Conformable Lagrangian Mechanics of Actuated Pendulum
Abstract
:1. Introduction
- Since, in general, dissipation described by non-conservative forces breaks the variational structure of classical mechanics, modifying the kinetic term with conformable derivatives preserves the action principle, which allows us to construct models with various symmetries, conserved charges, and phenomenological rich generalizations. This is crucial for maintaining compatibility with Noether’s theorem, symplectic structure, canonical quantization, and important tools of theoretical physics.
- Since the conformable derivative encodes scale-dependent time dynamics and breaks time-reversal symmetry when , embedding the irreversible behavior in the kinetic term itself can be viewed as geometrizing dissipation, similar to how general relativity encodes forces into the curvature of spacetime.
- Traditional damping models require inserting terms like , with arbitrary coefficients b. These models break energy conservation and lack universality. The conformable kinetic term replaces such ad hoc additions with a single parameter , offering a principled and systematic generalization of classical mechanics, which is important for theoretical construction. In phenomenological applications, should be related to physical parameters.
- Quantum systems with classical damping are notoriously hard to formulate. The conformable approach, with its modified kinetic term, allows the direct application of canonical quantization [9,33,34,37,38]:
- The dissipation is obtained in the conformable approach with a minimal and elegant modification, replacing with . No external functions or potentials are introduced. The resulting models connect naturally to memory effects, anomalous diffusion, and non-Markovian behavior.
2. Fundamentals of Conformable Calculus
2.1. Conformable Derivative and Integral
2.2. Conformable, Fractal, and Jackson Derivatives
3. Conformable Lagrangian Mechanics of Point Particle
3.1. Conformable Lagrangian Formalism
3.2. Brief Discussion of Generalized Integration Measures
- Positivity: for to define a bona fide measure.
- Regularity: should be sufficiently smooth, e.g., continuous or piecewise continuous, so that integration by parts holds
- Normalization (optional): For probability or thermodynamic interpretations, one may require that
- Exponential Decay: models a memory kernel with finite memory timescale [57].
- Stretched Exponential: captures a broad distribution of memory times.
- Power-Law: yields a long-memory kernel like in fractional integrals.
4. Conformable Actuated Pendulum
4.1. Classical Actuated Pendulum
4.2. Conformable Dynamics with No Force
- Case 1: .
- Case 2: .
- Case 3: with .
4.3. General Solution for
4.4. Power Series Solution of the Linearized Equation for
5. Conformable Actuated Pendulum with a Non-Conservative Force
5.1. Physical Interpretation of Parameters
5.2. Numerical Validation Example
5.3. Families of Conformable Equations
- Case 1: .
- Case 2: .
- Case 3: with .
5.4. General Solution for
5.5. Power Series Solution of Linearized Forced Equation for
6. Numerical Study of the General Conformable Actuated Pendulum
1.00 | 0.095 | 0.090 | 0.085 |
0.75 | 0.095 | 0.075 | 0.055 |
0.60 | 0.095 | 0.065 | 0.035 |
0.50 | 0.095 | 0.050 | 0.020 |
0.33 | 0.095 | 0.030 | 0.008 |
0.25 | 0.095 | 0.020 | 0.005 |
7. Discussion
7.1. Summary for Conformable Homogeneous Equations
- : Equation (56) displays a time-dependent inertia term proportional to t and a constant effective damping term. Its linearized form can be related to the Bessel-type equations.
- , : Equation (57) is a non-autonomous ODE with power-law time-dependent coefficients. In general, even linearized, these do not belong to the classical families, like Euler–Cauchy, with well-known closed-form solutions. Therefore, their analysis typically requires specialized techniques or numerical methods.
7.2. Summary for Conformable Actuated Pendulum with a Non-Conservative Force
- : Equation (71) is an autonomous nonlinear ODE (or its linearized version is a second-order linear ODE with constant coefficients). This is the standard model for an actuated pendulum with friction and external forces.
- : Equation (73) is a non-autonomous ODE with a time-dependent inertial term () and a constant effective damping. By linearization, it can often be reformulated as a Bessel-type or Euler-type ODE with known analytic solutions in terms of special functions (Mittag–Leffler functions, Wright function, Prabhakar function, generalized hypergeometric gunctions, Fox H-functions, Meijer G-functions, etc.).
- : Equation (74) represents a more general class of non-autonomous ODEs with power-law time-dependent coefficients. In general, these equations do not correspond to standard forms unless further transformations or approximations are applied. Their analysis typically relies on series methods (such as the Frobenius method) or numerical approaches.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Simple Numerical Comparison of Standard and Conformal Integration Measures
Difference from | ||
1.0 | 0 | |
0.75 | ||
0.5 | ||
0.25 |
Appendix B. Proof of the General Solution of Equation (60)
Appendix C. Proof of the Solution of Equation (62)
Appendix D. Proof of the Solution of Equation (76)
Appendix E. Power Series Solution of the Linearized Equation (96)
- (i)
- The Inertial Term:
- (ii)
- The First Part of the Damping Term:
- (iii)
- The Second Part of the Damping Term:
- (iv)
- The Restoring Term:
- Case 1: If then
- Case 2: If then
- For : The term from the first series (with exponent ) leads to
- For : The term from the second series (with exponent ) results in
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Crişan, A.V.; Godinho, C.F.d.L.; Porto, C.M.; Vancea, I.V. Conformable Lagrangian Mechanics of Actuated Pendulum. Mathematics 2025, 13, 1634. https://doi.org/10.3390/math13101634
Crişan AV, Godinho CFdL, Porto CM, Vancea IV. Conformable Lagrangian Mechanics of Actuated Pendulum. Mathematics. 2025; 13(10):1634. https://doi.org/10.3390/math13101634
Chicago/Turabian StyleCrişan, Adina Veronica, Cresus Fonseca de Lima Godinho, Claudio Maia Porto, and Ion Vasile Vancea. 2025. "Conformable Lagrangian Mechanics of Actuated Pendulum" Mathematics 13, no. 10: 1634. https://doi.org/10.3390/math13101634
APA StyleCrişan, A. V., Godinho, C. F. d. L., Porto, C. M., & Vancea, I. V. (2025). Conformable Lagrangian Mechanics of Actuated Pendulum. Mathematics, 13(10), 1634. https://doi.org/10.3390/math13101634