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Positive kinetic metric: the fibre map $p = M v$ , the Hamiltonian and Hamilton's equations on explicit one- and two-label data

Physical scope

A worked check of the Legendre transform on the model class with positive definite kinetic operator: a one-label particle of mass m in a uniform downward potential field, with the explicit momentum p = mv, Hamiltonian H = |p|^2/(2m) + mg x_3, Hamilton equations, and the exact falling solution through a prescribed initial datum; and a two-label free model with unequal masses, where the mass operator is not a scalar and the inverse branch is computed entrywise. All quantities are model values of the declared idealizations; no measurement is reported.

Example

(a) One label in a uniform field. Declare the one-label model of the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N = 1$ , mass $m > 0$ , time window $I_{0} = R$ , configuration domain $Q = R^{3}$ and potential $U (t, x) : = m g x_{3}$ with constant acceleration magnitude $g > 0$ , $[g] = ℓ τ^{- 2}$ , read in $m s^{- 2}$ , with no residual forces; the force map is $F = (0, 0, - m g)$ and the mechanical Lagrangian is $L (t, x, v) = \frac{1}{2} m ∣ v ∣_{2}^{2} - m g x_{3}$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). Then:

the momentum of the presentation is $p = M v = m v$ (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta), the presentation is hyperregular on the Cartesian chart with inverse branch $v = p / m$ (Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian), and the Hamiltonian is $H (t, x, p) = \frac{∣ p ∣_{2}^{2}}{2 m} + m g x_{3} .$
Hamilton's equations $\dot{x} = H_{p}$ , $\dot{p} = - H_{x}$ read $\dot{x} = p / m$ , $\dot{p} = (0, 0, - m g)$ , so that $m \ddot{x} = \dot{p} = (0, 0, - m g)$ : the system is equivalent to the model's equation of motion and the potential is the uniform field (Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart).
for every datum $(0, x_{0}, v_{0})$ , where $x_{0} \in R^{3}$ is the initial position (metres) and $v_{0} \in R^{3}$ the initial coordinate velocity (metres per second), the curve $x (t) : = x_{0} + v_{0} t - \frac{1}{2} g t^{2} e_{3}, e_{3} = (0, 0, 1),$ is the maximal motion of the model through that datum, defined for all real times; its velocity is $v_{0} - g t e_{3}$ and its momentum is $m (v_{0} - g t e_{3})$ .

(b) Two labels with unequal masses. Declare instead the free two-label model $N = 2$ , masses $m_{1}, m_{2} > 0$ , $I_{0} = R$ , $Q = R^{6}$ and zero force map, so that $U \equiv 0$ . Then $M (v_{1}, v_{2}) = (m_{1} v_{1}, m_{2} v_{2})$ , the Hamiltonian is $H = ∣ p_{1} ∣_{2}^{2} / (2 m_{1}) + ∣ p_{2} ∣_{2}^{2} / (2 m_{2})$ , the inverse branch is $v = M^{- 1} p = (p_{1} / m_{1}, p_{2} / m_{2})$ , and Hamilton's equations are ${\dot{x}}_{i} = p_{i} / m_{i}$ , ${\dot{p}}_{i} = 0$ : each label moves uniformly in a straight line, and the fibre map is a linear isomorphism whose inverse is not a scalar multiple of the identity when $m_{1} \neq m_{2}$ .

All quantities are model values of the declared idealizations; no measurement is reported.

Facts & Assumptions

Given: The one-label model of part (a) with its uniform potential and the two-label free model of part (b), together with their mechanical Lagrangians, momenta, Hamiltonians and initial data.

[F1]

Model and Lagrangian data (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): the mechanical Lagrangian of a declared potential presentation is $L = K - U$ with $K (v) = \frac{1}{2} v \cdot M v$ , $M$ symmetric positive definite, and for one label $M v = m v$ with $m > 0$ ; dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

[F2]

Momentum, hyperregularity and Hamiltonian (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Hyperregular potential presentations and their inverse Legendre branches in the model's charts, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian): the fibre derivative is the momentum $p = M v$ , the presentation is hyperregular on the Cartesian chart with $v = M^{- 1} p$ , and the Hamiltonian of the branch is $H (t, x, p) = \frac{1}{2} p \cdot M^{- 1} p + U (t, x)$ .

[F3]

Equivalence theorem (Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart): for the global potential branch a curve is a motion of the model if and only if, with $p = M x^{'}$ , the pair satisfies Hamilton's equations of the branch Hamiltonian.

[F4]

Initial-value theory (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder): every initial datum of the model has exactly one maximal motion, and a motion defined on all of $R$ through the datum is that maximal motion.

[F5]

Calculus of the displayed curves (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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 derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, 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, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ): differentiable curves are differentiated componentwise, finite sums and constant multiples obey the usual derivative rules, and the partial derivatives of the displayed quadratic and linear functions are computed by the standard rules.

Verification

technique · direct; compute the momentum, inverse branch and Hamiltonian from the cited general results, read Hamilton's equations on the explicit data, and verify the displayed solutions against the model's equation

1.1F1F2algebra

Part (a), momentum and Hamiltonian: with $K (v) = \frac{1}{2} m ∣ v ∣_{2}^{2}$ and $U (x) = m g x_{3}$ , the fibre derivative is $L_{v} (t, x, v) = m v$ , so $p = m v$ and $v = p / m$ ; [F2] gives hyperregularity and $H (t, x, p) = \frac{1}{2} p \cdot (p / m) + m g x_{3} = ∣ p ∣_{2}^{2} / (2 m) + m g x_{3}$ .

1.2F2F3F5algebra

Part (a), equations: by [F5] the partial derivatives are $H_{p} = p / m$ and $H_{x} = (0, 0, m g)$ , so Hamilton's equations are $\dot{x} = p / m$ and $\dot{p} = (0, 0, - m g)$ ; differentiating the first and substituting the second gives $m \ddot{x} = \dot{p} = (0, 0, - m g)$ , which is the model's equation of motion of the uniform-field potential.

1.3F2F5algebra

Part (b): with $U \equiv 0$ and $M (v_{1}, v_{2}) = (m_{1} v_{1}, m_{2} v_{2})$ , [F2] gives $v = M^{- 1} p = (p_{1} / m_{1}, p_{2} / m_{2})$ and $H = \frac{1}{2} p \cdot M^{- 1} p = ∣ p_{1} ∣_{2}^{2} / (2 m_{1}) + ∣ p_{2} ∣_{2}^{2} / (2 m_{2})$ ; [F5] gives $H_{p} = M^{- 1} p$ and $H_{x} = 0$ , so Hamilton's equations are ${\dot{x}}_{i} = p_{i} / m_{i}$ and ${\dot{p}}_{i} = 0$ , whose solutions are the uniform straight motions $x_{i} (t) = x_{i} (0) + p_{i} (0) t / m_{i}$ .

2.1F3F4step 1.2algebra

Part (a), solution: the curve $x (t) = x_{0} + v_{0} t - \frac{1}{2} g t^{2} e_{3}$ has $\dot{x} = v_{0} - g t e_{3}$ , $\ddot{x} = - g e_{3}$ and satisfies $m \ddot{x} = (0, 0, - m g)$ with $x (0) = x_{0}$ , $\dot{x} (0) = v_{0}$ ; by [F4] it is the unique maximal motion through $(0, x_{0}, v_{0})$ and is defined for all real times, and its momentum $p = m \dot{x} = m (v_{0} - g t e_{3})$ solves the displayed Hamilton system by [F3].

3.1step 1.1step 1.2step 2.1step 1.3∎

Conclusion: parts (a) and (b) exhibit the fibre map, the inverse branch and the branch Hamiltonian on explicit data, and part (a) checks the equivalence with the model's equation and its maximal solution; all claims follow from the cited items without further hypotheses.

Remarks

The inverse branch is the mass operator's inverse. In part (a) the inverse branch is multiplication by $1 / m$ , and in part (b) by the two scalar factors $1 / m_{1}$ , $1 / m_{2}$ ; this is the model instance of the general formulas $v = M^{- 1} p$ and $v = G (q)^{- 1} p$ of Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian. The positivity of the masses is what makes both inversions global on the fibre.
Potential normalisation. The displayed $H$ uses the representative $U = m g x_{3}$ ; replacing it by $U + c (t)$ shifts $H$ by $c (t)$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions), so the numerical value of $H$ at a state is convention-dependent while the Hamilton system is unchanged by the shift together with the corresponding action convention. No energy value is compared with any measurement here.

Positive kinetic metric: the fibre map $p = M v$ , the Hamiltonian and Hamilton's equations on explicit one- and two-label data

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Positive kinetic metric: the fibre map p=Mvp=Mv, the Hamiltonian and Hamilton's equations on explicit one- and two-label data

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Positive kinetic metric: the fibre map $p = M v$ , the Hamiltonian and Hamilton's equations on explicit one- and two-label data