Translate Noether's theorem into Hamiltonian mechanics. That is, define a symmetry for Hamiltonian mechanics (by translating the Lagrangian-definition), and prove that symmetries give rise to conserved observables.

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Noether's theorem, in its original version, applies to theories described by a Lagrangian. There is also a version which applies to theories described by a Hamiltonian. Suppose there is a particle moving on a line with Lagrangian L(q,q'), where q is its position and q' = dq/dt is its velocity. The momentum of this particle is defined to be p = dL/dq'.
The force on it is defined to be F = ...

At 2:00 pm a car's speedometer reads 30 mi/h. At 2:10 pm it reads 50 mi/h. Use the mean value theorem to show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h^2. Please show line by line work and be as clear as possible.

Using the Fundamental Theorem of Calculus I need to find the solution of the following problems. Can you explain how?
Please see the attached file for the fully formatted problems.

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

Set up a short problem related to your work environment to calculate the probability(ies) of an event happening. Then use Bayes' Theorem to revise the probability. Show all your work.

Find the Inverse LaPlace Transform using different methods described in the attachment.
To see the description of the problem in its true format, please download the attached question file.