Chapter 37: Problem 13
Find the geodesics of a manifold whose arc element is \(d s^{2}=d x^{2}+\) \(d y^{2}+d z^{2}\).
Short Answer
Expert verified
The geodesics on a flat Euclidean space are straight lines given by the equations: \( x(\lambda) = x_0 + \lambda V_x \), \( y(\lambda) = y_0 + \lambda V_y \), \( z(\lambda) = z_0 + \lambda V_z \).
Step by step solution
01
Introduce the Euler-Lagrange equation
The Euler-Lagrange equation is used to find the geodesics of a manifold. For this problem, the Lagrangian \(L\) is the square root of the metric, that is \[ L = \sqrt{ \dot{x}^2 + \dot{y}^2 + \dot{z}^2 } \], where a dot over a variable denotes derivative with respect to a parameter.
02
Application of Euler-Lagrange equation
Let's differentiate the Lagrangian with respect to the coordinates \(x\), \(y\) and \(z\), and their derivatives. These should be zero for a geodesic. This operation results in three equations: \[ \dfrac{d}{d\lambda} ( \dfrac{\partial L}{\partial \dot{x} } ) - \dfrac{\partial L}{\partial x } = 0 \] \[ \dfrac{d}{d\lambda} ( \dfrac{\partial L}{\partial \dot{y} } ) - \dfrac{\partial L}{\partial y } = 0 \] \[ \dfrac{d}{d\lambda} ( \dfrac{\partial L}{\partial \dot{z} } ) - \dfrac{\partial L}{\partial z } = 0 \] Since the Lagrangian does not depend explicitly on \(x\), \(y\) and \(z\), these equations simplify to: \[ \dfrac{d}{d\lambda} ( \dfrac{\partial L}{\partial \dot{x} } ) = 0 \] \[ \dfrac{d}{d\lambda} ( \dfrac{\partial L}{\partial \dot{y} } ) = 0 \] \[ \dfrac{d}{d\lambda} ( \dfrac{\partial L}{\partial \dot{z} } ) = 0 \] These are the equations of motion.
03
Solution of the equations
The solution to the equations of motion provides the geodesics of the manifold. They correspond to straight lines in this Euclidean flat space case: \[ x(\lambda) = x_0 + \lambda V_x \] \[ y(\lambda) = y_0 + \lambda V_y \] \[ z(\lambda) = z_0 + \lambda V_z \] where \( V_{x}\), \( V_{y}\) , \(V_{z} \) are the components of the initial velocity, and \( x_{0}\), \( y_{0}\) , \( z_{0}\) are the initial coordinates.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Euler-Lagrange Equation
When grappling with the dynamics of physical systems, the Euler-Lagrange equation stands as a cornerstone of classical mechanics, particularly within the framework of Lagrangian mechanics. This fundamental equation provides us with a systematic approach to determine the path, or trajectory, that a system will follow over time, given its energy properties.
To deeply understand how this equation comes into play, let's consider it as a mathematical recipe that determines the shortest or most efficient path between two points — similar to finding the best route to reach your destination in a terrain. In the context of the exercise, the shortest path on a manifold is known as a geodesic, akin to a straight line on a flat surface, or a great circle on a sphere. The Euler-Lagrange equation is employed to identify these geodesics after defining a Lagrangian, which in our case, represents the kinetic energy of a particle moving through the manifold minus any potential energy, which happens to be zero in a flat manifold.
For the given manifold, with the arc element defined as \(d s^{2}=d x^{2}+d y^{2}+d z^{2}\), the Lagrangian simplifies to the square root of the sum of the squares of the derivatives of the coordinates with respect to an arbitrary parameter, which is often time or in the case of this problem, another parameter \(\lambda\). The Euler-Lagrange equation thus sets our sights on determining the minima of the action integral, which represents the weighted sum of all possible paths, guiding us toward the exact trajectory the object will traverse — the geodesic.
To deeply understand how this equation comes into play, let's consider it as a mathematical recipe that determines the shortest or most efficient path between two points — similar to finding the best route to reach your destination in a terrain. In the context of the exercise, the shortest path on a manifold is known as a geodesic, akin to a straight line on a flat surface, or a great circle on a sphere. The Euler-Lagrange equation is employed to identify these geodesics after defining a Lagrangian, which in our case, represents the kinetic energy of a particle moving through the manifold minus any potential energy, which happens to be zero in a flat manifold.
For the given manifold, with the arc element defined as \(d s^{2}=d x^{2}+d y^{2}+d z^{2}\), the Lagrangian simplifies to the square root of the sum of the squares of the derivatives of the coordinates with respect to an arbitrary parameter, which is often time or in the case of this problem, another parameter \(\lambda\). The Euler-Lagrange equation thus sets our sights on determining the minima of the action integral, which represents the weighted sum of all possible paths, guiding us toward the exact trajectory the object will traverse — the geodesic.
Differential Geometry
Differential geometry is an elegant dance between calculus and geometry, where curves, surfaces, and more abstract spaces are examined using the techniques of differentiation and integration. Imagine it as an advanced toolkit that allows mathematicians and physicists to stretch, bend, and twist space in order to study its shape and the paths within it — a crucial endeavor, especially when trying to understand the fabric of our universe.
In the given exercise, differential geometry offers the stage upon which the concept of geodesics can be articulated with precision. A geodesic can be thought of as the version of a straight line when the idea is extended into curved space, such as the surface of a globe or the undulating terrain of a manifold.
With the problem at hand, the simple manifold described by the arc element \(d s^{2}=d x^{2}+d y^{2}+d z^{2}\) serves as the playground where differential geometry demonstrates its relevance. In this realm, we use differential calculus, specifically the partial derivatives of the Lagrangian with respect to coordinates and their derivatives, to find the geodesics. As through a microscope, differential geometry empowers us to inspect the local properties of our manifold and discern the straightest possible lines, even in a potentially curved multidimensional landscape.
In the given exercise, differential geometry offers the stage upon which the concept of geodesics can be articulated with precision. A geodesic can be thought of as the version of a straight line when the idea is extended into curved space, such as the surface of a globe or the undulating terrain of a manifold.
With the problem at hand, the simple manifold described by the arc element \(d s^{2}=d x^{2}+d y^{2}+d z^{2}\) serves as the playground where differential geometry demonstrates its relevance. In this realm, we use differential calculus, specifically the partial derivatives of the Lagrangian with respect to coordinates and their derivatives, to find the geodesics. As through a microscope, differential geometry empowers us to inspect the local properties of our manifold and discern the straightest possible lines, even in a potentially curved multidimensional landscape.
Equations of Motion
Equations of motion are the backbone of classical mechanics, crucial for predicting the future positions and velocities of objects. These mathematical expressions are derived from Newton's Laws of Motion, but they also emerge naturally from the framework of Lagrangian mechanics where the paths taken by physical systems are determined by minimizing the action through the Euler-Lagrange equation.
By examining the problem's solution, we can see how the Euler-Lagrange equation transforms into equations of motion. These equations are essentially recipes that tell us exactly how an object moves within a space given its starting point and its initial momentum. For our manifold, because the arc element does not include terms that involve the actual coordinates themselves, the resulting equations of motion are remarkably straightforward. They state that the rate of change of momentum is zero, implying that the momentum, and consequently velocity, is conserved.
The end result is a clear set of instructions or 'equations of motion' for the geodesics, revealing that an object moves in a straight line at a constant speed, which is expected in a flat, Euclidean space. Equipped with these equations, one can easily predict the object’s path from any given point in space.
By examining the problem's solution, we can see how the Euler-Lagrange equation transforms into equations of motion. These equations are essentially recipes that tell us exactly how an object moves within a space given its starting point and its initial momentum. For our manifold, because the arc element does not include terms that involve the actual coordinates themselves, the resulting equations of motion are remarkably straightforward. They state that the rate of change of momentum is zero, implying that the momentum, and consequently velocity, is conserved.
The end result is a clear set of instructions or 'equations of motion' for the geodesics, revealing that an object moves in a straight line at a constant speed, which is expected in a flat, Euclidean space. Equipped with these equations, one can easily predict the object’s path from any given point in space.
Lagrangian Mechanics
Lagrangian mechanics is a reimagining of classical mechanics that offers a powerful alternative to the force-centric approach of Newtonian mechanics. Instead of focusing on the forces, it recasts the dynamics of systems in terms of energy, more specifically the difference between kinetic and potential energy, which makes up the Lagrangian of the system. This shift provides a broadened perspective, one that's especially advantageous when dealing with complex motions and constrained systems.
In the context of our exercise, we engage with Lagrangian mechanics by first constructing a Lagrangian for the manifold. In this flat, potential-free landscape, our Lagrangian is a scalar field representing total kinetic energy associated with the motion. It's the basis upon which we apply the Euler-Lagrange equation to churn out the equations that govern motion.
Lagrangian mechanics isn't just about finding answers to specific problems; it's also a gateway to deeper insights into the principles of physics. It links closely with symmetries and conservation laws, revealing why certain quantities remain constant over time. In this case, due to the homogeneity of the manifold, we observe the conservation of momentum, a signature trait that the Lagrangian perspective captures elegantly. For students, understanding the beauty and utility of Lagrangian mechanics paves the way for exploring more advanced concepts in physics, including the realms of quantum mechanics and general relativity.
In the context of our exercise, we engage with Lagrangian mechanics by first constructing a Lagrangian for the manifold. In this flat, potential-free landscape, our Lagrangian is a scalar field representing total kinetic energy associated with the motion. It's the basis upon which we apply the Euler-Lagrange equation to churn out the equations that govern motion.
Lagrangian mechanics isn't just about finding answers to specific problems; it's also a gateway to deeper insights into the principles of physics. It links closely with symmetries and conservation laws, revealing why certain quantities remain constant over time. In this case, due to the homogeneity of the manifold, we observe the conservation of momentum, a signature trait that the Lagrangian perspective captures elegantly. For students, understanding the beauty and utility of Lagrangian mechanics paves the way for exploring more advanced concepts in physics, including the realms of quantum mechanics and general relativity.
