Find a function M

Question

Let $x(t)$ be a real valued vector. Can you find a function M such that

$\dot{M}=\frac{\text{d}M}{\text{d}t}=\dot{x}^T\dot{x}$.

I have tried

$M=\dot{x}^Tx, M=x^Tx$

and many more which don't work. I know that if $x(t)$ is continuous then so will

$\dot{M}=\dot{x}^T\dot{x}$

be and thus M will exist. But how do i find this M?

$\dot{x}$ continuous $\iff$ $x$ is $C^1$. – Martín-Blas Pérez Pinilla Mar 23 '19 at 17:00 — Martín-Blas Pérez Pinilla, Mar 23 '19 at 17:00

score 1 · Answer 1 · answered Mar 23 '19 at 16:59

1

Impossible a general formula even in dimension 1. In the case, you are asking for $$\int(x'(t))^2\,dt$$ and no such formula exists. In concrete cases maybe you can do the integral.

answered Mar 23 '19 at 16:59

Martín-Blas Pérez Pinilla

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Wouldn't this integral exist if x'(t) is continuous? – hola Mar 23 '19 at 17:01
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@hola, it will exist. But you are asking for a formula. – Martín-Blas Pérez Pinilla Mar 23 '19 at 17:03
@hola, see https://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral/202. – Martín-Blas Pérez Pinilla Mar 23 '19 at 17:33

Find a function M

1 Answers1