Consider a mass m constrained to move in a vertical line under the influence of gravity. Using the coordinate x measured vertically down from a convenient origin O, write down the Lagrangian L and find the generalized momentum p = ∂xL/∂ẋ. Find the Hamiltonian H as a function of x and p, and write down Hamilton's equations of motion. It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)

Design the class linetype to implement a line so that the class lineType:

1. Overloads the stream insertion operator, $<<$, for easy output.
2. Overloads the stream extraction operator, $>>$, for easy intput. (The line $a x+b y=c$ is input as $(a, b, c)$.
3. Overloads the assignment operator to copy a line into another line.
4. Overloads the unary operator $+$, as a member function, so that it returns true if a line is vertical; false otherwise.
5. Overloads the unary operator -, as a member function, so that it returns true if a line is horizontal; false otherwise.
6. Overloads the operator $==$, as a member function, so that it returns true if two lines are equal; false otherwise.
7. Overloads the operator $| |$, as a member function, so that it returns true if two lines are parallel; false otherwise.
8. Overloads the operator $\& \&$, as a member function, so that it returns true if two lines are perpendicular; false otherwise. Write a program to test your class.

Find the Lagrangian, the generalized momentum, and the Hamiltonian for a free particle (no forces at all) confined to move along the x axis. (Use x as your generalized coordinate.) Find and solve Hamilton's equations.

(a) Confirm that the kinetic energy operator, $- \left( \hbar ^ { 2 } / 2 m \right) \mathrm { d } ^ { 2 } / \mathrm { d } x ^ { 2 }$, is hermitian. (b) The operator corresponding to the angular momentum of a particle is $( \hbar / \mathrm { i } ) \mathrm { d } / \mathrm { d } \phi ,$ where $\phi$ is an angle. Is this operator hermitian?

Briefly describe how the && operator works.

Briefly describe how the and operator works.

The __________ operator is used to dynamically allocate memory.

The _______ operator is used to dynamically allocate memory.

Briefly describe how the or operator works.

Let $G$ be a graph. The line graph of $G$ is a new graph $L(G)$ whose vertices are the edges of $G$; two vertices of $L(G)$ are adjacent if, as edges of $G$, they share a common end point. In symbols:

$$V[L(G)]=E(G)\text{ and }E[L(G)]=\{e_1e_2:|e_1\cap e_2|=2\}$$

Prove or disprove the following statements about the relationship between a graph and its line graph : a. If $G$ is Eulerian, then $L(G)$ is also Eulerian. b. If $G$ has a Hamiltonian cycle, then $L(G)$ is Eulerian. (See Exercise 50.16 for the definition of a Hamiltonian cycle.) c. If $L(G)$ is Eulerian, then $G$ is also Eulerian. d. If $L(G)$ is Eulerian, then $G$ has a Hamiltonian cycle.

Briefly describe how a ternary operator works.

Using the fact that the radial momentum operator is given by $\hat{p}_r=-i \hbar \frac{1}{r} \frac{\partial}{\partial r} r$, calculate the commutator $\left[\hat{r}, \hat{p}_r\right]$ between the position operator, $\hat{r}$, and the radial momentum operator.

Determine whether the statement is true or false, and justify your answer. If there is a nonzero vector in the kernel of the matrix operator

$$T_A: R^n→R^n$$

then this operator is not one-to-one.

A ferry operator takes tourists to an island. The operator carries an average of 500 people per day for a round-trip fare of $20. The operator estimates that for each$1 increase in fare, 20 fewer people will take the trip. What fare will maximize the number of people taking the ferry?