# Questions tagged [canonical-transformation]

Use this tag for questions related to canonical transformations, which are changes of canonical coordinates that preserve the form of Hamilton's equations.

57 questions
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### Basis for that $T: \mathbb{R}^3 \to \mathbb{R}^3$ is in rational canonical form

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T(x,y,z) = (x+y+z, x+y+z, x+y+z)$ Find a basis for $T$ such that your matrix$(A_T)$ is in rational canonical form. I ...
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### Solving $u_{\epsilon \eta} = \frac{ u_\epsilon - u_\eta}{4 (\epsilon - \eta) } .$

I'm trying to solve the following PDE $$y^2 u_{xx} - u_{yy} = 0.$$ I've found its canonical form as $$u_{\epsilon \eta} = \frac{ u_\epsilon - u_\eta}{4 (\epsilon - \eta) } .$$ However, I'm having ...
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### Using the characteristics to get the canonical form of a pde

I've been asked to consider this parabolic equation. $3\frac{∂^2u}{∂x^2} + 6\frac{∂^2u}{∂x∂y} +3\frac{∂^2u}{∂y^2} - \frac{∂u}{∂x} - 4\frac{∂u}{∂y} + u = 0$ I calculated the characteristic ...
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### Sample mean and sample covariance canonical forms

Let $(x_1, x_2,\dotsc, x_n)$ be a sequence of vectors: ($\forall i=1,\dots,n$) \begin{pmatrix} x_i^1 \\ x_i^2\\ \vdots\\ x_i^m \end{pmatrix} In statistics, one often has to compute the ...
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### Understanding Y U Egorov's theorem

I was reading the paper "Hörmander, L., Fourier integral operators. I, Acta Math. 127, 79-183 (1971). ZBL0212.46601." and on page 3, there is a statement saying the following thing: Can someone ...
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### Minimal polynomial for any power of Jordan block is same as the minimal polynomial of the Jordan block.

Let $J$ be the $n \times n$ Jordan block corresponding to the eigen value $1$. For any natural number $r$ is it true that the minimal polynomial for $J^r$ is $(X-1)^n$ ? Another way to think about it ...
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### Reduce the equation to canonical form. $u_{xx}+2ayu_{xy}+e^{2x}u_{yy}-u=0$

Consider the equation $$u_{xx}+2ayu_{xy}+e^{2x}u_{yy}-u=0$$ where $a$ is a real constant. Determine the value(s) of $a$ when this equation is elliptic everywhere in the $xy$-plane in which case find ...
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### Canonical equations (PDE)

Consider the equation $$u_{xx} +2ay u_{xy} +e^{2x} u_{yy} −u = 0,$$ where $a$ is a real constant. Determine the value(s) of a when this equation is elliptic everywhere in the $xy$-plane in which case ...
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### Find a General solution to the equation, leading him to the canonical form.

enter image description hereHere it is my question. uxx + 10uxy + 25uyy + ux + 5uy = 0 I recognize that it is a hyperbolic PDE. . I don't know how to proceed further to get the canonical form. can ...
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### Deriving the equations defining a symplectic map

The objective is to derive $$p_k=\frac{\partial\phi}{\partial q_k},\quad Q_k=\frac{\partial\phi}{\partial P_k},\quad \tilde{H}=H+\frac{\partial\phi}{\partial t},$$ where $\phi=\phi(t,q,P)$. I ...
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