Solving the two mass system with linear algebra LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= Here is the system of equations: JSFH QyQ+SSRlcTFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JS1JI3UxR0YlNiNJInRHRiVGLkYuLCZGKyEiJi1JI3UyR0YlRi0iIiMiIiI=

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print(); # input placeholder

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print(); # input placeholder

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First we'll just have Maple solve the system: QyQtSSdkc29sdmVHNiI2IzwkSSRlcTFHRiVJJGVxMkdGJSIiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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print(); # input placeholder

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Here is the coefficient matrix for the system: JSFH QyQ+SSJBRzYiLUknTWF0cml4R0YlNiM3JDckISImIiIjNyRGLCEiIyIiIg==

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print(); # input placeholder

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JSFH This matrix is symmetric, so it can be diagonalized and it has real eigenvalues. JSFH QyQ+SSJURzYiLUktRWlnZW52ZWN0b3JzR0YlNiNJIkFHRiUiIiI=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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JSFH Let's call P the matrix whose columns are independent eigenvectors of A. P will diagonalize the matrix A. JSFH QyQ+SSJQRzYiJkkiVEdGJTYjIiIjIiIi

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKV9cJXkn

U will be the column vector of the original functions we are solving for. QyQ+SSJVRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM3JDcjLUkjdTFHRiU2I0kidEdGJTcjLUkjdTJHRiVGMCIiIg==

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print(); # input placeholder

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V will be the column vector of the new variables. JSFH QyQ+SSJWRzYiLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHNiRGKUkoX3N5c2xpYkdGJTYkKiRJIlBHRiUhIiJJIlVHRiUiIiI=

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print(); # input placeholder

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JSFH The sysyem of equations for the new functions V corresponds to the diagonal matrix D=P^(-1)AP. QyQ+SSRlcTNHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JS1JI3YxR0YlNiNJInRHRiVGLkYuLCRGKyEiIiIiIg==

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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JSFH Let's check this by solving the system with some initial conditions. First we'll just have Maple solve for us. JSFH QyQ+SSNTMUc2Ii1JJ2Rzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkNyhJJGVxMUdGJUkkZXEyR0YlLy1JI3UxR0YlNiMiIiEiIiIvLUkjdTJHRiVGMiIiJC8tLUkiREdGKDYjRjFGMiMhIiIiIiMvLS1GPDYjRjdGMkY0PCQtRjE2I0kidEdGJS1GN0ZHRjQ=

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print(); # input placeholder

PCQvLUkjdTFHNiI2I0kidEdGJiwqKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IyomIiInIyIiIiIiI0YoRjRGNEYyRjMjISIjIiM6LUkkY29zR0YtRjAjRjciIiYtRixGJyMiIiQiIzUtRjpGJyMiIihGPC8tSSN1MkdGJkYnLCpGKiNGNEY4RjkjRjRGPEY9I0Y/RjxGQSMiIzlGPA==

JSFH Here are the solutions for u1 and u2 extracted as expressions. JSFH QyQ+SSNGMUc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQmSSNTMUdGJTYjIiIiLUkjdTFHRiU2I0kidEdGJUYt

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print(); # input placeholder

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print(); # input placeholder

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Now let's do it ourselves. First we get the initial conditions in ters of the new variables V. Remember that V=P^(-1)*U. First we compute the column vector giving v1(0) and v2(0), and then we compute the column vector giving v1'(0), v2'(0). JSFH JSFH QyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQqJEkiUEdGKSEiIi1JJ01hdHJpeEdGKzYjNyQ3IyIiIjcjIiIkRjY=

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print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKSVbNGcn

QyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQqJEkiUEdGKSEiIi1JJ01hdHJpeEdGKzYjNyQ3IyNGMCIiIzcjIiIiRjk=

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print(); # input placeholder

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Now we solve the diagonal system with these initial conditions for the new variables. JSFH QyQ+SSNNMUc2Ii1JJ2Rzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjNyVJJGVxM0dGJS8tSSN2MUdGJTYjIiIhIyIjOSIiJi8tLUkiREdGKDYjRjBGMSMiIiRGNSIiIg==

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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JSFH Finally, we convert back to the original functions u1, u2 and see that we get the same solution. JSFH QyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiRJIlBHRiktSSdNYXRyaXhHRis2IzckNyNJI3YxR0YpNyNJI3YyR0YpIiIi

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print(); # input placeholder

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Here is a plot of the solutions. JSFH QyQtSSVwbG90RzYiNiQ3JEkjRjFHRiVJI0YyR0YlL0kidEdGJTsiIiEiIz8iIiI=

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