Problem B.3 Simple harmonic oscillator. We study the simple harmonic oscillator of an object of mass m in one dimension. iv) Given the force F(x) = -kx, where k> 0 is a constant, discuss the corresponding potential V(x). v) Find the corresponding Lagrangian. vi) Derive the Euler-Lagrange equation(s) and show the equivalence with Newton's equa- tion(s). vii) Solve the equation of motions, i.e. find x(t) and (t), and discuss the choice of initial conditions. viii) Discuss the conservation of energy. ix) What happens if F(x) = kx, with k > 0 ?
Problem B.3 Simple harmonic oscillator. We study the simple harmonic oscillator of an object of mass m in one dimension. iv) Given the force F(x) = -kx, where k> 0 is a constant, discuss the corresponding potential V(x). v) Find the corresponding Lagrangian. vi) Derive the Euler-Lagrange equation(s) and show the equivalence with Newton's equa- tion(s). vii) Solve the equation of motions, i.e. find x(t) and (t), and discuss the choice of initial conditions. viii) Discuss the conservation of energy. ix) What happens if F(x) = kx, with k > 0 ?
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Transcribed Image Text:Problem B.3 Simple harmonic oscillator.
We study the simple harmonic oscillator of an object of mass m in one dimension.
iv) Given the force F(x) = -kx, where k> 0 is a constant, discuss the corresponding
potential V(x).
v) Find the corresponding Lagrangian.
vi) Derive the Euler-Lagrange equation(s) and show the equivalence with Newton's equa-
tion(s).
vii) Solve the equation of motions, i.e. find x(t) and (t), and discuss the choice of initial
conditions.
viii) Discuss the conservation of energy.
ix)
What happens if F(x) = kx, with k > 0 ?
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