Hamilton-Jacobi equation - 2023 Penn Bowl

Question

This equation emerges as the low wavelength limit of the Schrödinger equation. For 10 points each:

[10h] Identify this nonlinear partial differential equation in classical mechanics. This equation is the result of a canonical transformation in terms of generalized position and momenta that are constants of motion.

ANSWER: Hamilton-Jacobi equation [reject “Jacobi-Hamilton equation”]

[10m] The Hamilton principle function in the Hamilton-Jacobi equation represents this quantity, the time integral of the Lagrangian. Hamilton’s principle states that the equation of motion of a system minimizes this quantity.

ANSWER: action

[10e] More general than Hamilton’s principle, D’Alembert’s principle states that the equation of motion minimizes the virtual type of this quantity. The rate of this quantity is power, and it is defined as force time distance.

ANSWER: work [or virtual work]

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Summary


2023 Penn Bowl (Harvard)	10/21/2023	Y	3	13.33	67%	67%	0%
2023 Penn Bowl (Mainsite)	10/21/2023	Y	7	21.43	100%	86%	29%
2023 Penn Bowl @ Waterloo	10/28/2023	N	4	17.50	100%	75%	0%
2023 Penn Bowl @ FSU	10/28/2023	N	2	15.00	100%	50%	0%
2023 Penn Bowl (Norcal)	10/28/2023	N	1	20.00	100%	100%	0%
2023 Penn Bowl (South Central)	10/28/2023	N	3	16.67	100%	67%	0%
2023 Penn Bowl (UK)	10/28/2023	N	5	20.00	100%	80%	20%

Data


Boston College	Brandeis	0	0	0	0
Brown	Harvard	0	10	10	20
MIT	Tufts	0	10	10	20
Columbia A	Swarthmore	0	10	10	20
Cornell B	JHU A	10	10	10	30
Cornell A	JHU B	0	10	10	20
PItt	NYU	0	10	10	20
Columbia B	RIT	0	10	10	20
John Jay A	Rutgers	10	10	10	30
UMD B	UMD A	0	0	10	10