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Existence, uniqueness and conservation of energy for a one-dimensional conservative model: phase curves lie in energy level sets and the velocity is determined by the position and energy up to sign

Physical scope

The basic dynamical content of a conservative one-dimensional model: every datum has exactly one maximal motion; the mechanical energy m x-dot^2/2 + U(x) is constant along every motion; a motion of initial energy E_0 keeps its position in the sublevel set {U <= E_0}, and where U is strictly below E_0 its velocity is one of the two square-root values determined by the position and E_0; every non-turning phase point of the motion lies on the energy level set of E_0; and the constant motions are exactly the rest motions at equilibria. The result is a conditional deduction from the declared potential presentation; it asserts no periodicity, no boundedness of the motion, no asymptotic behaviour and no measured value.

Statement

Adopt a conservative one-dimensional model: the class of clause (a) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria with a time- and velocity-independent force law presented through a potential, that is $F (x) = - U^{'} (x) (x \in Q)$ for a function $U : Q \to R$ of class $C^{2}$ (so the residual force of clause (d) of that definition vanishes identically and the model is a global potential model). Write $E (x, v) = \frac{1}{2} m v^{2} + U (x)$ for the energy function of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points clause (b). Then:

Existence and uniqueness. Every datum $(t_{0}, x_{0}, v_{0})$ with $t_{0} \in I_{0}$ , $x_{0} \in Q$ , $v_{0} \in R$ has exactly one maximal motion: a motion $x : J_{\max} \to Q$ with $x (t_{0}) = x_{0}$ and $x^{'} (t_{0}) = v_{0}$ that has no extension to a motion on a strictly larger interval. Its interval $J_{\max}$ is open, contains $t_{0}$ and is contained in $I_{0}$ , and every motion of the model with the same datum is the restriction of $x$ to a subinterval.
Energy conservation. For every motion $x : J \to Q$ the function $t \mapsto E (x (t), x^{'} (t))$ is constant on $J$ .
Level-set containment and velocity determination. Let $x : J \to Q$ be a motion with datum $(t_{0}, x_{0}, v_{0})$ and let $E_{0} : = E (x_{0}, v_{0})$ . Then $U (x (t)) \leq E_{0}$ for every $t \in J$ ; if $U (x (t)) < E_{0}$ then $x^{'} (t) = \pm \sqrt{\frac{2 (E_{0} - U (x (t)))}{m}},$ the two signs giving the only two possible values; and $x^{'} (t) = 0$ holds exactly when $U (x (t)) = E_{0}$ . In particular the phase curve of $x$ is contained in the energy level set $Σ_{E_{0}}$ of clause (b) of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points.
Rest motions. A point $x_{*} \in Q$ is an equilibrium of the model if and only if $U^{'} (x_{*}) = 0$ ; in that case the constant curve $x \equiv x_{*}$ is the maximal motion with datum $(t_{0}, x_{*}, 0)$ , and conversely the maximal motion with datum $(t_{0}, x_{*}, 0)$ is constant exactly when $U^{'} (x_{*}) = 0$ .

The statement is conditional on the declared potential $U$ : replacing $U$ by $U + c$ with a constant $c$ changes $E$ on the whole phase plane by $c$ and leaves clauses 1, 3 and 4 unchanged. No claim of periodicity, boundedness of the position or of the velocity, or approach to an equilibrium is made.

Facts & Assumptions

Given: The conservative one-dimensional model $(m, I_{0}, Q, F)$ with $F = - U^{'}$ , its phase system $G_{F} (x, v) = (v, - U^{'} (x) / m)$ , its energy $E$ , and a datum $(t_{0}, x_{0}, v_{0})$ .

[L1]

Model vocabulary (The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria clauses (a)-(e)): $Q$ is nonempty open, $F$ is continuous and locally Lipschitz in the state variable; motions are the $C^{2}$ curves $x : J \to Q$ with $m x^{''} = F (x)$ on an interval $J \subseteq I_{0}$ with nonempty interior; $x$ is a motion if and only if $t \mapsto (x (t), x^{'} (t))$ is a solution of $z^{'} = G_{F} (z)$ ; and $x_{*}$ is an equilibrium if and only if $F (x_{*}) = 0$ , equivalently $U^{'} (x_{*}) = 0$ because $U$ is $C^{1}$ and $F = - U^{'}$ .

[L2]

Phase-field regularity: $G_{F} (x, v) = (v, - U^{'} (x) / m)$ is continuous on the open set $Q \times R$ , and it is locally Lipschitz in the state variable: the first component is linear, and $U^{'}$ is of class $C^{1}$ because $U$ is $C^{2}$ , hence locally Lipschitz on the compact coordinate subsets of the open set $Q$ by A $C^{1}$ map is locally Lipschitz on compact coordinate subsets with the identity charts; a locally Lipschitz map is locally Lipschitz in the sense of Local Lipschitz continuity in the state variable, locally uniform in time and parameters.

[L3]

Maximal solutions (Picard-Lindelöf local existence and uniqueness for first-order systems, Extensions, maximal solutions, maximal intervals, and global solutions, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, First-order systems, initial value problems, and solutions on intervals): for a continuous field on an open domain that is locally Lipschitz in the state variable, every initial datum has exactly one maximal solution; its maximal interval is open; and every solution with the same datum is a restriction of it.

[L4]

Differentiation rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set): a sum of differentiable real functions is differentiable with the sum of the derivatives; a constant multiple of a differentiable function is differentiated by the same factor; the product $x^{'} x^{'}$ has derivative $2 x^{'} x^{''}$ ; and the composite $U \circ x$ of a $C^{1}$ function $U$ with a differentiable curve $x$ is differentiable with derivative $U^{'} (x (t)) x^{'} (t)$ .

[L5]

A continuous real function on an interval that is differentiable with zero derivative at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L6]

Square roots: for $a \geq 0$ all solutions of $y^{2} = a$ are $y = \pm \sqrt{a}$ , and those two values are distinct exactly when $a > 0$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Proof

Proof technique: direct; apply the maximal-solution theorem to the phase system, differentiate the energy along a motion, and read the consequences off the conservation identity.

1.1givenL1L2L3

The phase field $G_{F}$ satisfies the hypotheses of [L3] on the open time-state domain $I_{0} \times Q \times R$ by [L2]. Hence every initial state $(x_{0}, v_{0})$ has exactly one maximal solution $z : J_{\max} \to Q \times R$ of $z^{'} = G_{F} (z)$ with $z (t_{0}) = (x_{0}, v_{0})$ , with $J_{\max}$ open and containing $t_{0}$ , and every solution of the phase system with the same initial state is a restriction of $z$ ; the maximal interval is contained in the time projection $I_{0}$ of this domain. By the biconditional of [L1] the first component $x$ of $z$ is a motion and the second is $x^{'}$ ; and if $y$ is any motion with the same datum, then $t \mapsto (y (t), y^{'} (t))$ is a solution of the phase system with the same initial state and hence a restriction of $z$ , so $y$ is a restriction of $x$ . This is clause 1.

1.2givenL1L4L5algebra

Let $x : J \to Q$ be a motion and put $e (t) : = E (x (t), x^{'} (t)) = \frac{1}{2} m x^{'} (t)^{2} + U (x (t))$ for $t \in J$ . At every interior point $t$ of $J$ , $x^{'}$ is differentiable with derivative $x^{''}$ and $x$ is differentiable, so by [L4] the function $e$ is differentiable there with $e^{'} (t) = m x^{'} (t) x^{''} (t) + U^{'} (x (t)) x^{'} (t) = x^{'} (t) (m x^{''} (t) + U^{'} (x (t))) = x^{'} (t) (F (x (t)) + U^{'} (x (t))) = 0,$ where the equation of motion $m x^{''} (t) = F (x (t))$ and $F = - U^{'}$ were used. The curve $e$ is continuous on the interval $J$ , so [L5] gives that $e$ is constant on $J$ . This is clause 2.

2.1givenstep 1.2L6algebra

Let $x : J \to Q$ be a motion with datum $(t_{0}, x_{0}, v_{0})$ and $E_{0} = E (x_{0}, v_{0})$ . By step 1.2, $E (x (t), x^{'} (t)) = E_{0}$ for every $t \in J$ ; subtracting $U (x (t))$ gives $\frac{1}{2} m x^{'} (t)^{2} = E_{0} - U (x (t))$ . The left side is nonnegative, so $U (x (t)) \leq E_{0}$ ; and $x^{'} (t)^{2} = 2 (E_{0} - U (x (t))) / m$ . If $U (x (t)) < E_{0}$ then the right side is strictly positive, and [L6] applied to $a = 2 (E_{0} - U (x (t))) / m > 0$ gives $x^{'} (t) = \pm \sqrt{a}$ ; while $x^{'} (t) = 0$ holds exactly when $U (x (t)) = E_{0}$ . Finally $(x (t), x^{'} (t)) \in Σ_{E_{0}}$ for every $t \in J$ because $E (x (t), x^{'} (t)) = E_{0}$ , by the definition of the level set. This is clause 3.

3.1givenstep 1.1L1∎

Clause 4: by [L1], $x_{*}$ is an equilibrium exactly when $F (x_{*}) = 0$ , that is $U^{'} (x_{*}) = 0$ . If so, the constant curve $x \equiv x_{*}$ has $x^{''} \equiv 0$ and $m \cdot 0 = F (x_{*}) = 0$ , so it is a motion with datum $(t_{0}, x_{*}, 0)$ ; by the uniqueness clause of step 1.1 it is the maximal motion with that datum. Conversely, if the maximal motion with datum $(t_{0}, x_{*}, 0)$ is the constant curve $x \equiv x_{*}$ , then substituting into the equation of motion gives $0 = m \cdot 0 = F (x_{*})$ , so $U^{'} (x_{*}) = 0$ . This is clause 4.

Remarks

What conservation does and does not give. Clause 3 confines the motion to the sublevel set ${U \leq E_{0}}$ and fixes the speed at each position where the level is crossed, up to sign; it does not by itself show that the motion oscillates between turning positions, that a period exists, or that the motion stays bounded for all forward times. Those require additional hypotheses on $U$ and are not claimed here.
The sample-independent sign. On an interval on which the velocity does not vanish, its sign is constant, because a continuous function with no zero has constant sign; this is why the phase curve travels along one branch of the level set between consecutive turning points. No turning points are asserted to exist for a general $U$ .
Normalization. Adding a constant to $U$ shifts $E$ by that constant on the whole phase plane; $Σ_{E_{0}}$ moves to $Σ_{E_{0} + c}$ and the velocity formula of clause 3 is unchanged, as it must be, since the force is unchanged. The energy conservation statement is a statement about the declared representative $U$ .

Existence, uniqueness and conservation of energy for a one-dimensional conservative model: phase curves lie in energy level sets and the velocity is determined by the position and energy up to sign

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources