Hamilton-Jacobi Theory: Finding the Best Canonical Transformation + Examples | Lecture 9

Lecture 9, course on Hamiltonian and nonlinear dynamics. Hamilton-Jacobi theory for finding the best canonical transformation to solve the Hamilton's equations. Hamilton-Jacobi equation, a partial differential equation for the Hamilton's principal function S with n+1 variables and n parameters. Common simplification if original H is independent of time, solving for W, Hamilton's characteristic function. Geometric interpretation: solutions in original coordinates map to equilibria in the Hamilton-Jacobi coordinates. S is also the action integral. Examples: simple harmonic oscillator and central force problem, e.g., Kepler's 2-body problem.

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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)

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📚Attitude Dynamics and Control

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► Chapters
0:00 Hamilton-Jacobi theory introduction
4:18 Every point in phase space is an equilibrium point
5:51 Derivation of Hamilton-Jacobi equation
13:06 Example: Hamilton-Jacobi for simple harmonic oscillator
31:00 Simplification: if Hamiltonian is time-independent
34:23 Hamilton's Principal function S is the action integral
39:00 Example: Hamilton-Jacobi for Kepler problem
50:07 Simplification: if Hamiltonian is separable

► Class notes in PDF form

► in OneNote form

► See the entire playlist for this online course:
Advanced Dynamics - Hamiltonian Systems and Nonlinear Dynamics

This course gives the student advanced theoretical and semi-analytical tools for analysis of dynamical systems, particularly mechanical systems (e.g., particles, rigid bodies, continuum systems). We discuss methods for writing equations of motion and the mathematical structure they represent at a more sophisticated level than previous engineering dynamics courses. We consider the sets of possible motion of mechanical systems (trajectories in phase space), which leads to topics of Hamiltonian systems (canonical and non-canonical), nonlinear dynamics, periodic & quasi-periodic orbits, driven nonlinear oscillators, resonance, stability / instability, invariant manifolds, energy surfaces, chaos, Poisson brackets, basins of attraction, etc.

► Background on Lagrangian systems, which have their own lecture series, 'Analytical Dynamics'

► Continuation of this course on a related topic
Center manifolds, normal forms, and bifurcations

► A simple introductory course on Nonlinear Dynamics and Chaos

► References
The class will largely be based on the instructor’s notes.
In addition, references are:
A Student’s Guide to Lagrangians and Hamiltonians by Hamill
Analytical Dynamics by Hand & Finch
Classical Dynamics: A Contemporary Approach by José & Saletan
Classical Mechanics, 3rd Edition by Goldstein, Poole, & Safko

Lecture 2021-07-15

action angle cyclic variables in classical mechanics statistical physics quasiperiodic online course Hamilton Hamilton-Jacobi theory three-body problem orbital mechanics Symplectic Geometry topology

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[arnabdasphysics]

The best introductory level discourse I ever came across!

[meetghelani]

i've gone through about 1/2 of the length of this lecture and it's beautiful. Thanks for posting it.

[byrongibby]

This series and the series on Lagrangian and 3D Rigid Body Dynamics have been invaluable to me, thank you very much professor!

[giuseppefasanella]

Thanks for this super clear explanation of HJ. When I was graduating N year ago, this was a question during my exam and I hadn't the faintest idea on what it was :) Now I finally understood it, although it seems more of a beautiful thing rather than a useful one. In fact, I actually never used HJ in any practical application.

[Eta_Carinae__]

I think it makes sense that the Hamilton-Jacobi theorem is nonunique, given the dependence of the generating function of the Lagrangian, which is also nonunique. Several Lagrangians would imply several generating functions. It's probably also worth keeping in mind that all of this business around the constraints for generating functions being canonical to begin with is built on a specific example of a nonunique Lagrangian where the function is added to the orginal Lagrangian and is of the form dF/dt. Maybe there are other functions that could work as canonical, which will have differing constraints, and then perhaps the constraints would then be different, and then the generating function that gives us back equillibrium points along all the points in ${p} \times {q}$ could be something other than the action...

[henk_iii]

I'm glad to have you in my life - Thank you!

[guidogallopyn]

Very clear and enjoyable presentation. Note that in "Tensors, Differential Forms and Variational Principles; David Lovelock and Hanno Rund" (chapter 6 again coincidentally) provide a different solution to the HJ eq for the harmonic oscillator namely S(t, q) = -1/2 w q^2 tan (w t + Phi) with H(q, p) = 1/2 p^2 +1/2 w q^2. The difference is attributed to the fact that the equation in the video entails a choice of the integration parameter alpha which corresponds directly to total energy E and as such restricts oneself to a particular member of the one parameter (alpha) family of possible extremal curves (determined by initial conditions). As such the S(q, alpha, t) only reflects the value of S on one particular extremal, where S(t, q) has the correct value in the whole of R2, for all extremals, without the need to parameterize. On the extremal corresponding to specific initial conditions it is easily verified that the solution from Lovelock and Rund equals the solution in this video. Note also the comment in Lovelock and Rund that S should be a function of S(t, q) and should not be represented as a function S(t, q, parameters) where parameters merely serve to label individual members of the family of extremals.

[vinbo]

So the "independent Hamiltonian example" at 52:50, is not necessarily totally decoupled, since V_theta term has r, the last term has r and phi, the "feeling like decouped in the sequencial solving procedure" can also be seen as independent, right?

[mygreneyhyportnex]

at 47:00 the integration should be with respect to dr (not dq)

[luisgarabito]

Amazing video! This helped understand part of Schrödingers paper for qm mechanics

[BigMoneyPauper]

I am curious about the connection between the action integral and Hamilton's principle function. If I was doing the functional derivative of the action I am treating the action integral as a map between a set of possible paths and the real numbers, whereas here Hamilton's principle function is a map from the R^d to R. How can I reconcile these two ideas? I am also curious as well, for a given Hamiltonian or initial conditions on the motion, does that imply boundary conditions/initial conditions on the solution to the PDE given by the HJE?

[jiaqigan]

Thank you professor Rose, your video helped me a lot!

[user_]

Much better explanation than Landau. Thank you.

[Eta_Carinae__]

If the GF is the action, and the path taken is the one s.t. dS/dt = 0, then according to HJ: S = W - tE
=> E = W/t - S/t, for the interval t of interest. Taking differential intervals then, wouldn't:
=> E = dW/dt - dS/dt = dW/dt
Meaning that W would essentially be a sum of total energies kind of term?

EDIT: in the time-independent case ofc.

[petermetric]

I do not see how S2(t) disappeared on the left hand side (18 mins in). It looks like they should be there but then divide both sides by S2 squared, and then set to constant.

[schmetterling]

Finding the best transformation for what purpose? There are only two handfuls of integrable Hamiltonians and they have all been worked out a long time ago. Is this class working towards perturbation theory?

[sudiptoborun]

This is the theory which was used by Erwin Schrodinger to solve H-atom as an eigenvalue problem.