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Hi all,

I have a (hopefully) quick conceptual question that I'd like to clear up with your help.

Is the following argument for why we treat position and velocity as independent variables in the Lagrangian correct? (referring to classical mechanics) :

The Lagrangian, [itex] \mathcal{L}[/itex] of a given physical system contains a full description of the dynamics of the system. As space and time are considered to be homogeneous (and in addition, space to be isotropic), if one can fully specify the configuration of a system at a given instant in time, then one can (in principle) determine its configuration at any later time. This is done by finding a specific form for the Lagrangian of the system from which one can determine the evolution of the system.

In order to fully describe the state of the system at a given instant in time [itex] t[/itex], and hence, be able to determine its state at some later time [itex]t+\delta t[/itex], it is sufficient to specify the coordinates and velocities of all of the particles constituting the system at the initial time [itex]t[/itex]. Specifying the coordinates of each particle alone is insufficient, as one cannot determine the dynamics of the system (and hence, how it will evolve) without knowing the corresponding velocities of the constituent particles.

At a given instant in time [itex]t [/itex], the [itex]i^{th} [/itex] particle within a given system, has a given coordinate [itex]q_{i}\left(t\right) [/itex] of fixed value (for that instant in time. Where [itex]q_{i} [/itex] is a generalised coordinate, working in 1-dimension for simplicity). In principle, for a coordinate value [itex]q_{i} [/itex] at time [itex]t[/itex], a particle can have any number of different velocities [itex]\dot{q}_{i}\left(t\right) [/itex] (in essence, the coordinates and velocities that specify the initial conditions of the system are independent). As such, one treats both coordinates [itex] q_{i}(t)[/itex] and velocity [itex] \dot{q}_{i}(t)[/itex] as independent variables in order to fully describe the state of the system at a given instant in time [itex] t[/itex] and hence determine its dynamics.

Sorry, I know this argument is a bit convoluted, but hopefully you get the gist of it.

I have a (hopefully) quick conceptual question that I'd like to clear up with your help.

Is the following argument for why we treat position and velocity as independent variables in the Lagrangian correct? (referring to classical mechanics) :

The Lagrangian, [itex] \mathcal{L}[/itex] of a given physical system contains a full description of the dynamics of the system. As space and time are considered to be homogeneous (and in addition, space to be isotropic), if one can fully specify the configuration of a system at a given instant in time, then one can (in principle) determine its configuration at any later time. This is done by finding a specific form for the Lagrangian of the system from which one can determine the evolution of the system.

In order to fully describe the state of the system at a given instant in time [itex] t[/itex], and hence, be able to determine its state at some later time [itex]t+\delta t[/itex], it is sufficient to specify the coordinates and velocities of all of the particles constituting the system at the initial time [itex]t[/itex]. Specifying the coordinates of each particle alone is insufficient, as one cannot determine the dynamics of the system (and hence, how it will evolve) without knowing the corresponding velocities of the constituent particles.

At a given instant in time [itex]t [/itex], the [itex]i^{th} [/itex] particle within a given system, has a given coordinate [itex]q_{i}\left(t\right) [/itex] of fixed value (for that instant in time. Where [itex]q_{i} [/itex] is a generalised coordinate, working in 1-dimension for simplicity). In principle, for a coordinate value [itex]q_{i} [/itex] at time [itex]t[/itex], a particle can have any number of different velocities [itex]\dot{q}_{i}\left(t\right) [/itex] (in essence, the coordinates and velocities that specify the initial conditions of the system are independent). As such, one treats both coordinates [itex] q_{i}(t)[/itex] and velocity [itex] \dot{q}_{i}(t)[/itex] as independent variables in order to fully describe the state of the system at a given instant in time [itex] t[/itex] and hence determine its dynamics.

Sorry, I know this argument is a bit convoluted, but hopefully you get the gist of it.

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