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Consider a particle of mass m moving in two dimensions,
Chapter 13, Problem 13.10(choose chapter or problem)
Consider a particle of mass m moving in two dimensions, subject to a force \(\mathbf{F}=-k x \hat{\mathbf{x}}+K \hat{\mathbf{y}}\), where k and K are positive constants. Write down the Hamiltonian and Hamilton's equations, using x and y as generalized coordinates. Solve the latter and describe the motion.
Questions & Answers
QUESTION:
Consider a particle of mass m moving in two dimensions, subject to a force \(\mathbf{F}=-k x \hat{\mathbf{x}}+K \hat{\mathbf{y}}\), where k and K are positive constants. Write down the Hamiltonian and Hamilton's equations, using x and y as generalized coordinates. Solve the latter and describe the motion.
ANSWER:Step 1 of 4
Consider the given data as follows.
The force, \(\overrightarrow F = - kx\widehat x + K\widehat y\)
To find the potential \(\left( U \right)\), use the following equation.
\(\overrightarrow F = - \nabla U\)
Thus,
\({F_x} = - \frac{{\partial U}}{{\partial x}}\)
And
\({F_y} = - \frac{{\partial U}}{{\partial y}}\)
Therefore, the potential is,
\(U = - \int {{F_x}dx} - \int {{F_y}dy} \)
\( = - \left( { - k} \right)\int {xdx} - K\int {dy} \)
\( = \frac{{k{x^2}}}{2} – Ky\)
Now, kinetic energy can be defined as follows.
\(T = \frac{1}{2}m\left( {{{\dot x}^2} + {{\dot y}^2}} \right)\)