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PHYS 705: Classical Mechanics Canonical Transformation 2 1 PHYS 705: Classical Mechanics Canonical Transformation 2 Canonical Variables and Hamiltonian Formalism As we have seen, in the Hamiltonian Formulation of Mechanics, qj, p j are independent variables in phase space on equal footing The Hamilton’s Equation for q j , p j are “symmetric” (symplectic, later) H H qj and p j pj q j This elegant formal structure of mechanics affords us the freedom in selecting other appropriate canonical variables as our phase space “coordinates” and “momenta” - As long as the new variables formally satisfy this abstract structure (the form of the Hamilton’s Equations. 3 Canonical Transformation Recall (from hw) that the Euler-Lagrange Equation is invariant for a point transformation: Qj Qqt j ( ,) L d L i.e., if we have, 0, qj dt q j L d L then, 0, Qj dt Q j Now, the idea is to find a generalized (canonical) transformation in phase space (not config. space) such that the Hamilton’s Equations are invariant ! Qj Qqpt j (, ,) (In general, we look for transformations which Pj Pqpt j (, ,) are invertible.) 4 Invariance of EL equation for Point Transformation First look at the situation in config. space first: dL L Given: 0, and a point transformation: Q Qqt( ,) j j dt q j q j dL L 0 Need to show: dt Q j Q j L Lqi L q i Formally, calculate: (chain rule) Qji qQ ij i qQ ij L L q L q i i Q j i qiQ j i q i Q j From the inverse point transformation equation q i qQt i ( ,) , we have, qi q q q q 0 and i i qi Q i Q i k j Q j Qj k Qk t 5 Invariance of EL equation for Point Transformation dL L Forming the LHS of EL equation with Q : LHS j dt Q j Q j d Lqi L q i L q i LHS dt iqQij i qQ ij i qQ ij Ldqi dL q i Lqi L q i q dt Q dt q Q qQ qQ i i j ij iiij ij d Lqi L qi L d qi L q i dt qQ q Q q dt Q q Q i i j i j i j i j 6 Invariance of EL equation for Point Transformation (exchange order of diff) d L Lqi L d qi q i LHS dt q q Q q dt Q Q i i i j i jj dq q i i 0 Qj dt Q j (Since that’s what given !) q q i i 0 Qj Q j dL L LHS 0 0 dt Q j Q j 7 Canonical Transformation Now, back to phase space, we need to find the appropriate (canonical) transformation Qjj Qqpt(,,)and P jj Pqpt (,,) such that there exist a transformed Hamiltonian KQPt( , ,) with which the Hamilton’s Equations are satisfied: K K Q j and P j Pj Q j (The form of the EOM must be invariant in the new coordinates.) ** It is important to further stated that the transformation considered must also be problem-independent meaning that ( Q , P ) must be canonical coordinates for all system with the same number of dofs. 8 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., t2 I Ldt 0 t1 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 conjugate variable q j , p j and their derivatives qj, p j 2. We will consider variations in all 2n phase space coordinates 9 Hamilton’s Principle in Phase Space 1. To rewrite the integrant in terms of q j ,,, p jj q p j , we will utilize the definition for the Hamiltonian (or the inverse Legendre Transform): H pqjj L Lpq jj Hqpt(, ,) (Einstein’s sum rule) Substituting this into our variation equation, we have t2 t 2 I Ldt p q H( q , p , t ) dt 0 j j t1 t 1 2. The variations are now for nqj ' s and n pj ' s : (all q’s and p’s are independent) The rewritten integrant is formally a (,,)qqp pqj j Hqpt (,,) q,,, p q p p function of j j j j but in fact it does not depend on j , i.e. p j 0 This fact will proved to be useful later on. again, we will required the variations for the q j to be zero at ends 10 q q Hamilton’s Principle in Phase Space j0 j j pj p0 j j Affecting the variations on all 2n variables q j , p j , we have, t2 I qj q j q' s d d d j q q j j j t1 pj p j p' s d ddt 0 j p p j j j As in previous discussion, the second term in the sum for q j ' s can be rewritten using integration by parts: t t2 2 t 2 q j q j q j d dt dt q q dt q jj t j t 1 1 t1 11 Hamilton’s Principle in Phase Space Previously, we have required the variations for the q j to be zero at end pts t t2 2 t 2 q j q j q j q j d so that, 0 dt dt q q dt q t t, t j j t j 1 2 t 1 1 t1 So, the first sum with q j ' s can be written as: t2 q q d jd j d dt q dt q j j j t1 t2 d q q dtwhere q j d q dt q j j j j j t1 12 Hamilton’s Principle in Phase Space Now, perform the same integration by parts to the corresponding term for pj ' s t t2 2 t 2 we have, p j p j p j d dt dt p p dt p jj t j t 1 1 t1 t2 p Note, since , j without enforcing the p j 0 0 p j t1 variations for p j to be zero at end points. nd This gives the result for the 2 sum in the variation equation for p j ' s : t2 d p p dtwhere p j d p dt p j j j j j t1 13 Hamilton’s Principle in Phase Space Putting both terms back together, we have: t2 I d d d qj p j dt 0 q dt q p dt p jjj j jj t1 1 2 Since both variations are independent, 1 and2 must vanish independently ! d (,,)qqp pq Hqpt (,,) 0 j j 1 dt q j q j H p j and q j q jj q H p 0 j H q j (one of the Hamilton’s p j q j equations) 14 Hamilton’s Principle in Phase Space t2 I d d d qj p j dt 0 q dt q p dt p jjj j jj t1 1 2 d (,,)qqp pq Hqpt (,,) 0 j j 2 dt p j p j H 0 and q j p j p j p j H 0q j 0 H nd p j (2 Hamilton’s q j p j equations) 15 Hamilton’s Principle in Phase Space So, we have just shown that applying the Hamilton’s Principle in Phase Space, the resulting dynamical equation is the Hamilton’s Equations. H p j q j H q j p j 16 Hamilton’s Principle in Phase Space Notice that a full time derivative of an arbitrary function F of (, q pt ,) can be put into the integrand of the action integral without affecting the variations: t2 dF pj q j H(, q p ,) t dt dt t 1 t t 2 2 dF p q H(, q p ,) t dt dt j j dt t1 t1 t2 dF F t2 const t1 t1 Thus, when variation is taken, this constant term will not contribute ! 17 Canonical Transformation Now , we come back to the question: When is a transformation to Q , P canonical? We need Hamilton’s Equations to hold in both systems This means that we need to have the following variational conditions: p q H( q , p , t ) dt 0 AND P Q K( Q , P , t ) dt 0 j j j j For this to be true simultaneously, the integrands must equal And, from our previous slide, this is also true if they are differed by a full time derivative of a function of any of the phase space variables involved + time: dF pq Hqpt(,,)(,,),,,, PQ KQPt qpQPt jj jj dt 18 Canonical Transformation dF pq Hqpt(,,) PQ KQPt (,,) qpQPt ,,,, (1)(9.11) G jj jj dt F is called the Generating Function for the canonical transformation: Qj Qqpt j (, ,) (,)qpjj QP jj ,: Pj Pqpt j (, ,) As the name implies, different choice of F give us the ability to generate different Canonical Transformation to get to different Qj, P j F is useful in specifying the exact form of the transformation if it contains half of the old variables and half of the new variables. It, then, acts as a bridge between the two sets of canonical variables. 19 Canonical Transformation dF pq Hqpt( , , ) PQ KQPt ( , , ) qpQPt , , , , (*) ( G 9.11) jj jj dt F is called the Generating Function for the canonical transformation: Qj Qqpt j (, ,) (,)qpjj QP jj ,: Pj Pqpt j (, ,) Depending on the form of the generating functions (which pair of canonical variables being considered as the independent variables for the Generating Function), we can classify canonical transformations into four basic types.
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