We begin by defining the differential operator \(\mathrm{D}\). For a single-variable function \(f(t)\), we let \(\mathrm{D}f = \frac{df}{dt}\). In the multivariable context, such as \(f(t, x)\), the partial operators are denoted by \(\mathrm{D}_t f = \frac{\partial f}{\partial t}\) and \(\mathrm{D}_x f = \frac{\partial f}{\partial x}\).
Consider the motion of a particle in \(\mathbb{R}^1\). Let \(x(t)\) denote the position of the particle as a function of time. The velocity \(v(t)\) and acceleration \(a(t)\) are defined through the successive application of the differential operator: \[v(t) = \mathrm{D}x(t), \quad a(t) = \mathrm{D}^2x(t).\]
When the particle is subject to a force \(F\), which we assume to be a function of position \(x\) alone, its motion is governed by the fundamental law of dynamics.
NoteNewton’s Second Law
The motion of a particle of mass \(m\) under the influence of a force \(F(x)\) satisfies the second-order ordinary differential equation: \[F(x(t)) = m\mathrm{D}^2x(t),\] with the initial state prescribed by \(x(t_0) = x_0\) and \(\mathrm{D}x(t_0) = v_0\).
Assuming \(F\) is a smooth function, the standard existence and uniqueness theorem for ordinary differential equations implies that there exists a unique local solution for any given initial condition. Such a trajectory \(x(t)\) in the configuration space is formally referred to as an orbit.
Tipblow up
To illustrate the qualitative behavior of such systems, we consider a nonlinear case where \(F(x) = x^2\). For any solution initiated with \(x_0 > 0\) and \(v_0 > 0\), the particle will reach infinity within a finite temporal duration.
Caution
To analyze the divergence, we multiply the equation \(m\ddot{x} = x^2\) by \(\dot{x}\) and integrate with respect to time, yielding the conservation of energy \(\frac{1}{2}m\dot{x}^2 - \frac{1}{3}x^3 = E\). For the specific case where \(E=0\) and \(m=1\), the equation reduces to \(\dot{x} = \sqrt{2/3}x^{3/2}\). Separating variables and integrating from \(t=0\) to \(t\) gives: \[\int_{x_0}^{x} u^{-3/2} du = \int_{0}^{t} \sqrt{2/3} dt \implies -2(x^{-1/2} - x_0^{-1/2}) = \sqrt{2/3}t.\] As \(x \to \infty\), the time \(t\) approaches the finite value \(T_{blowup} = 2x_0^{-1/2}/\sqrt{2/3}\), confirming the finite-time singularity 1.
1 For general energy levels \(E \neq 0\), the solution \(t(x)\) involves the inversion of an incomplete elliptic integral of the first kind, and the particle’s trajectory can be expressed in terms of the Weierstrass \(\wp\)-function.
TipHarmonic Oscillator
Alternatively, if we consider a linear restoring force \(F(x) = -kx\), the governing equation becomes \(m\mathrm{D}^2x + kx = 0\). This is the classical harmonic oscillator, whose general solution is expressed as: \[x(t) = A \cos(\omega t + \phi), \quad \omega = \sqrt{\frac{k}{m}},\] where \(\omega\) represents the natural frequency of the oscillation.
2 Energy and Phase Space
We now introduce the integral operator \(\mathrm{I}\). For a single-variable function \(F(x)\), the indefinite integral is denoted by \(\mathrm{I}(F) := \int F(x) dx\), while the definite integral over the interval \([a, b]\) is expressed as \(\mathrm{I}(F, a, b) := \int_a^b F(x) dx\).
In the study of classical mechanics, we define the kinetic energy of a system as \(\frac{1}{2}mv^2\). The potential energy \(V(x)\) is defined through the work done by the force \(F(x)\).
TipDefinition: Energy
The potential energy \(V(x)\) is defined as the negative integral of the force function: \[V(x) = -\mathrm{I}(F(x)),\] which implies the relation \(F(x) = -\mathrm{D} V\). The total energy \(E\) of the system is the sum of kinetic and potential energies: \[E(x, v) = \frac{1}{2}mv^2 + V(x).\]
A fundamental characteristic of energy in such systems is its conservation, meaning its value remains invariant along any trajectory. This leads us to the following theorem.
NoteConservation of Energy
Assume a particle satisfies Newton’s law \(F(x) = m\mathrm{D}^2x\). The total energy \(E\) is conserved; that is, for any solution \(x(t)\) of Newton’s equation, the quantity \(E(x(t), \mathrm{D}x(t))\) is independent of time \(t\).
Caution
To verify this invariance, we differentiate the total energy with respect to time \(t\) along a trajectory \(x(t)\): \[\mathrm{D}_t E = \mathrm{D}_t \left( \frac{1}{2}m(\mathrm{D}x)^2 + V(x) \right).\] Applying the chain rule, we obtain: \[\mathrm{D}_t E = m(\mathrm{D}x)(\mathrm{D}^2x) + (\mathrm{D}_x V)(\mathrm{D}x).\] Substituting the relations \(m\mathrm{D}^2x = F(x)\) and \(\mathrm{D}_x V = -F(x)\), the expression becomes: \[\mathrm{D}_t E = (\mathrm{D}x)F(x) - F(x)(\mathrm{D}x) = 0.\] Since the time derivative vanishes identically, we conclude that \(E\) is constant along the motion.
In mechanics, such a conserved quantity is often referred to as an integral of motion. As illustrated in the provided text, although Newton’s second law is a second-order differential equation in \(x\), the energy \(E\) depends only on the position \(x\) and velocity \(\dot{x}\) (or \(\mathrm{D}x\)), and not on the acceleration \(\ddot{x}\). Consequently, the conservation law allows us to effectively “integrate” the original second-order equation once, transforming it into a first-order differential equation: \[\frac{m}{2}(\mathrm{D}x(t))^2 + V(x(t)) = E_0.\] By solving for \(\mathrm{D}x\), we obtain the separable form \(\mathrm{D}x(t) = \pm \sqrt{\frac{2}{m}(E_0 - V(x(t)))}\), which can be solved explicitly.
Beyond the perspective of energy levels, we can interpret the dynamics within a geometric framework. We observe that the state of the system is characterized by the pair \((x, v)\), which satisfies the following system of first-order equations: \[\begin{cases} \mathrm{D}x = v \\ \mathrm{D}v = \frac{1}{m}F(x) \end{cases}\] Substituting the first equation into the second recovers Newton’s second law, \(m\mathrm{D}^2x = F(x)\).
TipDefinition: Phase Space
We define the pair \((x, v)\) as the state of the particle in the phase space\(\mathbb{R}^2\). The initial state of the system is given by the conditions \(x(0)=x_0\) and \(v(0)=v_0\).
The conservation of energy implies that the evolution of the state \((x, v)\) in phase space is constrained to a specific level set: \[\{(x, v) \in \mathbb{R}^2 \mid E(x, v) = E(x_0, v_0)\}.\] If the force \(F\) and potential \(V\) are smooth, then \(E\) is a smooth function of \(x\) and \(v\). By the Implicit Function Theorem, this level set corresponds to a smooth curve in \(\mathbb{R}^2\), representing the physical orbit of the particle.
3 Systems with Damping
In many physical contexts, we must consider forces that depend not only on the position of the particle but also on its velocity. A primary example is a system experiencing linear resistance, where the dissipative force is proportional to the velocity but acts in the opposite direction.
TipLinear Damped Oscillator
Consider a system where the total force is given by \(F = -kx - \gamma \mathrm{D}x\) with \(\gamma > 0\). The governing kinematic equation for this system is: \[m \mathrm{D}^2x + \gamma \mathrm{D}x + kx = 0.\]
The qualitative behavior of the solutions to this second-order linear equation depends on the discriminant of the characteristic equation \(mr^2 + \gamma r + k = 0\).
Underdamped (\(\gamma^2 < 4mk\)): The system oscillates with an exponentially decaying amplitude.
Overdamped (\(\gamma^2 > 4mk\)): The damping is so strong that the system returns to equilibrium without oscillating.
Critically Damped (\(\gamma^2 = 4mk\)): The system returns to equilibrium as quickly as possible without oscillating.
When damping is present, energy is no longer a conserved quantity. In fact, no non-constant continuous function \(E(x, \mathrm{D}x)\) exists that remains invariant along all trajectories of the damped system. However, we can still characterize the evolution of the mechanical energy.
NoteProposition: Dissipation of Energy
Suppose a particle moves under the influence of a force field \(F(x, \mathrm{D}_t x) = F_1(x) - \gamma \mathrm{D}_t x\) with \(\gamma > 0\). Let the energy \(E\) be defined as: \[E(x, \mathrm{D}_t x) = \frac{1}{2}m(\mathrm{D}_t x)^2 + V(x)\] where \(\mathrm{D}_x V = -F_1(x)\). Then, along any trajectory \(x(t)\), the time derivative of the energy satisfies: \[\mathrm{D}_t E(x(t), \mathrm{D}_t x(t)) = -\gamma(\mathrm{D}_t x(t))^2 \leq 0.\] This result indicates that while energy is not conserved, it is a non-increasing function of time, reflecting the physical reality of energy dissipation due to friction or resistance.
CautionProof
To prove this proposition, we compute the total derivative of \(E\) with respect to \(t\) along the path \(x(t)\). Using the chain rule, we have: \[\mathrm{D}_t E = \mathrm{D}_t \left( \frac{1}{2}m(\mathrm{D}_t x)^2 + V(x) \right) = m(\mathrm{D}_t x)(\mathrm{D}_t^2 x) + (\mathrm{D}_x V)(\mathrm{D}_t x).\] Substituting the equation of motion \(m(\mathrm{D}_t^2 x) = F_1(x) - \gamma \mathrm{D}_t x\) and the relation \(\mathrm{D}_x V = -F_1(x)\) into the expression above, we obtain: \[\mathrm{D}_t E = (\mathrm{D}_t x)(F_1(x) - \gamma \mathrm{D}_t x) - F_1(x)(\mathrm{D}_t x).\] Simplifying the terms yields: \[\mathrm{D}_t E = F_1(x)(\mathrm{D}_t x) - \gamma (\mathrm{D}_t x)^2 - F_1(x)(\mathrm{D}_t x) = -\gamma (\mathrm{D}_t x)^2.\] Given that \(\gamma > 0\) and \((\mathrm{D}_t x)^2 \geq 0\), it follows that \(\mathrm{D}_t E \leq 0\) for all \(t\). \(\square\)
For the specific case of the damped harmonic oscillator, we can visualize the trajectories in phase space as inward-spiraling paths that eventually converge to the origin \((0,0)\), representing the loss of energy until the particle comes to rest.