Classical mechanics and Geommetry

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  • #1
Klaus_Hoffmann
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1
Given the Hamiltonian of a system [tex] \mathcal H [/tex] , could we obtain the curves solution to Hamilton equations X(t) Y(t) Z(t) as the Geodesic of a certain surface with Christoffle symbols [tex] \Gamma ^{i} _{jk} [/tex] i mean the curve X(t) satisfies the equation:

[tex] \nabla _{x(t)} X(t)=0 [/tex] (covariant derivative vanishes)

Also given the 1-form [tex] \theta =p^{i}dq^{i}-Hdt [/tex] and some elements of Diff. Geommetry how could solve our physical system ?? or reduce the solutions for x,y,z to 'Quadratures' ?? thanks.
 
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  • #2


Hello,

Thank you for your interesting question. The answer is yes, we can obtain the curves solution to Hamilton's equations as the geodesic of a certain surface with Christoffel symbols. This is known as the Hamilton-Jacobi formalism, which is a powerful tool in solving classical mechanics problems.

To understand this, let's first look at the Hamiltonian of a system, denoted as \mathcal H . Hamilton's equations describe the time evolution of a system in terms of its Hamiltonian and the system's canonical coordinates and momenta. In other words, they tell us how the position and momentum of a system change over time. These equations can be written as:

\frac{dq^{i}}{dt} = \frac{\partial \mathcal H}{\partial p_{i}}

\frac{dp_{i}}{dt} = -\frac{\partial \mathcal H}{\partial q^{i}}

Now, let's consider the geodesic equation in differential geometry, which describes the shortest path between two points on a curved surface. This equation can be written as:

\nabla _{x(t)} x(t) = 0

where \nabla is the covariant derivative.

Here comes the interesting part. By using the Hamilton-Jacobi formalism, we can transform Hamilton's equations into a set of equations that look similar to the geodesic equation. This transformation involves introducing a new set of variables, known as the action-angle variables, which are related to the canonical coordinates and momenta. These new variables eliminate the time dependence in the Hamiltonian and reduce the equations to first-order differential equations. These equations can then be interpreted as the geodesic equation on a curved surface, with the Christoffel symbols determined by the Hamiltonian.

So, in summary, by using the Hamilton-Jacobi formalism, we can map the solutions of Hamilton's equations onto the geodesic equation on a certain surface with Christoffel symbols determined by the Hamiltonian. This allows us to use the tools of differential geometry to solve classical mechanics problems.

As for your question about reducing the solutions for x, y, z to 'quadratures', this would depend on the specific system and its Hamiltonian. In some cases, it may be possible to find analytic solutions using the action-angle variables, but in most cases, numerical methods would be needed. I hope this helps answer
 
  • #3


Thank you for your question. It is an interesting one, as it combines the concepts of classical mechanics and geometry.

Classical mechanics is a branch of physics that describes the motion of objects under the influence of forces. It is based on the principles of Newton's laws of motion and the conservation of energy and momentum. On the other hand, geometry is a branch of mathematics that deals with the properties and relationships of shapes and spaces.

In the context of your question, the Hamiltonian of a system is a mathematical function that represents the total energy of the system. It is derived from the Lagrangian, which describes the dynamics of the system. The Hamiltonian is an important concept in classical mechanics, as it allows us to formulate the equations of motion for a system.

Now, in terms of geometry, the Hamiltonian can be viewed as a function on a space called the Hamiltonian phase space. This space is equipped with a metric, which is a mathematical tool that measures distances and angles. This metric is related to the Hamiltonian through the Hamilton-Jacobi equation.

The Christoffel symbols that you mentioned are related to the covariant derivative, which is a mathematical tool that describes how a vector field changes as we move along a curve in a space. In this case, the curves X(t), Y(t), and Z(t) are solutions to Hamilton's equations of motion. The covariant derivative vanishes because these curves are geodesics, which are the shortest paths between two points in a space equipped with a metric.

To answer your question about obtaining the curves X(t), Y(t), and Z(t) as geodesics of a certain surface, it is possible if we define a metric on the surface that is related to the Hamiltonian. This is known as the Hamiltonian-Jacobi theory of geodesics.

As for your second question, the 1-form theta is related to the Hamiltonian through the Legendre transformation. This transformation allows us to rewrite the equations of motion in terms of the Hamiltonian instead of the Lagrangian. In terms of differential geometry, the 1-form theta can be seen as a connection on the Hamiltonian phase space.

To solve a physical system using differential geometry, we can use the tools of differential geometry to study the properties of the Hamiltonian phase space. This can help us understand the dynamics of the system and potentially find solutions to our physical system. However, it is important
 

1. What is classical mechanics?

Classical mechanics is a branch of physics that deals with the motion of macroscopic objects, such as particles, bodies, and systems of particles. It is based on Newton's laws of motion and can be used to describe the behavior of objects in everyday life.

2. How does classical mechanics differ from quantum mechanics?

Classical mechanics and quantum mechanics are two different theories that describe the behavior of matter and energy. Classical mechanics deals with macroscopic objects, while quantum mechanics deals with subatomic particles and their interactions.

3. What is the role of geometry in classical mechanics?

Geometry plays a crucial role in classical mechanics as it provides a framework for describing the motion of objects in space. It is used to calculate distances, angles, and positions of objects, and to analyze the forces acting on them.

4. How does classical mechanics explain the motion of planets?

Classical mechanics explains the motion of planets using Kepler's laws of planetary motion and Newton's law of universal gravitation. It describes how the planets orbit around the sun in elliptical paths due to the gravitational pull of the sun.

5. Can classical mechanics be applied to all systems?

Classical mechanics can be applied to most systems that are not at the atomic or subatomic level. It is often used to explain and predict the motion of objects in our everyday world, such as cars, airplanes, and even the movement of the human body.

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