If already known the Action and unkown the Metric, how to get geodesic equation?

Thread starter yicong2011
Start date Jun 22, 2011
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Geodesic Geodesic equation Metric

In summary, the geodesic equation is a fundamental concept in differential geometry that describes the shortest path between two points on a curved surface. It is derived from the metric tensor and can be applied to any surface with a defined metric. In physics, it is used to describe the motion of objects in space-time and has practical applications in navigation, robotics, and computer graphics.

Jun 22, 2011

yicong2011

If already known the form of Action and unkown the Metric, how to derive the geodesic equation?

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Jun 22, 2011

yicong2011

Another question, if already known the Action (unknown the metric), how to derive Riemann Curvature from the Action?

Jun 22, 2011

atyy

Science Advisor

http://arxiv.org/abs/1102.0529, section 19.1 derives the geodesic equation from the action

Related to If already known the Action and unkown the Metric, how to get geodesic equation?

1. What is the geodesic equation?

The geodesic equation is a fundamental equation in differential geometry that describes the shortest path between two points on a curved surface, also known as a geodesic. It is commonly used in the study of space-time and is an essential concept in general relativity.

2. How is the geodesic equation related to the metric?

The geodesic equation is derived from the metric tensor, which is a mathematical object that describes the curvature of a surface. The metric tensor contains information about the distances and angles between points on a surface, which is used to calculate the shortest path between two points.

3. Can the geodesic equation be used to find the shortest path on any surface?

Yes, the geodesic equation can be applied to any surface with a defined metric. This includes curved surfaces such as spheres, cones, and even more complex shapes. It can also be used in higher dimensions, such as in the study of space-time.

4. What is the significance of the geodesic equation in physics?

The geodesic equation has significant importance in physics, particularly in the field of general relativity. It is used to describe the motion of objects in space-time and to predict the path of particles under the influence of gravity. It also has applications in other areas of physics, such as optics and electromagnetism.

5. How is the geodesic equation used in practical applications?

The geodesic equation has practical applications in various fields, including navigation, robotics, and computer graphics. For example, it is used in GPS systems to calculate the shortest path between two points on Earth's surface. In robotics, it is used to plan the most efficient path for a robot to follow. In computer graphics, it is used to create realistic animations of objects moving through space.