1. State Hamilton's Principle. Starting from Hamilton's Principle derive the Euler-Lagrange Equations. Generalize the Euler-Lagrange equations for N independent coordinates. 15 Points
Added by Joseph S.
Close
Step 1
Hamilton's Principle states that the path taken by a system between two points in time is the one that minimizes the action integral, which is defined as the integral of the Lagrangian over time. Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 87 other Physics 103 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Hamilton's principle states that the dynamics of a physical system is determined by a variational problem based on the Lagrangian function. The Lagrangian function can often be found in the form L = K - V, where K is the kinetic energy and V is the potential energy. Consider a one-dimensional oscillator moving in potential V(x), x ∈ R. If the oscillator has mass m, then the kinetic energy is K = 1/2mv², where v is the velocity of the oscillator. The Lagrangian is L(m,x) = 1/2m(dx/dt)² - V(x), where the notation dx/dt is used. Hamilton's principle states that the equation of motion for x(t) is obtained by minimizing the integral of L(m,x)dt. By using the Euler-Lagrange equation, show that the equation of motion is m(d²x/dt²) + V'(x) = 0, where V'(x) represents the derivative of V(x) with respect to x. (b) The quadratic potential V(x) = 1/2kx² gives the simple harmonic oscillator. Solve the equation of motion in this case, with the initial conditions x(0) = 1 and dx/dt(0) = 0. What is the frequency of the oscillations?
Sri K.
Madhur L.
Write down the Hamiltonian function and Hamilton's canonical equations for a simple Atwood machine.
Shaiju T.
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD