Write the Hamiltonian function and find Hamilton's canonical equations for the three-dimensional motion of a projectile in a uniform gravitational field with no air resistance. Show that these equations lead to the same equations of motion as found.

Complete each sentence by adding a personal pronoun that agrees with the antecedent. Underline the antecedent. Can you return the jacket and hat to ______ rightful owner?

Use the method of Lagrange multipliers to find the tensions in the two strings of the double Atwood machine.

Set up the Hamiltonian and Hamilton's equations for a projectile of mass m, moving in a vertical plane and subject to gravity but no air resistance. Use as your coordinates x measured horizontally and y measured vertically up. Comment on each of the four equations of motion.

Find the differential equations of motion of a projectile in a uniform gravitational field without air resistance.

Underline the pronoun in parentheses that best completes each sentence. The teachers in charge of the video yearbook are Ms. Ramey and (he, him).

Underline the verb in parentheses that best completes the sentence. Berries (is, are) a mainstay in the diets of many birds.

Work earlier problem by using the method of Lagrange multipliers. (a) Show that the acceleration of the ball is $\frac{5}{7}$g. (b) Find the tension in the string.

The method of Lagrange multipliers works perfectly well with non-Cartesian coordinates. Consider a mass m that hangs from a string, the other end of which is wound several times around a wheel (radius R, moment of inertia I) mounted on a frictionless horizontal axle. Use as coordinates for the mass and the wheel x, the distance fallen by the mass, and φ he angle through which the wheel has turned (both measured from some convenient reference position). Write down the modified Lagrange equations for these two variables and solve them (together with the constraint equation) for ẍ and φ̈ and the Lagrange multiplier. Write down Newton's second law for the mass and wheel, and use them to check your answers for ẍ and φ̈. Show that λ∂f/∂x is indeed the tension force on the mass. Comment on the quantity λ∂f/∂φ.

Calculate the product: (x - 7)(x + 8)

(a) Suppose that f is continuous on (a, b) and $\lim _{x \rightarrow a^{+}} f(x)=\lim _{x \rightarrow b^{-}} f(x)=\infty$. Prove that f has a minimum on all of (a, b). (b) Prove the corresponding result when $a=-\infty$ and/or $b=\infty$.

Explain how you decide whether a formula is explicit or recursive.

A particle is free to slide along a smooth cycloidal trough whose surface is given by the parametric equations

$$\begin{array}{l}{x=\frac{a}{4}(2 \theta+\sin 2 \theta)} \\ {y=\frac{a}{4}(1-\cos 2 \theta)}\end{array}$$

where $0 \leq \theta \leq \pi$ and a is a constant. Find the Lagrangian function and the equation of motion of the particle.

Underline the verb in parentheses that best completes the sentence. Concrete lions (stands, stand) guardian over the library entrance.