Review of Classical Mechanics

Slide 1: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Review of Classical Mechanics Wenjun Deng April 2008

Slide 2: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Contents 1 Basic facts about classical mechanics 2 Lagrange’s equations 3 Hamilton’s equations 4 Conservation theorems 5 Canonical transformations 6 Simple harmonic oscillator 7 Summary

Slide 4: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Why is classical mechanics useful? F = ma is not well-suited to constraints. Many math methods are enabled to solve physics problems Lagrange multipliers Calculus of variations Matrices Underlies quantum mechanics and statistical mechanics

Slide 5: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary The chart of classical mechanics F=ma D'Alembert's Lagrange's principle equations Principle of virtual work Hamilton's Hamilton's principle equations Statistical mechanics Quantum Hamilton-Jacobi Canonical mechanics theory transformations

Slide 8: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Hamilton’s principle Lagrangian: L = T −V = L(q1 , q2 , · · · , qn , q1 , q2 , · · · , qn , t) ˙ ˙ ˙ = L(q, q, t) ˙ Hamilton’s principle: Deﬁne line integral: t2 I= L dt t1 Physical motion is the motion that satisﬁes: t2 δI = δ L dt = 0 t1

Slide 9: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Derivation of Lagrange’s equations Consider a set of possible motions labeled by α: q1 (t, α) = q1 (t, 0) + αη1 (t) q2 (t, α) = q2 (t, 0) + αη2 (t) . . . qn (t, α) = qn (t, 0) + αηn (t) qi (t, 0) describe the physical motion. ηi (t) are arbitrary independent functions that vanish at end points: ηi (t1 ) = ηi (t2 ) = 0

Slide 10: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Derivation of Lagrange’s equations (con’d) t2 ∂I ∂L ∂qi ∂L ∂ qi ˙ δI = δα = + dt δα ∂α t1 ∂qi ∂α ∂ qi ∂α˙ // qi (t, α) = qi (t, 0) + αηi (t) t2 t2 ∂L ∂L = ηi dt + ηi dt δα ˙ t1 ∂qi t1 ∂ qi˙

Slide 11: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Derivation of Lagrange’s equations (con’d) t2 ∂I ∂L ∂qi ∂L ∂ qi ˙ δI = δα = + dt δα ∂α t1 ∂qi ∂α ∂ qi ∂α˙ // qi (t, α) = qi (t, 0) + αηi (t) t2 t2 ∂L ∂L = ηi dt + ηi dt δα ˙ t1 ∂qi t1 ∂ qi˙ t2 t2 t2 ∂L ∂L d ∂L = ηi dt + ηi − ηi dt δα t1 ∂qi ∂ qi ˙ t1 t1 ˙ dt ∂ qi // ηi (t1 ) = ηi (t2 ) = 0 t2 ∂L d ∂L = − ηi dt δα t1 ∂qi dt ∂ qi ˙

Slide 12: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Derivation of Lagrange’s equations (con’d) From last page: t2 ∂L d ∂L δI = − ηi dt δα t1 ∂qi dt ∂ qi ˙ δI should be 0 for arbitrary independent functions ηi , so ∂L d ∂L − =0 ∂qi dt ∂ qi ˙ We get Lagrange’s equations!

Slide 15: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Legendre transformation Consider a 2-variable function f (x, y) with such a diﬀerential: df = u dx + v dy where ∂f ∂f u= ,v= ∂x ∂y

Slide 16: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Legendre transformation Consider a 2-variable function f (x, y) with such a diﬀerential: df = u dx + v dy where ∂f ∂f u= ,v= ∂x ∂y Deﬁne g = ux − f dg = u dx + x du − df

Slide 17: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Legendre transformation Consider a 2-variable function f (x, y) with such a diﬀerential: df = u dx + v dy where ∂f ∂f u= ,v= ∂x ∂y Deﬁne g = ux − f dg = u dx + x du − df = x du − v dy We get g = g(u, y)

Slide 18: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Legendre transformation of Lagrangian f = f (x, y) ∂f ∂f df = dx + dy ∂x ∂y = u dx + v dy g = ux − f dg = x du − v dy g = g(u, y)

Slide 19: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Legendre transformation of Lagrangian f = f (x, y) L = L(q, q, t) ˙ ∂f ∂f ∂L ∂L ∂L df = dx + dy dL = dqi + dqi + ˙ dt ∂x ∂y ∂qi ∂ qi ˙ ∂t = u dx + v dy g = ux − f dg = x du − v dy g = g(u, y)

Slide 20: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Legendre transformation of Lagrangian f = f (x, y) L = L(q, q, t) ˙ ∂f ∂f ∂L ∂L ∂L df = dx + dy dL = dqi + dqi + ˙ dt ∂x ∂y ∂qi ∂ qi ˙ ∂t = u dx + v dy ∂L g = ux − f = pi ∂ qi ˙ dg = x du − v dy ∂L d ∂L = = pi ˙ g = g(u, y) ∂qi dt ∂ qi ˙

Slide 21: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Legendre transformation of Lagrangian f = f (x, y) L = L(q, q, t) ˙ ∂f ∂f ∂L ∂L ∂L df = dx + dy dL = dqi + dqi + ˙ dt ∂x ∂y ∂qi ∂ qi ˙ ∂t = u dx + v dy = pi dqi + pi dqi + Lt dt ˙ ˙ g = ux − f H = qi pi − L ˙ dg = x du − v dy dH = qi dpi − pi dqi − Lt dt ˙ ˙ g = g(u, y) H = H(q, p, t)

Slide 22: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Hamilton’s equations The diﬀerential of the Hamiltonian: ∂L dH = qi dpi − pi dqi − ˙ ˙ dt ∂t It can also be written as: ∂H ∂H ∂H dH = dpi + dqi + dt ∂pi ∂qi ∂t

Slide 23: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Hamilton’s equations The diﬀerential of the Hamiltonian: ∂L dH = qi dpi − pi dqi − ˙ ˙ dt ∂t It can also be written as: ∂H ∂H ∂H dH = dpi + dqi + dt ∂pi ∂qi ∂t Hamilton’s equations: ∂H ∂H qi = ˙ , − pi = ˙ ∂pi ∂qi and ∂L ∂H − = ∂t ∂t

Slide 24: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary The formal procedure to get the Hamiltonian 1 Construct the Lagrangian by L(q, q, t) = T − V ; ˙ ∂L 2 Get the conjugate momenta pi = ∂ q˙i ; 3 Construct the Hamiltonian by H = qi pi − L(q, q, t); ˙ ˙

Slide 25: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary An alternative way to get the Hamiltonian In many problems, the Lagrangian can be written as: (1) (2) L = L(0) (q, t) + Lj (q, t)qj + Lj,k (q, t)qj qk ˙ ˙ ˙

Slide 26: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary An alternative way to get the Hamiltonian In many problems, the Lagrangian can be written as: (1) (2) L = L(0) (q, t) + Lj (q, t)qj + Lj,k (q, t)qj qk ˙ ˙ ˙ If the equations deﬁning the generalized coordinates don’t (2) depend on time explicitly, then Lj,k (q, t)qj qk = T , and ˙ ˙ qi pi = 2T . ˙ If the forces are derivable from a conservative potential, then L(0) = −V . In this case, H = qi pi − L = T + V = total energy. ˙

Slide 29: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Cyclic coordinates Deﬁnition: a generalized coordinate qj is called a cyclic coordinate, if it does not appear explicitly in the Lagrangian.

Slide 30: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Cyclic coordinates Deﬁnition: a generalized coordinate qj is called a cyclic coordinate, if it does not appear explicitly in the Lagrangian. From the Lagrange’s equation and the Hamilton equation: ∂L d ∂L ∂H = = pj = − ˙ ∂qj dt ∂ qj ˙ ∂qj The conjugate momentum pj of a cyclic coordinate qj is a constant. A coordinate that is cyclic will also be absent from the Hamiltonian.

Slide 31: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Energy conservation When deriving the Hamilton equations, we had: ∂L ∂H − = ∂t ∂t

Slide 32: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Energy conservation When deriving the Hamilton equations, we had: ∂L ∂H − = ∂t ∂t The total time derivative of the Hamiltonian: dH ∂H ∂H ∂H = qi + ˙ pi + ˙ dt ∂qi ∂pi ∂t Using the Hamilton equations, the ﬁrst 2 terms on the RHS cancel out,

Slide 33: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Energy conservation When deriving the Hamilton equations, we had: ∂L ∂H − = ∂t ∂t The total time derivative of the Hamiltonian: dH ∂H ∂H ∂H = qi + ˙ pi + ˙ dt ∂qi ∂pi ∂t Using the Hamilton equations, the ﬁrst 2 terms on the RHS cancel out, so dH ∂H ∂L = =− dt ∂t ∂t If t doesn’t appear explicitly in L, it will also be absent from H, and H will be constant in time.

Slide 36: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Deﬁnition Consider an invertible transformation of coordinates and momenta Qi = Qi (q, p, t) Pi = Pi (q, p, t) Suppose there exists a function K(Q, P, t) such that the motion equations are in the Hamiltonian form: ˙ ∂K ˙ ∂K Qi = , Pi = − ∂Pi ∂Qi

Slide 37: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Deﬁnition (con’d) Hamilton’s principle about (Q, P ): t2 δ ˙ (Pi Qi − K) dt = 0 t1 Meanwhile, Hamilton’s principle about (q, p): t2 δ (pi qi − H) dt = 0 ˙ t1 Both statements will be satisﬁed if: ˙ dF λ(pi qi − H) = Pi Qi − K + ˙ dt

Slide 38: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Deﬁnition (con’d) From last page: ˙ dF λ(pi qi − H) = Pi Qi − K + ˙ dt F = F (q, p, Q, P, t) has continuous 2nd derivatives.

Slide 39: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Deﬁnition (con’d) From last page: ˙ dF λ(pi qi − H) = Pi Qi − K + ˙ dt F = F (q, p, Q, P, t) has continuous 2nd derivatives. The transformation (q, p) → (Q, P ) is called a canonical transformation if λ = 1 an extended canonical transformation if λ = 1 Restricted canonical transformation: canonical transformation with no explicit time dependence: Qi = Qi (q, p) Pi = Pi (q, p)

Slide 40: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Generating function F Canonical transformation: ˙ dF pi qi − H = Pi Qi − K + ˙ dt If half of the variables of F are from (q, p), and half are from (Q, P ), then F is useful for specifying the exact form of the transformation, and is called the generating function. For example: F = F1 (q, Q, t)

Slide 41: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary An example of the generating function F F = F1 (q, Q, t) Plug into the transformation relation: ˙ dF1 pi qi − H = Pi Qi − K + ˙ dt ˙ ∂F1 ∂F1 ∂F1 ˙ = Pi Qi − K + + qi + ˙ Qi ∂t ∂qi ∂Qi ˙ ˙ Match the coeﬃcients of qi , Qi , and the constant: ∂F1 pi = ∂qi ∂F1 Pi = − ∂Qi ∂F1 K = H+ ∂t

Slide 42: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Four basic canonical transformations Generating function Generating function derivatives F = F1 (q, Q, t) pi = ∂Fi1 ∂q ∂F Pi = − ∂Q1 i F = F2 (q, P, t) − Qi Pi pi = ∂Fi2 ∂q Qi = ∂F2 ∂Pi F = F3 (p, Q, t) + qi pi qi = − ∂Fi ∂p 3 ∂F Pi = − ∂Q3 i F = F4 (p, P, t) + qi pi − Qi Pi qi = − ∂Fi ∂p 4 Qi = ∂F4 ∂Pi

Slide 43: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Why are canonical transformations useful? If we choose the generating function wisely, the K will contain more cyclic coordinates than H, which makes the canonical equations about (Q, P ) easier to solve. Hamilton-Jacobi theory provides a method to achieve this goal, which is a long story.

Slide 46: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Newton’s law approach F = −mω 2 q = m¨ q 2 q+ω q =0 ¨ End up solving a 2nd-order diﬀerential equation, although the solution is easy to get in this problem: q = q0 sin(ωt + φ)

Slide 47: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Lagrange’s approach Lagrangian: 1 1 L = mq 2 − mω 2 q 2 ˙ 2 2 Lagrange’s equation: d ∂L ∂L − = 0 dt ∂q ˙ ∂q m¨ + mω 2 q = 0 q End up with the same diﬀerential equaiton q + ω 2 q = 0. ¨

Slide 48: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Canonical transformation approach Hamiltonian: p2 1 H= + mω 2 q 2 2m 2 Choose generating function: 1 F1 = mω 2 q 2 cot Q 2 Get the canonical transformation equations: ∂F1 p = = mωq cot Q ∂q ∂F1 1 1 P = − = mωq 2 2 ∂Q 2 sin Q

Slide 49: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Canonical transformation approach (con’d) Canonical transformation equations from last page: ∂F1 p = = mωq cot Q ∂q ∂F1 1 1 P = − = mωq 2 2 ∂Q 2 sin Q Solve for (q, p) in terms of (Q, P ): 2P q = sin Q √ mω p = 2mωP cos Q

Slide 50: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Canonical transformation approach (con’d) (q, p) from last page: 2P q = sin Q √ mω p = 2mωP cos Q Get K: ∂F1 K = H+ =H ∂t p2 1 = + mω 2 q 2 2m 2 = ωP cos2 Q + ωP sin2 Q = ωP

Slide 51: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Canonical transformation approach (con’d) K = ωP Apparently Q is a cyclic coordinate of K, so P is a constant and can be written as P = E . ω The canonical equation: ˙ ∂K Q= =ω ∂P Thus, Q = ωt + φ

Slide 52: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Canonical transformation approach (con’d) Solution in (Q, P ): Q = ωt + φ E P = ω Get the solution in (q, p): 2P 2E q = sin Q = sin(ωt + φ) √ mω mω 2 √ p = 2mωP cos Q = 2mE cos(ωt + φ) No need to solve 2nd-order diﬀerential equation.

Slide 54: Basic facts L’s eqns H’s eqns Conservations Canonical xforms Oscillator Summary Summary F=ma D'Alembert's Lagrange's principle equations Principle of virtual work Hamilton's Hamilton's principle equations Statistical mechanics Quantum Hamilton-Jacobi Canonical mechanics theory transformations

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