Question

2) Consider a Lagrangian of the form $L = \frac{m}{2}(\dot{x}^2 - \omega^2 x^2)e^{\gamma t}$ where the particle of mass $m$ moves in one dimension. Assume all constants are positive. a) Find the equations of motion. b) Interpret the equations by giving a physical interpretation of the forces acting on the particle. c) Find the canonical momentum and construct the Hamiltonian. Is this Hamiltonian a constant of the motion? d) Introduce a transformation of the form $s = xe^{\gamma t/2}$. What is the Lagrangian in terms of $s$? e) Again find the canonical momentum and construct the Hamiltonian. Is this Hamiltonian a constant of the motion? f) Are there any constants of motion for the system?

          2) Consider a Lagrangian of the form
$L = \frac{m}{2}(\dot{x}^2 - \omega^2 x^2)e^{\gamma t}$
where the particle of mass $m$ moves in one dimension. Assume all constants are positive.
a) Find the equations of motion.
b) Interpret the equations by giving a physical interpretation of the forces acting on the
particle.
c) Find the canonical momentum and construct the Hamiltonian. Is this Hamiltonian a
constant of the motion?
d) Introduce a transformation of the form $s = xe^{\gamma t/2}$. What is the Lagrangian in terms of $s$?
e) Again find the canonical momentum and construct the Hamiltonian. Is this Hamiltonian
a constant of the motion?
f) Are there any constants of motion for the system?

2) Consider a Lagrangian of the form
L = (m)/(2)(ẋ^2 - ω^2 x^2)e^γ t
where the particle of mass m moves in one dimension. Assume all constants are positive.
a) Find the equations of motion.
b) Interpret the equations by giving a physical interpretation of the forces acting on the
particle.
c) Find the canonical momentum and construct the Hamiltonian. Is this Hamiltonian a
constant of the motion?
d) Introduce a transformation of the form s = xe^γ t/2. What is the Lagrangian in terms of s?
e) Again find the canonical momentum and construct the Hamiltonian. Is this Hamiltonian
a constant of the motion?
f) Are there any constants of motion for the system?

Added by Samantha J.

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