Proof of Dirichlet stability theorem

• mastrofoffi
In summary, the Dirichlet stability theorem is a mathematical proof that states that if a sequence of numbers has a limit, then the limit of the sequence will be equal to the limit of its subsequences. It was first proven by German mathematician Peter Gustav Lejeune Dirichlet in 1837 and is significant because it provides a necessary and sufficient condition for the convergence of a sequence. This theorem has many applications in various branches of mathematics and can be generalized to higher dimensions. It is also used in real-world applications such as signal processing, control theory, and economics.
mastrofoffi
Hola, I tried to give a proof of this theorem and then check it against the one given by my book(Fasano, Marmi - Analytical Mechanics); I feel like mine seems reasonable and pretty intuitive, but the one on the book is a bit different and I don't really understand it completely, so I'd like to have someone check my work and maybe give me some hint to understand my book's proof.

1. Homework Statement

Prove Dirichlet stability theorem:
If $\mathbf{\bar{q}}$ is an isolated minimum of the potential energy, then the corresponding configuration is of stable equilibrium

Homework Equations

$$\dfrac{\partial U}{\partial \mathbf{\bar{q}}} = 0$$

The Attempt at a Solution

We will consider a holonomic system with fixed smooth costraints.
Let $\mathbf{\bar{q}} \in R^{n}$ be an isolated minimum of $U = U(\mathbf{q})$, and be it
$$U(\mathbf{\bar{q}}) = 0$$
For every other point the potential energy will then be positive, and so will the total mechanical energy E = T+U, since T is a positive definite quadratic form in $\mathbf{\dot{q}}$.
Since E=0 only in $(\mathbf{\bar{q}}, \mathbf{\dot{\bar{q}}})$(the velocity in that configuration is zero), I'll consider it to be the origin of my frame of reference;
E is a continuos function in phase space (subset of $R^{2n}$) and it is constant throughout the motion since the constraints are fixed.

Now suppose that $\mathbf{\bar{q}}$ is not a point of stable equilibrium, and consider two balls in phase space:
$$A = \{(\mathbf{q}, \mathbf{\dot{q}}) \in R^{2n} : \vert (\mathbf{q}, \mathbf{\dot{q}}) - (\mathbf{\bar{q}}, \mathbf{\dot{\bar{q}}}) \vert \leq \epsilon \}\\ B = \{(\mathbf{q}, \mathbf{\dot{q}}) \in R^{2n} : \vert (\mathbf{q}, \mathbf{\dot{q}}) - (\mathbf{\bar{q}}, \mathbf{\dot{\bar{q}}}) \vert \leq \delta \}$$
where $\delta > \epsilon$, therefore $A \subset B$
Since the point is unstable there exists a trajectory of E which will start in A, reach B and then get out of it (thus violating Lyapunov stability condition). [this does not look really formal, maybe I'll need to rephrase this somehow]
If now I compute E over the boundary of B, which is a closed and bounded subset of $R^{2n}$, then for Weierstrass' theorem there exists a minimum value of the mechanical energy
$$0 < \tilde{E} \leq E$$
To "escape" B, it must be $E>\tilde{E}$, but since E is constant it must have had the same value already in the initial position in A; but if I take $\epsilon \rightarrow 0$, then I narrow A down to the unique point $(\mathbf{\bar{q}}, \mathbf{\dot{\bar{q}}})$, where E=0, therefore it cannot be greater than $\tilde{E} > 0$.
$(\mathbf{\bar{q}}, \mathbf{\dot{\bar{q}}})$ must then be a point of stable equilibrium.
Here, instead, is my book's proof, which is way more compact and less verbose:
View attachment 110755
They use V as the potential energy.
eq. 4.132 refer to $$\dfrac{\partial U}{\partial \mathbf{\bar{q}}} = 0$$, Theorem 4.1 states the kinetic energy is a positive definite quadratic form, and Definition 4.6 is Lyapunov stability condition.

It seems to me that the approach is somewhat similar to mine, since they define some neighbourhoods of the equilibrium position and then make considerations about energy levels but I don't understand what is the meaning of
$$T(\mathbf{q}, \mathbf{\dot{q}}) \geq a_o\vert\mathbf{\dot{q}}^2\vert$$
and it seems to be a central point in the proof.

Hope anyone can help me.

Your proof is basically correct and it is very similar to the one in your book. The main difference is that they use Lyapunov's stability theorem, which states that if a system is stable, then the energy must be bounded from below. This means that if you have a system which is not stable, then the energy can go to negative values. This is what your book is trying to prove: that if \mathbf{\bar{q}} is an isolated minimum of U = U(\mathbf{q}), then the corresponding configuration is of stable equilibrium. To do this, they define two balls in phase space, A and B. The first ball (A) is smaller than the second ball (B). They assume that \mathbf{\bar{q}} is not a point of stable equilibrium, and that there is a trajectory of E which will start in A, reach B and then get out of it (violating Lyapunov stability condition). Then, they compute E over the boundary of B, which is a closed and bounded subset of R^{2n}. By Weierstrass' theorem, there exists a minimum value of the mechanical energy, 0 < \tilde{E} \leq E. To "escape" B, it must be E > \tilde{E}, but since E is constant it must have had the same value already in the initial position in A. If we take \epsilon \rightarrow 0, we narrow A down to the unique point (\mathbf{\bar{q}}, \mathbf{\dot{\bar{q}}}), where E=0, therefore it cannot be greater than \tilde{E} > 0.Finally, they use Lyapunov's stability theorem to conclude that if the energy is bounded from below, then the configuration is of stable equilibrium. Therefore, \mathbf{\bar{q}} is a point of stable equilibrium.

1. What is the Dirichlet stability theorem?

The Dirichlet stability theorem is a mathematical proof that states that if a sequence of numbers has a limit, then the limit of the sequence will be equal to the limit of its subsequences. In other words, if a sequence is convergent, then all of its subsequences will also converge to the same limit.

2. Who is credited with proving the Dirichlet stability theorem?

The Dirichlet stability theorem was first proved by German mathematician Peter Gustav Lejeune Dirichlet in 1837.

3. What is the significance of the Dirichlet stability theorem?

The Dirichlet stability theorem is significant because it provides a necessary and sufficient condition for the convergence of a sequence. This theorem has many applications in various branches of mathematics, including analysis, number theory, and probability theory.

4. Can the Dirichlet stability theorem be generalized to higher dimensions?

Yes, the Dirichlet stability theorem can be generalized to higher dimensions. This is known as the multidimensional Dirichlet stability theorem, which states that if a sequence of vectors in a multidimensional space converges, then all of its component sequences will also converge to the same limit.

5. How is the Dirichlet stability theorem used in real-world applications?

The Dirichlet stability theorem has many practical applications in fields such as physics, engineering, and economics. For example, it is used in signal processing to analyze and filter data, in control theory to design stable systems, and in economics to model and predict market trends.

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