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Classical Mechanics Examples (Hamiltonian Formalism) Classical Mechanics Examples (Hamiltonian Formalism) Dipan Kumar Ghosh Physics Department, Indian Institute of Technology Bombay Powai, Mumbai 400076 October 16, 2016 1 Introduction In Lagrangian formalism we described mechanical systems in terms of generalised coordi- nates and generalized velocities (qi; q_i). An alternative formalism is to describe the system in terms of generalized coordinates and generalized momenta. 1.1 Legendre Transformation Going over from one set of independent variables to another is achieved by means of Legendre Transformation. We are already familiar with such transformation in ther- modynamics. Suppose we are working in a micro canonical ensemble in which we work with constant (N; V; E). A state function that depends on these variables is the entropy S which can be regarded as a function of (N; V; E). Alternatively, we could consider the energy E itself as a thermodynamic potential which is a function of (N; V; S). The prob- lem with a microcanonical ensemble is that it is difficult to keep total energy as constant. It is therefore desirable to have a canonical ensemble in which we use a different set of control parameter, viz. (N; V; T ) which can be easily achieved by keeping the system in thermal contact with a heat reservoir. The new state function which is appropriate to @E this condition is the Helmholtz Free Energy F = E −TS. Note that T = . This @S V;N was an example of a Legendre transformation. Most experiments are performed under condition of fixed pressure rather than of constant volume. Instead one could think of a Legendre transform to enthalpy H = U + PV which is a function of (S; P; N). In this @E case we observe that P = − . One can make yet another transformation where @V S;N 1 c D. K. Ghosh, IIT Bombay 2 we we take the control variable to be (T; P; N). We define in this case Gibb's Free Energy G(T; P; N) = H − TS = E + PV − TS = F + PV All these are examples of Legendre of transformation. The essential idea behind Legendre transformation is to subtract off a conjugate pair to eliminate that variable. Suppose we have a system of spring with potential energy kx2=2 1 and thermal energy . The energy can be written as U(S; x) = kx2 + TS. Suppose we 2 @U wish to change the variable from x to the force F = − = −kx. The new energy @x function E(S; F ) can be written as follows: E(S; F ) = U(S; x) − F x 1 = (TS + kx2) + kx2 2 3 = TS + F 2 2k The new energy function is the difference between the old function and the product of the old and the new variables. Mathematically, if f = f(x; y), we can write @f @f df = dx + dy @x @y = udx + vdy where @f @f u = ; v = @x @y Suppose we wish to change over to a new set of control variables variable u and y, We define g = f − ux, where x is the variable going out and u (being the conjugate of x) is coming in. We then have dg = df − udx − xdu = (udx + vdy) − (udx + xdu) = vdy − xdu Note that dx does not appear in the above. We have @g @g x = − ; v = @u @y 2 Legendre Transformation in Classical Mechanics- Hamilton's Equations We use essentially the same technique to go over from the Lagrangian description in terms of (q; q_) to the Hamiltonian description in terms of (p; q). Recall that the canonically c D. K. Ghosh, IIT Bombay 3 conjugate momentum is defined through @L pi = (1) @q_i The definition (1) gives the known definition for the linear momentum in case of the Cartesian coordinates (p = mx_) The Hamiltonian (which is the functional that corresponds to the thermodynamic potential ) is given in terms of the Lagrangian through X H = piq_i − L (2) Note that we have X @L X @L dL = dq + dq_ @q i @q_ i i i i i X X = p_idqi + pidq_i (3) i i where we have used the Euler Lagrange equation d @L @L = dt @q_i @q_i which is equivalent to d @L pi = dt @qi to simplify the first term. Using the fact that ! X X X d piq_i = q_idpi + pidq_i i i i and using (3), we can write X X X dH = d( piq_i − L) = q_idpi − p_idqi i i i which gives the Hamilton's Canonical Equations @H q_i = (4a) @pi @H p_i = − (4b) @qi These are a set of 2n first order equations for 2n unknowns pi and qi which replace the n second order Lagrange equations. [Note that for Cartesian coordinates, in case @V of H = T + V , these give the standard equationsp _ = F = − and the velocity @x @T x_ = = p=m]. @p The dynamics of a system with n generalized coordinates and n generalized momenta can be considered to take place in a 2n dimensional phase space. A point in the phase space at a particular time represents the positions and momentum of all all the particles. As the system develops, the point in the phase space has a trajectory in this phase space. c D. K. Ghosh, IIT Bombay 4 2.1 Conservation of Energy Consider a system of n particles in a position dependent potential 1 X L = m q_2 − V (q ; q ; : : : q ) 2 i i 1 2 n i The moments are pi = miq_i so that 1 X p2 H = i + V (q ; q ; : : : ; q ) 2 2m 1 2 n i i Now, dH X @H X @H @H = p_ + q_ + dt @p i @q i @t i i i i X @H @H X @H @H = − + @p @q @q @p i i i i i i = 0 where we have put the partial derivative of H with respect to time to be zero as the Hamiltonian does not have explicit time dependence. Thus for systems where the Hamil- tonian does not have explicit time dependence the total energy is conserved. If the energy is conserved, the trajectory in the phase space lies on 2n − 1 dimensional hyper surface. 2.2 Example: Simple Pendulum Take the reference of potential energy at the point of suspension. The Lagrangian is 1 L = ml2θ_2 − mgl(1 − cos θ) 2 @L 2 _ In this case, the canonical momentum p = pθ = = ml θ. The Hamiltonian is given @θ_ by 1 p2 H = pθ_ − L = ml2θ_2 + (1 − mgl cos θ) = + (1 − mgl cos θ) 2 2ml2 c D. K. Ghosh, IIT Bombay 5 θ l l−lcos θ For a conservative system such as the pendulum, the energy remains constant. In case of the pendulum, two types of motion is possible with a critical value of energy separating the two. If the energy is below the critical energy (which corresponds to the potential energy corresponding to the highest point (θ = π) of the trajectory the motion is oscillatory (though not necessarily simple harmonic). This is known as libration. When the energy exceeds the critical energy the pendulum executes rotational motion about the hinge. The Hamilton's equations are @H p_ = − = −mgl sin θ @θ @H p θ_ = = @p ml2 In the phase space(θ; p) a point corresponds to the state of the pendulum at a particular time and the state develops as per the above equations. The trajectory is called a flow. c D. K. Ghosh, IIT Bombay 6 We represent the flow by means of phase portraits, which shows the way the momentum (or the velocity) of the phase points change as we go away from a fixed point. Fixed point of a trajectory is a point at which the velocity of the phase point is zero. Such points are equilibrium points of the differential (Hamilton's) equation. Fixed points are of two types: stable and unstable. Stable fixed points (also called attractors or sink) are points where the flow is moving towards the fixed points and unstable fixed points (repellers or sources) are those at which the flow is moving away from the fixed points. (There are also saddles where the flow is attracted in one direction are is repealed in another. There are also stable orbits which correspond to periodic solutions of the differential equation, which need not pass through the fixed points). The phase plot for different values of energy are shown (figure from web) with the blue curve having pointed edges representing the critical curve separating the libration and rotation. It is known as the separatrix of the motion. We have chosen x axis as θ and the y axis as p = ml2θ_. For convenience, let us scale by taking ml2 = 1 so that θ_ = p and p2 H(θ; p) = − a2θ 2 p where a = mgl. dx dp Thus = θ_ = p=ml2 = p and = −mgl sin θ ≡ −a2 sin θ. If the particle is at dt dt (x; p), it has a velocity (x; _ p_) = (p; −a2 sin θ). Our analyses starts by determining the region of the phase plane where dx=dt = 0; > 0 or < 0. On the x-axis p = 0, i.e. θ_ = 0, i.e. there is no horizontal component. In the upper half plane p > 0 so that θ_ > 0 so that θ increases with time and the trajectory moves rightward. The phase space is 2n dimensional. We can choose such that the first n components of a phase space vector (known as the \state vector") are the n generalized coordinates followed by n components c D. K. Ghosh, IIT Bombay 7 q which are the conjugate momenta. Let us represent the state vector by ξ = .
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