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Hamilton’s principle Hamiltonian Dynamics W E I based on FW-18 V E R the action is stationary under small virtual displacements about the actual motion of the system Equivalent to Newton’s laws! fixed initial and final configurations Euler-Lagrange equations New set of coordinates (transformations are assumed nonsingular and invertible): Variational statement of mechanics: (for conservative forces) action based on FW-32 Hamilton’s principle: a different function of new coordinates and velocities Hamilton’s principle for the new set of coordinates: the particle takes the path that minimizes the integrated difference of the kinetic and potential energies 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 174 Generalization to a system with n degrees of freedom: V E R if all the generalized coordinates are independent 176 Generalized momenta and the Hamiltonian W E I W E based on FW-20 Let’s define generalized momentum (canonical momentum): I V E R for independent generalized coordinates Lagrange’s equations can be written as: for k holonomic constraints: if the lagrangian does not depend on some coordinate, cyclic coordinate the corresponding momentum is a constant of the motion, a conserved quantity. 175 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 Hamiltonian Dynamics (coordinates and momenta equivalent variables): If the lagrangian does not depend explicitly on the time, then the hamiltonian is a constant of the motion: generalized momentum: W E I relations are assumed nonsingular and invertible Hamiltonian: time shift invariance implies that the hamiltonian is conserved Proof: V E R Legendre transformation from to Hamilton’s equations: also: (equivalent to Lagrange’s equations) 2n coupled first-order differential equations for coordinates and momenta 178 180 If there are only time-independent potentials and time-independent constraints, then the hamiltonian represents the total energy. V E R W E I Taking time derivative: { Proof: 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: 179 181 Modified Hamilton’s principle: How can we guarantee independent variables subject to independent variations ? with fixed endpoints: We can automatically guarantee this form if we set coefficients of velocities to 0: 0 and the new Hamiltonian is: variations of all ps and qs are independent Hamilton’s equations from Hamilton’s principle: 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! (the modified Hamilton’s principle may be taken to be the basic statement of mechanics, equivalent to Newtons laws) we will use this freedom to construct a transformation so that all Q and P are cyclic, i.e. constants of the motion! 182 Canonical Transformations 184 Hamilton-Jacobi theory based on FW-34 First let’s introduce another function S: Under what conditions do the transformations to new set of coordinates and momenta, based on FW-35 Legendre transformation relations are assumed nonsingular and invertible from to preserve the form of Hamilton’s equations? Such transformations should satisfy: ns atio equ on’s tonian t l i Ham Hamil s to lead ith new w (canonical transformations) the total derivative of any function can be added because it will not contribute to the modified Hamilton’s principle S generates canonical transformation, the Hamilton’s equations hold for new coordinates and momenta with the new Hamiltonian! 183 185 We want to use the freedom to choose S so that ! Hamilton’s principal function S is the action: Then Hamilton’s eqns. imply that all the P and Q are cyclic, i.e. constants of the motion! !s are constants the action evaluated along the dynamic trajectory Such S must satisfy: If the Hamiltonian does not explicitly depend on time, H is constant, and we can separate off the time dependence: 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) General form of S: Hamilton-Jacobi equation for Hamilton’s characteristic function W: overall integration constant (irrelevant) Sometimes the solution W can be separated in a sum of independent additive functions: any n independent non-additive integration constants 186 General form of S: 188 Example (a particle in one-dimensional potential): overall integration constant (irrelevant) any n independent non-additive integration constants Hamilton-Jacobi equation: Let’s look at a particular solution: 1 1 Hamilton’s principal function By assumption: Hamiltonian is independent of time so we can look for a solution of the form: It generates following transformation: inv Solution to the mechanical problem: er tin g Hamilton-Jacobi equation for Hamilton’s characteristic function: 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 Solution: 2n constants, !s and "s, are determined from 2n initial conditions 187 189 Example (a particle in one-dimensional potential) continued: at this point the trajectory is not determined The 2nd constant of motion: provides relation between q and t (constants ! and " determined from initial conditions) For harmonic oscillator: as expected 190 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