#### Problem

A mass $m$ is situated at the point $(x, y)$ in the plane; a mass $M$ is situated at the point $(X, Y)$. The two masses are joined by a linear spring with spring constant $k$ but are otherwise unconstrained. Assume the spring to have zero length when relaxed.

Write the equations of motion for this system as canonical Euler-Lagrange equations and as canonical Hamiltonian equations. Show explicitly that the quantity $p_x + p_y + p_X + p_Y$ Poisson commutes with the total energy $H$ and is thus conserved.

#### Solution

The first step is to determine the the Lagrangian equation, $L = T - V$, where $T$ is the total Kinetic Energy and $V$ is the total Potential Energy. For this system:

(1)Using the Lagrangian, the canonical Euler-Lagrange equations can be developed using:

(4)where $q^i$ is used to refer to the generalized coordinates. Because there are four degrees of freedom for this system $\left( x, y, X, Y\right)$, there will be four Euler-Lagrange equations. For this problem, the four equations are:

(5)Now for the canonical Hamiltonian equations. First, define the momentum conjugate to $q^i$ as:

(9)and the Hamiltonian, $H$, as:

(10)Applied to this problem, these equations are:

(11)Using these equations, the canonical Hamiltonian equations are defined as:

(16)Because there are four degrees of freedom, there will be eight Hamiltonian equations in canonical form. For this problem, those equations are:

(18)Hamilton's equations in canonical form are equivalent to the statement that $\dot f = \left\{f,H \right\}$ for all functions $f \left(q, p\right)$. If $\dot f = \left\{f,H\right\}=0$, then f is a conserved quantity. In order to check if the function $f = p_x+p_y+p_X+p_Y$ Poisson commutes with the total energy $H$, $\dot f$ was determined and was found to be:

(26)Because $\dot f=0$, the quantity $f =p_x+p_y+p_X+p_Y$ is conserved.

Scott agrees, having corrected a few typographical errors — although the idea was to show that $\frac{d}{dt} (p_x+p_y+p_X+p_Y)$ equals zero by explicitly computing $\{p_x+p_y+p_X+p_Y, H\}$ to be zero, and not the other way around…

#### Discussion

Evidently I can't read/count… I guess I originally did this as problem 4, when it was supposed to be problem 3. I know I did the problem I claimed, I just mislabeled it. Thanks avakis2 for moving it for me :)