinates.) ** It is important to further stated that the transformation considered ( QP, ) must also be problem-independent meaning that must be canonical coordinates for all system with the same number of dofs.

12 12 Canonical Transformation To see what this condition might say about our canonical transformation, we need to go back to the Hamilton s Principle: Hamilton s Principle: The motion of the system in configuration space is such that the action I has a stationary value for the actual path,.i.e., I t t 2 1 Ldt 0 Now, we need to extend this to the 2n-dimensional phase space 1. The integrant in the action integral must now be a function of the independent conugate variable q, p and their derivatives q, p 2. We will consider variations in all 2n phase space coordinates

13 13 Hamilton s Principle in Phase Space To rewrite the integrant in terms of q, p, q, p, we will utilize the definition for the Hamiltonian (or the inverse Legendre Transform): H p q L L p q H( q, p, t) (Einstein s sum rule) Substituting this into our variation equation, we have t 2 2 I Ldt p q H ( q, p, t) dt 0 t 1 1 t t

14 14 Hamilton s Principle in Phase Space Applying the Hamilton s Principle in Phase Space, the resulting dynamical equation is the Hamilton s Equations. p q H q H p