ODEs in Physics - Core Definitions

Ordinary differential equations govern the time evolution of physical systems, from classical mechanics to quantum dynamics, providing mathematical models for deterministic processes.

Classification of ODEs

DefinitionOrder and Linearity

An ODE of order $n$ involves derivatives up to the $n$ -th order:

$F\left(t, y, y', y'', \ldots, y^{(n)}\right) = 0$

A linear ODE has the form:

$a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \cdots + a_1(t)y' + a_0(t)y = f(t)$

where $f(t)$ is the forcing term. If $f(t) = 0$ , the equation is homogeneous; otherwise it's inhomogeneous.

Physical laws are predominantly linear or linearizable, making linear ODE theory central to mathematical physics.

ExampleSimple Harmonic Oscillator

The undamped oscillator equation:

$m\frac{d^2x}{dt^2} + kx = 0$

is second-order, linear, and homogeneous with constant coefficients. The general solution is:

$x(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) = C\cos(\omega_0 t + \phi)$

where $\omega_0 = \sqrt{k/m}$ is the natural frequency, and $A, B$ (or $C, \phi$ ) are determined by initial conditions $x(0)$ and $\dot{x}(0)$ .

Existence and Uniqueness

TheoremPicard-Lindelöf Theorem

For the initial value problem:

$\frac{dy}{dt} = f(t, y), \quad y(t_0) = y_0$

if $f(t,y)$ and $\partial f/\partial y$ are continuous in a region containing $(t_0, y_0)$ , then there exists a unique solution in some interval $|t - t_0| < \delta$ .

This theorem guarantees that physical systems described by ODEs have well-defined evolution given initial conditions, fundamental to deterministic physics.

ExampleExponential Growth

The equation $dy/dt = \lambda y$ with $y(0) = y_0$ has unique solution:

$y(t) = y_0 e^{\lambda t}$

This models radioactive decay ( $\lambda < 0$ ), population growth ( $\lambda > 0$ ), and RC circuit discharge.

Systems of ODEs and Phase Space

DefinitionPhase Space Formulation

An $n$ -th order ODE can be rewritten as a system of $n$ first-order ODEs. For Newton's equation $\ddot{x} = F(x,\dot{x},t)/m$ , define:

$\mathbf{z} = \begin{pmatrix}x \\ v\end{pmatrix}, \quad \frac{d\mathbf{z}}{dt} = \begin{pmatrix}v \\ F(x,v,t)/m\end{pmatrix}$

The space of $(\mathbf{x}, \mathbf{v})$ is phase space, where trajectories represent system evolution.

ExampleDamped Harmonic Oscillator

The equation $\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = 0$ becomes:

$\frac{d}{dt}\begin{pmatrix}x \\ v\end{pmatrix} = \begin{pmatrix}0 & 1 \\ -\omega_0^2 & -2\gamma\end{pmatrix}\begin{pmatrix}x \\ v\end{pmatrix}$

Eigenvalues $\lambda = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}$ determine the solution type:

Underdamped ( $\gamma < \omega_0$ ): Oscillatory decay
Critically damped ( $\gamma = \omega_0$ ): Fastest return to equilibrium
Overdamped ( $\gamma > \omega_0$ ): Exponential decay without oscillation

Green's Functions for ODEs

DefinitionGreen's Function

For the linear operator $\mathcal{L}y = y'' + p(t)y' + q(t)y$ , the Green's function $G(t,t')$ satisfies:

$\mathcal{L}G(t,t') = \delta(t - t')$

with appropriate boundary conditions. The solution to $\mathcal{L}y = f(t)$ is:

$y(t) = \int G(t,t')f(t')dt'$

ExampleDriven Oscillator

For $\ddot{x} + \omega_0^2 x = F(t)/m$ with zero initial conditions, the Green's function is:

$G(t,t') = \frac{\sin[\omega_0(t-t')]}{m\omega_0}\theta(t-t')$

The solution is:

$x(t) = \frac{1}{m\omega_0}\int_0^t F(t')\sin[\omega_0(t-t')]dt'$

This is the convolution of the forcing with the impulse response, fundamental to linear response theory.

RemarkIntegrating Factors

For first-order linear equations $y' + p(t)y = q(t)$ , the integrating factor:

$\mu(t) = e^{\int p(t)dt}$

transforms the equation to:

$\frac{d}{dt}[\mu(t)y] = \mu(t)q(t)$

giving:

$y(t) = \frac{1}{\mu(t)}\left[y_0\mu(t_0) + \int_{t_0}^t \mu(s)q(s)ds\right]$

These foundational concepts enable systematic solution of ODEs governing time-dependent phenomena in physics, from orbital mechanics to quantum state evolution.