I guess, time is not a quantity in the sense of your book. It is the parameter parameterizing the dynamics.Visceral said:The title pretty much says it. According to my book, Classical Dynamics by Thornton and Marion, generalized coordinates can be quantities other than position such as energy or length squared, but what about time?
A. Neumaier said:I guess, time is not a quantity in the sense of your book. It is the parameter parameterizing the dynamics.
The main stream view is indeed ''no''. But there is a trickle of literature (I don't remember precise references now) which works in extended pase spase where time and energy are anotherr pair of conjugate variables. However, this approach has its own pitfalls and is not widely used because of that.nlsherrill said:I was just working on a problem the other day that had force as a function of position and time, and had to derive the potential and then the lagrangian and hamiltonian. So essentially, the answer is no?
Yes, time can be treated as a generalized coordinate in certain physical systems. In classical mechanics, time is often considered as a parameter rather than a coordinate. However, in other areas of physics such as quantum mechanics, general relativity, and thermodynamics, time can be used as a generalized coordinate to describe the dynamics of a system.
A generalized coordinate is a variable that describes the position or state of a physical system. In classical mechanics, generalized coordinates are typically position and momentum, while in other areas of physics they may include other variables such as temperature, entropy, or electric charge.
Time is unique as a generalized coordinate because it is a unidirectional variable that only moves forward. In contrast, other generalized coordinates such as position or momentum can move in both directions. Additionally, time is often considered as a continuous variable, while other generalized coordinates may be discrete.
Using time as a generalized coordinate allows for a more comprehensive understanding of a physical system's dynamics. It also allows for the application of mathematical tools such as Lagrangian and Hamiltonian mechanics, which can simplify the analysis of complex systems. Additionally, time as a generalized coordinate can help bridge the gap between different areas of physics, as it is a fundamental concept in many theories.
While time can be a useful generalized coordinate, there are some limitations to its application. In some physical systems, time may not be the most appropriate variable to describe the dynamics, and other generalized coordinates may be more suitable. Additionally, there may be cases where time is not a continuous variable, such as in quantum mechanics, which can complicate its use as a generalized coordinate.