For the [[Harmonic Oscillator]] in three dimensions we write [vectors](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vector) in place of the variables and we obtain component-wise solutions to the equations of motion. This follows from the [one-dimensional model](1D%20Harmonic%20Oscillator.md) and is modeled with vectors in [Euclidean space](Euclidean%20space.md). # Lagrangian The [Lagrangian](Harmonic%20Oscillator.md#Lagrangian) is as follows $\mathcal{L}(\mathbf{q},\dot{\mathbf{q}})=\frac{1}{2}m\dot{\mathbf{q}}^2-\frac{k}{2}\mathbf{q}^2$ # Hamiltonian The [Hamiltonian](Harmonic%20Oscillator.md#Hamiltonian) is obtained from the [Lagrangian](Harmonic%20Oscillator.md#Lagrangian) through a [Legendre transform](Hamiltonians.md#Legendre%20transform). This gives $\mathcal{H}(\mathbf{q},\mathbf{p})=\frac{1}{2}\frac{\mathbf{p}^2}{m}+\frac{k}{2}\mathbf{q}^2$ # Equations of motion ## Lagrange Equation of motion Where the [Euler-Lagrange equation](Euler-Lagrange%20equation.md) of motion, a [[2nd order ODE]], is $\ddot{\mathbf{q}}=-\frac{k}{m}\mathbf{q} = -\omega^2 \mathbf{q}$ where $\omega^2=\frac{k}{m}$ is the [[Frequency]]. ### Solution The solution is a classical [superposition](Superposition%20principle.md) of $\sin$ and $\cos$ functions: $\mathbf{q}(t) = \begin{pmatrix}A_1\cos{\omega t}+B_1\sin{\omega t}\\ A_2\cos{\omega t}+B_2\sin{\omega t}\\ A_3\cos{\omega t}+B_3\sin{\omega t}\end{pmatrix}$ ## Hamilton Equations of Motion The [Hamilton equations of motion](Hamilton%20equations%20of%20motion.md#In%203%20spatial%20dimensions) in terms of [[conjugate momentum]] are $\dot{\mathbf{q}}=\frac{\mathbf{p}}{m}\;\;\;\;\mbox{and}\;\;\;\;\dot{\mathbf{p}}=-k\mathbf{q}$ where here we again simply replace the variables with vectors. ### solution The pair of solutions is as follows where we can treat these equations as [[2nd order ODE]]s. $(\mathbf{q}(t),\mathbf{p}(t))=\Bigg(\begin{pmatrix}q_{i1}\cos{\omega t}+q_{s1}\sin{\omega t}\\ q_{i2}\cos{\omega t}+q_{s2}\sin{\omega t}\\ q_{i3}\cos{\omega t}+q_{s3}\sin{\omega t}\end{pmatrix}, \begin{pmatrix}-\omega m q_{i1}\cos{\omega t}+\omega m q_{s1}\sin{\omega t}\\ -\omega m q_{i2}\cos{\omega t}+\omega m q_{s2}\sin{\omega t}\\ -\omega m q_{i3}\cos{\omega t}+\omega m q_{s3}\sin{\omega t}\end{pmatrix}\Bigg)$ #Mechanics #Mechanics/WaveMechanics #PhysicalExamples/PhysicalExamplesInMechanics