Physlib.ClassicalMechanics.HarmonicOscillator.Solution

For any two sets of initial conditions $IC_1$ and $IC_2$ for a classical harmonic oscillator, if their initial positions are equal ($IC_1.x_0 = IC_2.x_0$) and their initial velocities are equal ($IC_1.v_0 = IC_2.v_0$), then the initial conditions themselves are equal ($IC_1 = IC_2$).

For any two sets of initial conditions at an arbitrary time, $IC_1$ and $IC_2$, if their reference times are equal ($IC_1.t_0 = IC_2.t_0$), their positions at those times are equal ($IC_1.x_{t_0} = IC_2.x_{t_0}$), and their velocities at those times are equal ($IC_1.v_{t_0} = IC_2.v_{t_0}$), then the initial conditions themselves are equal ($IC_1 = IC_2$).

For a harmonic oscillator $S$ with angular frequency $\omega$ and a state specified at time $t_0$ by position $x_{t_0}$ and velocity $v_{t_0}$, this function calculates the equivalent initial conditions at time $t=0$. The resulting initial position $x_0$ and initial velocity $v_0$ are defined as: $$x_0 = \cos(\omega t_0) x_{t_0} - \frac{\sin(\omega t_0)}{\omega} v_{t_0}$$ $$v_0 = \omega \sin(\omega t_0) x_{t_0} + \cos(\omega t_0) v_{t_0}$$ This conversion effectively runs the trajectory backward from $t_0$ to $0$ to determine the starting state.

The zero initial condition for a harmonic oscillator is defined as the state where both the initial position $x_0$ and the initial velocity $v_0$ are equal to $0$. This defines the zero element (additive identity) for the `InitialConditions` structure.

For the zero initial conditions of a classical harmonic oscillator, the initial position $x_0$ is equal to $0$. This confirms that the zero initial condition state corresponds to a particle starting at the origin.

For the zero initial conditions of a classical harmonic oscillator, the initial velocity $v_0$ is equal to $0$. This indicates that the zero initial condition corresponds to a state with zero starting velocity.

Given a set of initial conditions $IC$ with initial position $x_0$ and initial velocity $v_0$, the trajectory of the classical harmonic oscillator is the function that maps each time $t$ to its position in 1D Euclidean space, defined by: $$x(t) = \cos(\omega t) x_0 + \frac{\sin(\omega t)}{\omega} v_0$$ where $\omega$ is the angular frequency of the oscillator system.

For a classical harmonic oscillator system $S$ with angular frequency $\omega$ and initial conditions $IC$ consisting of an initial position $x_0$ and an initial velocity $v_0$, the trajectory $x(t)$ is given by: $$x(t) = \cos(\omega t) x_0 + \frac{\sin(\omega t)}{\omega} v_0$$

For a classical harmonic oscillator system with zero initial conditions (where the initial position $x_0$ and the initial velocity $v_0$ are both zero), the resulting trajectory $x(t)$ is identically zero for all times $t$.

For a classical harmonic oscillator system $S$ and a given set of initial conditions $IC$, the resulting trajectory $x(t)$ is $n$-times continuously differentiable ($C^n$) for any $n \in \mathbb{N} \cup \{\infty\}$. That is, the trajectory is infinitely differentiable.

For a classical harmonic oscillator with angular frequency $\omega$ and initial conditions $IC$ specifying an initial position $x_0$ and an initial velocity $v_0$, let $x(t)$ be the trajectory of the oscillator. The velocity at time $t$, defined as the time derivative $\frac{d}{dt}x(t)$, is given by: $$\frac{d}{dt}x(t) = -\omega \sin(\omega t) x_0 + \cos(\omega t) v_0$$

For a classical harmonic oscillator with angular frequency $\omega$ and initial conditions $IC$ specifying an initial position $x_0$ and an initial velocity $v_0$, let $x(t)$ be the trajectory of the oscillator. The acceleration at time $t$, defined as the second time derivative $\frac{d^2}{dt^2}x(t)$, is given by: $$\frac{d^2}{dt^2}x(t) = -\omega^2 \cos(\omega t) x_0 - \omega \sin(\omega t) v_0$$

For a classical harmonic oscillator system $S$ with a given set of initial conditions $IC$, let $x(t)$ denote the trajectory of the oscillator. If $x_0$ is the initial position specified in $IC$, then the position at time $t = 0$ is $x(0) = x_0$.

For a classical harmonic oscillator system $S$ with initial conditions $IC$ specifying an initial position $x_0$ and an initial velocity $v_0$, let $x(t)$ be the trajectory of the oscillator. The velocity at time $t = 0$, defined as the time derivative $\frac{d}{dt}x(t)$ evaluated at $t=0$, is equal to the initial velocity $v_0$. That is, $$\frac{d}{dt}x(t) \Big|_{t=0} = v_0$$

For a classical harmonic oscillator system $S$ and a given set of initial conditions $IC$ specifying an initial position $x_0$ and an initial velocity $v_0$, let $x(t)$ be the trajectory of the oscillator. The trajectory $x(t)$ satisfies the equation of motion of the harmonic oscillator system $S$.

For a classical harmonic oscillator system $S$ and a given set of initial conditions $IC$ specifying an initial position $x_0$ and an initial velocity $v_0$ at time $t=0$, let $x: \text{Time} \to \mathbb{R}$ be an infinitely differentiable ($C^\infty$) function. If $x$ satisfies the oscillator's equation of motion and the initial conditions $x(0) = x_0$ and $\partial_t x(0) = v_0$, then $x$ is equal to the predefined trajectory of the system, given by: $$x(t) = \cos(\omega t) x_0 + \frac{\sin(\omega t)}{\omega} v_0$$ where $\omega$ is the angular frequency of the oscillator.

For a classical harmonic oscillator system $S$ with mass $m$ and spring constant $k$, let $x(t)$ be the trajectory defined by the initial conditions $IC$, where $x_0$ is the initial position and $v_0$ is the initial velocity at time $t=0$. The total energy $E(t)$ of the trajectory is constant for all time $t$ and is equal to the initial energy: $$E(t) = \frac{1}{2} (m \|v_0\|^2 + k \|x_0\|^2)$$

For a harmonic oscillator $S$ and a set of conditions $IC$ specifying a position $x_{t_0}$ and a velocity $v_{t_0}$ at an arbitrary time $t_0$, the trajectory $x(t)$ defined by the equivalent initial conditions at $t=0$ (obtained via the conversion function `toInitialConditions`) passes through the specified position at time $t_0$, such that $x(t_0) = x_{t_0}$.

For a harmonic oscillator $S$ and a set of conditions $IC$ specifying a position $x_{t_0}$ and a velocity $v_{t_0}$ at an arbitrary time $t_0$, let $x(t)$ be the trajectory defined by the equivalent initial conditions at $t=0$ (obtained via the conversion function `toInitialConditions`). The velocity of this trajectory at time $t_0$, defined as the time derivative $\frac{d}{dt}x(t)$ evaluated at $t = t_0$, is equal to the specified velocity: $$\left. \frac{d}{dt}x(t) \right|_{t=t_0} = v_{t_0}$$

Consider a classical harmonic oscillator $S$ with mass $m$ and spring constant $k$. Let $IC$ be a set of initial conditions specifying a position $x_{t_0}$ and a velocity $v_{t_0}$ at an arbitrary time $t_0$. Let $x(t)$ be the trajectory of the oscillator corresponding to these conditions. Then the total energy $E$ of the trajectory at time $t_0$ is given by: $$E(t_0) = \frac{1}{2} (m \|v_{t_0}\|^2 + k \|x_{t_0}\|^2)$$ where $\|\cdot\|$ denotes the norm in the oscillator's configuration space.

Consider a classical harmonic oscillator with angular frequency $\omega$ and initial conditions given by position $x_0$ and velocity $v_0$ at time $t=0$. For any time $t$, if the velocity of the trajectory is zero (i.e., $\frac{d}{dt}x(t) = 0$) and the initial conditions are non-trivial ($x_0 \neq 0$ or $v_0 \neq 0$), then the time $t$ satisfies the relation $$\tan(\omega t) = \frac{v_0}{\omega x_0}$$ where $x_0$ and $v_0$ are treated as the scalar values of the position and velocity in 1D space.

For a classical harmonic oscillator with angular frequency $\omega$ and initial conditions specifying an initial position $x_0$ and initial velocity $v_0$ at time $t=0$, if the component of the initial position $x_0(0)$ is non-zero, then the velocity $v(t) = \frac{d}{dt}x(t)$ is zero at time $$t = \frac{1}{\omega} \arctan \left( \frac{v_0(0)}{\omega x_0(0)} \right)$$ where $x_0(0)$ and $v_0(0)$ are the scalar components of the initial position and velocity vectors in 1D Euclidean space.

For a classical harmonic oscillator system with angular frequency $\omega$ and initial conditions given by position $x_0$ and velocity $v_0$ at time $t=0$, let $x(t)$ be the trajectory of the oscillator. The velocity of the trajectory at time $t$, denoted by $\frac{d}{dt}x(t)$, is zero if and only if the norm of the position at that time satisfies: $$\|x(t)\| = \sqrt{\|x_0\|^2 + \left(\frac{\|v_0\|}{\omega}\right)^2}$$ where $\|x_0\|^2 + (\|v_0\|/\omega)^2$ represents the square of the maximum displacement (amplitude) of the oscillation.

Consider a classical harmonic oscillator with angular frequency $\omega$. Let $IC$ be the initial conditions determined by specifying velocities at two times $t_1$ and $t_2$. If $\sin(\omega(t_2 - t_1)) \neq 0$, then the velocity of the resulting trajectory $x(t)$ at time $t_2$ is equal to the specified velocity $v_{t_2}$: $$\left. \frac{d}{dt}x(t) \right|_{t=t_2} = v_{t_2}$$ where $\frac{d}{dt}x(t)$ denotes the time derivative of the trajectory.

Given a classical harmonic oscillator $S$ with angular frequency $\omega$, and initial parameters specified by an amplitude $A$ and a phase $\phi$, this function defines the corresponding initial conditions $(x_0, v_0)$ at time $t = 0$ as: $$x_0 = A \cos \phi$$ $$v_0 = A \omega \sin \phi$$ These initial conditions result in a trajectory of the form $x(t) = A \cos(\omega t - \phi)$.

For a classical harmonic oscillator with angular frequency $\omega$, if the initial conditions are specified by an amplitude $A$ and a phase $\phi$, then the trajectory $x(t)$ at any time $t$ is given by: $$x(t) = A \cos(\omega t - \phi)$$ Here, the trajectory represents the position of the oscillator in 1-dimensional Euclidean space.

For a classical harmonic oscillator with angular frequency $\omega$, if the initial conditions are specified by an amplitude $A$ and a phase $\phi$, then the velocity $v(t)$ (the time derivative of the trajectory $x(t)$) at any time $t$ is given by: $$v(t) = -A \omega \sin(\omega t - \phi)$$ This velocity is expressed as a vector in 1-dimensional Euclidean space.

Given a classical harmonic oscillator $S$ with angular frequency $\omega$ and initial conditions $IC$ specifying an initial position $x_0$ and initial velocity $v_0$, this function calculates the corresponding amplitude $A$ and phase $\phi$. The values are determined by representing the state as a complex number $z = x_0 + i \frac{v_0}{\omega}$, where: - The amplitude is the magnitude of $z$, i.e., $A = |z| = \sqrt{x_0^2 + \left(\frac{v_0}{\omega}\right)^2}$. - The phase is the argument of $z$, i.e., $\phi = \arg(z)$. These parameters allow the trajectory to be expressed in the form $x(t) = A \cos(\omega t - \phi)$.

For a classical harmonic oscillator $S$ and any initial conditions $IC$ (specifying initial position and velocity), converting $IC$ into its amplitude-phase representation and then converting that representation back to initial conditions recovers the original $IC$.

For a classical harmonic oscillator $S$ with angular frequency $\omega$, let $IC$ be the initial conditions specifying an initial position $x_0$ and velocity $v_0$. Let $A$ and $\phi$ be the amplitude and phase calculated from these initial conditions. Then, for any time $t$, the trajectory $x(t)$ is given by: $$x(t) = A \cos(\omega t - \phi)$$ where $x(t)$ is represented as a vector in 1D Euclidean space whose single component at index 0 is the shifted cosine.

For a classical harmonic oscillator with angular frequency $\omega$, let $A$ and $\phi$ be the amplitude and phase derived from the initial conditions $(x_0, v_0)$. Then, for any time $t$, the velocity of the trajectory (the time derivative of the position) is given by the following expression in 1D Euclidean space: $$\frac{d}{dt} x(t) = -A \omega \sin(\omega t - \phi)$$ where $x(t)$ is the trajectory associated with the given initial conditions.

In a classical harmonic oscillator with angular frequency $\omega$, let $A$ and $\phi$ be the amplitude and phase determined by the initial conditions. If the amplitude $A$ is non-zero, then the velocity of the trajectory at time $t$ is zero if and only if the sine factor in the phase-shifted expression vanishes, i.e., $$\frac{d}{dt}x(t) = 0 \iff \sin(\omega t - \phi) = 0$$ where $x(t)$ represents the trajectory of the oscillator.

For a classical harmonic oscillator with angular frequency $\omega$, let $A$ and $\phi$ be the amplitude and phase determined by the initial conditions. If the amplitude $A$ is non-zero, then for any time $t$, the velocity of the trajectory $\dot{x}(t)$ is zero if and only if there exists an integer $n \in \mathbb{Z}$ such that $$t = \frac{\phi + n\pi}{\omega}$$

For a classical harmonic oscillator with angular frequency $\omega$ and initial conditions $IC$ (initial position $x_0$ and initial velocity $v_0$), let $x(t)$ be the resulting trajectory. Let $A = \sqrt{x_0^2 + (v_0/\omega)^2}$ be the amplitude of the oscillation. For any time $t$, the velocity $\partial_t x(t)$ is zero if and only if the norm of the position $\|x(t)\|$ is equal to the amplitude $A$.

For a classical harmonic oscillator with angular frequency $\omega$, let $A$ and $\phi$ be the amplitude and phase determined by the initial conditions. If the amplitude $A$ is non-zero, then for any time $t$, the trajectory $x(t)$ satisfies $x(t) = 0$ if and only if $\cos(\omega t - \phi) = 0$.

Consider a classical harmonic oscillator with angular frequency $\omega$. Let $IC$ be the initial conditions of the system, with $A$ and $\phi$ denoting the corresponding amplitude and phase respectively, such that the trajectory $x(t)$ is given by $x(t) = A \cos(\omega t - \phi)$. Provided the amplitude is non-zero ($A \neq 0$), the position of the oscillator is zero at time $t$ if and only if there exists an integer $n \in \mathbb{Z}$ such that: $$t = \frac{\phi + (2n + 1)\frac{\pi}{2}}{\omega}$$

For a classical harmonic oscillator $S$, the period $T$ is defined as the ratio of $2\pi$ to the angular frequency $\omega$: $$T = \frac{2\pi}{\omega}$$

The definition introduces the mathematical notation $T$ to represent the period of a harmonic oscillator.

For a classical harmonic oscillator $S$, the period $T$ and the angular frequency $\omega$ satisfy the relation $$T = \frac{2\pi}{\omega}$$

For any classical harmonic oscillator $S$, the period $T$ is strictly positive, satisfying $0 < T$.

For a classical harmonic oscillator $S$ with angular frequency $\omega$ and period $T = \frac{2\pi}{\omega}$, the trajectory $x(t)$ resulting from any set of initial conditions $IC$ is a periodic function of time with period $T$. That is, for all time $t$, $x(t + T) = x(t)$.

For a classical harmonic oscillator with period $T$, let $x(t)$ be the trajectory with initial position $x(0) = x_0$ and initial velocity $\dot{x}(0) = v_0$. Assuming that the initial position and velocity are not both zero (i.e., $(x_0, v_0) \neq (0, 0)$), if at some time $t$ the oscillator returns to its initial state such that $x(t) = x_0$ and $\dot{x}(t) = v_0$, then $t$ is an integer multiple of the period $T$. That is, there exists an integer $n \in \mathbb{Z}$ such that $t = nT$.