Lagrangians and Hamiltonians Assignment help
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Lagrangians and Hamiltonians Assignment
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Problem Set-Lagrangians and Hamiltonians
1. A box of mass m is on an inclined plane as shown in the below picture. Ignore friction.
a. Write the Lagrangian in terms of x and ݔሶ . (Hint: Write the height of the box off the
ground in terms of x and ߠ.)
b. Write the Hamiltonian in terms of p and x.
c. Find the differential equation of motion x(t) from both the Lagrangian and the
Hamiltonian. Your result should be the same using both methods.
2. A ball of mass m is suspended from the ceiling by a spring (spring constant k). It does not
swing side to side. See below picture.
y
a. Write the Langrangian in terms of y and ݕሶ .
b. Write the Hamiltonian in terms of p and y.
c. Find the differential equation of motion y(t) from both the Lagrangian and the
Hamiltonian. Your result should be the same using both methods.
ଵ
4. For the 2D harmonic oscillator with mass m in potential ܸ = ଶ ݇ሺ ݔଶ + ݕଶ ሻ
a. Write the Lagrangian in terms of x, ݔሶ , y, and ݕሶ .
b. Write the Hamiltonian in terms of px, py, x, y.
c. Find the differential equation of motion x(t) from both the Lagrangian and the
Hamiltonian. Your result should be the same using both methods.
ଵ
5. For the 2D harmonic oscillator with mass m in potential ܸ = ଶ ݇ ݎଶ .
a. Write the Lagrangian in terms of r, ݎሶ , θ, and ߠሶ .
b. Write the Hamiltonian in terms of pr, pθ, r, θ.
c. Find the differential equation of motion x(t) from both the Lagrangian and the
Hamiltonian. Your result should be the same using both methods.
6. A pendulum is attached to a block of mass M. The block is moving vertically. The length of
the pendulum is ℓ and the bob has mass m. (Hint: Look at Section 10.3)
a. Write the kinetic and potential energies of the system in terms of Cartesian
coordinates.
b. Write equations of constraint.
c. Use the constraints to transform the coordinates. How many degrees of freedom
are there?
d. Explain whether the new expressions for kinetic and potential energy make sense
conceptually.
e. Write the Lagrangian in terms of generalized coordinates.
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In a two-page paper, identify the physics principles contained within the following scenario. Explain how the ...
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3 pages
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The melting point of the synthesized product was 151.2-152.6oC. The theoretical melting point of (E, E)-1,4-diphenyl-1,3-b ...
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ERAU Meteorology Weather & Hurricanes Discussion
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Flager College Electricity Magnetism and Light Discussion
Instructions
In a two-page paper, identify the physics principles contained within the following scenario. Explain how the ...
Flager College Electricity Magnetism and Light Discussion
Instructions
In a two-page paper, identify the physics principles contained within the following scenario. Explain how these principals connect to electricity, magnetism, or light in modern applications in physics. Finally, consider the different concepts in which James Clerk Maxwell did research, and give an example of one of these concepts in use in your life. For instance, Maxwell's research led to the development of radio waves. If you listen to a radio, then you are using Maxwell's research. Provide another example from your own experience, compare, and contrast your scenario to the provided scenario below.
Scenario
Mandy took a trip to Rome, Italy. Once landed and inside the terminal, she turned her cell phone back on, but it was not charged. She found a charging station with a USB adaptor port. The USB was universal, providing 5 volts in any country you were in, and a small red LED next to her phone's screen told her the phone was successfully charging. This USB port seemed to have very high amperage, meaning it charged her phone quickly. She was aware, though, that almost all of Italy's electricity was generated by burning fossil fuels, and thus she was determined after this to use the portable solar charger she had bought rather than wall electricity.
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