REVIEW. Hamilton s principle. based on FW-18. Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws!

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1 Hamilton s principle Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws! based on FW-18 REVIEW the particle takes the path that minimizes the integrated difference of the kinetic and potential energies 174

2 Generalization to a system with n degrees of freedom: if all the generalized coordinates are independent REVIEW for k holonomic constraints: 175

3 Hamiltonian Dynamics Hamilton s principle: based on FW-32 the action is stationary under small virtual displacements about the actual motion of the system fixed initial and final configurations Euler-Lagrange equations New set of coordinates (transformations are assumed nonsingular and invertible): Hamilton s principle for the new set of coordinates: a different function of new coordinates and velocities Lagrange s equations remain invariant under the point transformations! we can choose any set of generalized coordinates and Lagrange s equations will correctly describe the dynamics 176

4 Generalized momenta and the Hamiltonian based on FW-20 Let s define generalized momentum (canonical momentum): Lagrange s equations can be written as: if the lagrangian does not depend on some coordinate, cyclic coordinate the corresponding momentum is a constant of the motion, a conserved quantity. for independent generalized coordinates REVIEW related to the symmetry of the problem - the system is invariant under some continuous transformation. For each such symmetry operation there is a conserved quantity! 177

5 If the lagrangian does not depend explicitly on the time, then the hamiltonian is a constant of the motion: Proof: time shift invariance implies that the hamiltonian is conserved REVIEW 178

6 If there are only time-independent potentials and time-independent constraints, then the hamiltonian represents the total energy. Proof: { REVIEW 179

7 Hamiltonian Dynamics (coordinates and momenta equivalent variables): generalized momentum: Hamiltonian: relations are assumed nonsingular and invertible Legendre transformation from to Hamilton s equations: (equivalent to Lagrange s equations) also: 2n coupled first-order differential equations for coordinates and momenta 180

8 Taking time derivative: If the lagrangian does not depend explicitly on the time, then the hamiltonian is a constant of the motion in addition we saw before, that for a conservative system with time-independent constraints: 181

9 Modified Hamilton s principle: independent variables subject to independent variations with fixed endpoints: 0 Hamilton s equations from Hamilton s principle: (the modified Hamilton s principle may be taken to be the basic statement of mechanics, equivalent to Newtons laws) variations of all ps and qs are independent 182

10 Canonical Transformations based on FW-34 Under what conditions do the transformations to new set of coordinates and momenta, Such transformations should satisfy: relations are assumed nonsingular and invertible preserve the form of Hamilton s equations? (canonical transformations) leads to Hamilton s equations with new Hamiltonian the total derivative of any function can be added because it will not contribute to the modified Hamilton s principle 183

11 How can we guarantee? We can automatically guarantee this form if we set coefficients of velocities to 0: and the new Hamiltonian is: whenever the transformations can be written in terms of some F, then the Hamilton s equations hold for new coordinates and momenta with the new Hamiltonian! F is the generator of the canonical transformation (in practice, not easy to determine if such a function exists) Any F generates some canonical transformation! we will use this freedom to construct a transformation so that all Q and P are cyclic, i.e. constants of the motion! 184

12 Hamilton-Jacobi theory First let s introduce another function S: based on FW-35 Legendre transformation from to S generates canonical transformation, the Hamilton s equations hold for new coordinates and momenta with the new Hamiltonian! 185

13 We want to use the freedom to choose S so that! Then Hamilton s eqns. imply that all the P and Q are cyclic, i.e. constants of the motion! Such S must satisfy: General form of S: Hamilton-Jacobi equation first order partial differential equation in n+1 variables (can imagine integrating it one variable at a time, keeping remaining variables fixed, introducing an integration constant each time) overall integration constant (irrelevant) any n independent non-additive integration constants 186

14 General form of S: overall integration constant (irrelevant) Let s look at a particular solution: any n independent non-additive integration constants By assumption: 1 1 Hamilton s principal function It generates following transformation: Solution to the mechanical problem: inverting Any set of 's in S represents n constants of motion; derivatives with respect to 's determine *s, another set of n constants of motion 2n constants, 's and *s, are determined from 2n initial conditions 187

15 Hamilton s principal function S is the action: 's are constants the action evaluated along the dynamic trajectory If the Hamiltonian does not explicitly depend on time, H is constant, and we can separate off the time dependence: Hamilton-Jacobi equation for Hamilton s characteristic function W: Sometimes the solution W can be separated in a sum of independent additive functions: 188

16 Example (a particle in one-dimensional potential): Hamilton-Jacobi equation: Hamiltonian is independent of time so we can look for a solution of the form: Hamilton-Jacobi equation for Hamilton s characteristic function: Solution: 189

17 Example (a particle in one-dimensional potential) continued: The 2nd constant of motion: at this point the trajectory is not determined For harmonic oscillator: provides relation between q and t (constants ' and * determined from initial conditions) as expected 190

18 Connection with quantum mechanics: wave function Schrödinger equation We seek a wave-like solution: real function = 0 Hamilton-Jacobi equation The phase of the semiclassical wave function is the classical action evaluated along the path of motion! Separating off time dependance corresponds to looking for stationary states, and problem often allow a separation of variables: 191