Problem 12
Question
(a) A particle of mass \(m\) constrained to lie in a given plane has a potential energy which is a function only of its distance \(r\) from a fixed point in the plane. Use the principle of least action to derive the equation $$ \phi=c_{1} \int \frac{d r}{\sqrt{r^{4}(E-V)-c_{1}^{2} r^{2}}} $$ of the particle's orbit, where \((r, \phi)\) are plane polar coordinates. (b) Apply part \((a)\) to the special case \(V=-\left(k^{2} / r\right) .\) Identify the orbit in each of the cases \(E>0, E=0, E<0\) (c) Solve the problem of part \((a)\), and subsequently that of part \((b)\), by the Hamilton-Jacobi method. SoLUTION: \(T=\frac{1}{2} m\left(\hat{r}^{2}+r^{2} \phi^{2}\right), p_{r}=m \dot{r}, p_{\phi}=m r^{2} \dot{\phi}\), $$ \begin{gathered} H=\left(\frac{1}{2 m}\right)\left[p_{r}^{2}+\left(\frac{p_{\phi}}{r}\right)^{2}\right]+V(r) \\ S^{*}=\alpha_{1} \phi+\int \sqrt{2 m(E-V)-\left(\frac{\alpha_{1}}{r}\right)^{2}} d r \end{gathered} $$ use the second of \((35)\) with \(i=1\)
Step-by-Step Solution
Verified Answer
The Hamiltonian is given by \( H = \frac{1}{2 m}( p_{r}^{2} + \left( \frac{ p_{\phi} }{ r } \right)^{2} ) + V(r) \) which encapsulates the energy dynamics of the system, and the orbits of the particle are governed by the equation \( \phi = c_{1} \int \frac{d r}{\sqrt{r^{4}(E-V)-c_{1}^{2} r^{2}}} \). The actual nature of the orbits depends on whether \( E > 0, E = 0, or E < 0 \). The Hamilton-Jacobi method can be used to solve the same problem, allowing for a different way to analyze the system.
1Step 1: Determination of Lagrangian
The Lagrangian, which is the difference between the kinetic energy (\( T \)) and potential energy (\( V \)), is given by \( L = T - V \). In this case, \( T = \frac{1}{2} m(\dot{r}^{2} + r^{2} \dot{\phi}^{2}) \) and \( V = V(r) \). Thus, \( L = \frac{1}{2} m(\dot{r}^{2} + r^{2} \dot{\phi}^{2}) - V(r) \).
2Step 2: Derivation of Equations of Motion
Apply the Euler-Lagrange equations to find the equations of motion. The general form of the Euler-Lagrange equation is \( \frac{d}{dt}(\frac{dL}{d\dot{q}}) - \frac{dL}{dq} = 0 \), where \( q \) are the generalized coordinates (\( r, \phi \)). Applying these equations, we obtain the equations of motion.
3Step 3: Apply the Principle of Least Action
The principle of least action states that the action \( S = \int L \ dt \) is stationary for the actual path taken by a system. This principle allows us to derive the Euler-Lagrange equations and hence the equations of motion. Apply this principle to derive the given equation \( \phi = c_{1} \int \frac{d r}{\sqrt{r^{4}(E-V)-c_{1}^{2} r^{2}}} \) of the particle's orbit.
4Step 4: Application to specific potential
Apply the equation derived in Step 3 to the special case \( V = - \frac{ k^{2} }{ r } \). Evaluate the resulting integral and compare it with the three conditions \( E > 0, E = 0, E < 0 \). The difference in these conditions will affect the properties of the particle's orbit.
5Step 5: Use the Hamilton-Jacobi Method
The Hamilton's Principal Function \( S^{*} \) is given by \( S^{*} = \alpha_{1} \phi + \int \sqrt{2 m(E-V) - \left( \frac{ \alpha_{1} }{ r } \right)^{2}} d r \). In the Hamilton-Jacobi equation, use the given \( S^{*} \) and manipulate the equation to solve for .
Key Concepts
Principle of Least ActionEuler-Lagrange EquationsHamilton-Jacobi MethodParticle Orbit in Plane Polar Coordinates
Principle of Least Action
In the realm of classical mechanics, the principle of least action, also known as the principle of stationary action, holds a fundamental place. This principle states that out of all possible paths that a physical system may take between two states, the path followed is the one for which the action is stationary (usually a minimum).
The action, denoted as S, is a quantity that encapsulates the dynamics of the system and is defined as the integral over time of the Lagrangian, L, which itself is the difference between the kinetic and potential energies, T and V respectively. Mathematically, how the principle is applied can be expressed as:
\[ S = \int L \, dt \]
In the exercise given, the principle is utilized to derive an equation for the orbit of a particle in a plane, where the potential energy is a function of the distance from a fixed point. By finding where the action is stationary, we can determine the conditions that govern the behavior of the particle's motion within the plane.
The action, denoted as S, is a quantity that encapsulates the dynamics of the system and is defined as the integral over time of the Lagrangian, L, which itself is the difference between the kinetic and potential energies, T and V respectively. Mathematically, how the principle is applied can be expressed as:
\[ S = \int L \, dt \]
In the exercise given, the principle is utilized to derive an equation for the orbit of a particle in a plane, where the potential energy is a function of the distance from a fixed point. By finding where the action is stationary, we can determine the conditions that govern the behavior of the particle's motion within the plane.
Euler-Lagrange Equations
The Euler-Lagrange equations form the cornerstone of the calculus of variations and provide a method to find functions that maximize or minimize functionals. These equations arise from the principle of least action and are indispensable for obtaining the equations of motion for a system.
The general Euler-Lagrange equation for a variable q is given by:
\[ \frac{d}{dt}(\frac{\partial L}{\partial \dot{q}}) - \frac{\partial L}{\partial q} = 0 \]
Here, L is the Lagrangian, q is a generalized coordinate, and \(\dot{q}\) is the time derivative of q. In the example provided, by applying the Euler-Lagrange equation to the Lagrangian that incorporates kinetic and potential energies, we derive the equations that describe the motion of the particle when constrained to a plane, leading us to understand how the particle orbits in the given circumstances.
The general Euler-Lagrange equation for a variable q is given by:
\[ \frac{d}{dt}(\frac{\partial L}{\partial \dot{q}}) - \frac{\partial L}{\partial q} = 0 \]
Here, L is the Lagrangian, q is a generalized coordinate, and \(\dot{q}\) is the time derivative of q. In the example provided, by applying the Euler-Lagrange equation to the Lagrangian that incorporates kinetic and potential energies, we derive the equations that describe the motion of the particle when constrained to a plane, leading us to understand how the particle orbits in the given circumstances.
Hamilton-Jacobi Method
The Hamilton-Jacobi method is a powerful analytical tool in classical mechanics that transforms the problem of finding a particle's path into a partial differential equation problem. It helps simplify complex systems and offers a bridge to quantum mechanics via the Schrödinger equation.
The key to this method is the Hamilton's principal function, S*, which can be thought of as an action that satisfies the Hamilton-Jacobi equation. For our exercise, S* takes on a specific form that involves integrals over the radial distance and depends on various constants and the potential energy function V(r).
By carefully manipulating this function, one can reduce the mechanics problem to solving a differential equation, the Hamilton-Jacobi equation. The provided solution shows this as an integral expression involving the energy E, potential V, and constants of integration, which when solved, yields the particle's trajectory in terms of its coordinates.
The key to this method is the Hamilton's principal function, S*, which can be thought of as an action that satisfies the Hamilton-Jacobi equation. For our exercise, S* takes on a specific form that involves integrals over the radial distance and depends on various constants and the potential energy function V(r).
By carefully manipulating this function, one can reduce the mechanics problem to solving a differential equation, the Hamilton-Jacobi equation. The provided solution shows this as an integral expression involving the energy E, potential V, and constants of integration, which when solved, yields the particle's trajectory in terms of its coordinates.
Particle Orbit in Plane Polar Coordinates
When a particle moves in a plane, it is often convenient to describe its motion in terms of plane polar coordinates (r, \(\phi\)), where r represents the radial distance from a fixed point (the origin) and \(\phi\) is the angular coordinate. In physics and engineering, plane polar coordinates offer a natural and efficient way to describe circular and radial motion.
The equation derived from the principle of least action in the exercise pertains specifically to the trajectory of the particle in these coordinates. It relates the rate of change of the angular coordinate to the radial distance and hence encapsulates the essence of the particle's orbit. By solving the equation for different values of the energy E, we can discriminate between the types of orbits the particle might have—whether they are elliptical, parabolic, or hyperbolic trajectories—mirroring the solutions observed in celestial mechanics and orbital dynamics.
The equation derived from the principle of least action in the exercise pertains specifically to the trajectory of the particle in these coordinates. It relates the rate of change of the angular coordinate to the radial distance and hence encapsulates the essence of the particle's orbit. By solving the equation for different values of the energy E, we can discriminate between the types of orbits the particle might have—whether they are elliptical, parabolic, or hyperbolic trajectories—mirroring the solutions observed in celestial mechanics and orbital dynamics.